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1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 24 "Calculus Single Variable " }{TEXT 259 3 " \n" }{TEXT 257 35 "Brian E. Blank and Steven G. Kran tz" }{TEXT 258 20 "\n\nSection 8.5\nWork \n" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 45 "1. Diminishing Loads: Exercises from the Text" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 357 20 "Exercise 1 Page 601" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 310 "A steam shovel lifts a 500 pound load \+ of gravel from the ground to a point 80 feet above the ground. Howev er the gravel is fine, and it leaks from the shovel at the rate of 1 pound per second. If it takes the steam shovel one minute to lift it s load at a constant rate, then how much work is performed?." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 358 8 "Solution" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "At time " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 22 " seconds, the height " }{XPPEDIT 18 0 "y ;" "6#%\"yG" }{TEXT -1 16 " of the load is " }{XPPEDIT 18 0 "80*t/60; " "6#*(\"#!)\"\"\"%\"tGF%\"#g!\"\"" }{TEXT -1 19 " feet. Solving for \+ " }{XPPEDIT 18 0 "t;" "6#%\"tG" }{TEXT -1 14 " in terms of " } {XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "t = 3*y/4;" "6#/%\"tG*(\"\"$\"\"\"%\"yGF'\"\"%!\"\"" }{TEXT -1 29 ". \+ At that time, the load is " }{XPPEDIT 18 0 "500-t;" "6#,&\"$+&\"\"\"% \"tG!\"\"" }{TEXT -1 32 " pounds. Therefore, at height " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 16 ", the load is " }{XPPEDIT 18 0 "50 0-3*y/4;" "6#,&\"$+&\"\"\"*(\"\"$F%%\"yGF%\"\"%!\"\"F*" }{TEXT -1 45 " pounds. Divide the interval [0,80] into " }{XPPEDIT 18 0 "N;" "6# %\"NG" }{TEXT -1 32 " equal subintervals of length " }{XPPEDIT 18 0 "Delta*y;" "6#*&%&DeltaG\"\"\"%\"yGF%" }{TEXT -1 18 ". Choose a point " }{XPPEDIT 18 0 "y[j];" "6#&%\"yG6#%\"jG" }{TEXT -1 9 " in the " } {XPPEDIT 18 0 "j^th;" "6#)%\"jG%#thG" }{TEXT -1 43 " subinterval Lifti ng the load through the " }{XPPEDIT 18 0 "j^th;" "6#)%\"jG%#thG" } {TEXT -1 30 " subinterval requires about " }{XPPEDIT 18 0 "(500-3*y[ j]/4)*Delta*y;" "6#*(,&\"$+&\"\"\"*(\"\"$F&&%\"yG6#%\"jGF&\"\"%!\"\"F. F&%&DeltaGF&F*F&" }{TEXT -1 50 " foot-pounds of work. The total wor k is about " }{XPPEDIT 18 0 "Sum((500-3*y[j]/4)*Delta*y,j = 1 .. N); " "6#-%$SumG6$*(,&\"$+&\"\"\"*(\"\"$F)&%\"yG6#%\"jGF)\"\"%!\"\"F1F)%&D eltaGF)F-F)/F/;F)%\"NG" }{TEXT -1 18 " foot-pounds. As " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 60 " increases the approximation becomes more accurate. Letting " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 99 " tend to infinity we obtain the work as the limit of the sums, which \+ we recognize as Riemann sums:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 " Work = " }{XPPEDIT 18 0 "l imit(Sum((500-3*y[j]/4)*Delta*y,j = 1 .. N),N = infinity);" "6#-%&limi tG6$-%$SumG6$*(,&\"$+&\"\"\"*(\"\"$F,&%\"yG6#%\"jGF,\"\"%!\"\"F4F,%&De ltaGF,F0F,/F2;F,%\"NG/F8%)infinityG" }{TEXT -1 10 " = " } {XPPEDIT 18 0 "Int(500-3*y/4,y = 0 .. 80);" "6#-%$IntG6$,&\"$+&\"\"\"* (\"\"$F(%\"yGF(\"\"%!\"\"F-/F+;\"\"!\"#!)" }{TEXT -1 24 " = 37600 f oot-pounds." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 " \+ \+ " }{TEXT 359 1 " " }{XPPEDIT 256 0 "Omega;" "6#%&OmegaG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 354 20 "Exercise 2 Page 601" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "When a mas s " }{XPPEDIT 18 0 "M;" "6#%\"MG" }{TEXT -1 26 " measured in slugs \+ is " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 77 " feet above the s urface of the earth, its weight in pounds is \n\n " } {XPPEDIT 18 0 "1.4077;" "6#-%&FloatG6$\"&xS\"!\"%" }{TEXT -1 3 " " } {TEXT 370 1 "x" }{TEXT -1 2 " " }{XPPEDIT 18 0 "10^16;" "6#*$\"#5\"#; " }{TEXT -1 4 " " }{TEXT 371 1 "x" }{TEXT -1 6 " " }{XPPEDIT 18 0 "M/((2.0917*`*`*10^7+y)^2);" "6#*&%\"MG\"\"\"*$,&*(-%&FloatG6$\"& <4#!\"%F%%\"*GF%\"#5\"\"(F%%\"yGF%\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 372 1 "." }{TEXT -1 205 " \n\n(One pound of force accelerates one slug one foot per second squared.) How much work is done lifting a satelli te from the surface of the earth, where it weighs 800 pounds, to a h eight of 200 miles?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 355 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "According to Newton's law, the for ce of gravity is " }{XPPEDIT 18 0 "F = G*M*m/((R+y)^2);" "6#/%\"FG**% \"GG\"\"\"%\"MGF'%\"mGF'*$,&%\"RGF'%\"yGF'\"\"#!\"\"" }{TEXT -1 9 " \+ where " }{XPPEDIT 18 0 "G;" "6#%\"GG" }{TEXT -1 28 " is a universal c onstant, " }{XPPEDIT 18 0 "M;" "6#%\"MG" }{TEXT -1 29 " is the mass \+ of the earth, " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 32 " is the \+ mass of the satellite, " }{XPPEDIT 18 0 "R;" "6#%\"RG" }{TEXT -1 35 " \+ is the radius of the earth, and " }{XPPEDIT 18 0 "y;" "6#%\"yG" } {TEXT -1 60 " is the height of the satellite above the earth's surfac e. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "R := 6375.58*km; #Earth's \+ radius in kilometers (given on page 598)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RG,$*&$\"'evj!\"#\"\"\"%#kmGF*F*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 37 "R := convert(R, units, km, ft)*ft/km;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"RG,$*&$\"+Ads\"4#!\"#\"\"\"%#ftGF*F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The value of " }{XPPEDIT 18 0 "G*M;" "6#*&%\"GG\"\"\"%\"MGF%" }{TEXT -1 37 " \+ , as measured in the metric unit " }{XPPEDIT 18 0 "newton*meter^2/k ilogram;" "6#*(%'newtonG\"\"\"*$%&meterG\"\"#F%%)kilogramG!\"\"" } {TEXT -1 18 ", is given to be " }{XPPEDIT 18 0 "3.98621;" "6#-%&Float G6$\"'@')R!\"&" }{TEXT -1 2 " " }{TEXT 367 1 "x" }{TEXT -1 1 " " } {XPPEDIT 18 0 "10^14;" "6#*$\"#5\"#9" }{TEXT -1 111 " (see page 598). Writing the newton in terms of the units for length, mass, and time, namely\n\n\n " }{XPPEDIT 18 0 "1*newton = 1*kg;" "6#/*& \"\"\"F%%'newtonGF%*&F%F%%#kgGF%" }{TEXT -1 3 " " }{TEXT 366 2 "x " }{TEXT -1 1 " " }{XPPEDIT 18 0 "1*m/(second^2);" "6#*(\"\"\"F$%\"mGF$* $%'secondG\"\"#!\"\"" }{TEXT -1 28 ", \n\n we have\n " } {XPPEDIT 18 0 "G*M = 3.98621;" "6#/*&%\"GG\"\"\"%\"MGF&-%&FloatG6$\"'@ ')R!\"&" }{TEXT -1 2 " " }{TEXT 368 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^14;" "6#*$\"#5\"#9" }{TEXT -1 3 " " }{XPPEDIT 18 0 "meter^3 /(second^2);" "6#*&%&meterG\"\"$*$%'secondG\"\"#!\"\"" }{TEXT 369 2 ". " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Thus, in terms of the uni t " }{XPPEDIT 18 0 "ft^3/(second^2);" "6#*&%#ftG\"\"$*$%'secondG\"\" #!\"\"" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "G*M;" "6#*&%\"GG\"\"\"%\"MG F%" }{TEXT -1 5 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "eqn1 := G*M[e] = evalf(3.98621*10^1 4*convert(1, units, m, ft)^3 )*ft^3/s^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/*&%\"GG\"\"\"&%\"MG6#%\"eGF(,$*($\"+wnr29\"\"(F(%#ftG\" \"$%\"sG!\"#F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Solve for Newton's constant " }{XPPEDIT 18 0 "G;" "6#%\"G G" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn2 := G = \+ solve(eqn1, G);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%\"GG,$**$ \"+wnr29\"\"(\"\"\"%#ftG\"\"$&%\"MG6#%\"eG!\"\"%\"sG!\"#F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Substitute into Newton's Law of Gravitation:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "NewtonsLaw := F = G*M[e]*M[s]/(R+y)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+NewtonsLawG/%\"FG**%\"GG\"\"\"&%\"MG6#%\"eGF)&F+6#% \"sGF),&*&$\"+Ads\"4#!\"#F)%#ftGF)F)%\"yGF)F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqn3 := subs(eqn2,NewtonsLaw); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/%\"FG,$*,$\"+wnr29\"\"(\"\"\"%#ftG\"\"$% \"sG!\"#&%\"MG6#F/F,,&*&$\"+Ads\"4#F0F,F-F,F,%\"yGF,F0F," }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Let us now find th e mass of the satellite. The unit will be the slug. " }{TEXT 364 1 "( " }{TEXT -1 2 " " }{XPPEDIT 18 0 "1*pound = 1*slug*1*ft/(s^2);" "6#/* &\"\"\"F%%£GF%*,F%F%%%slugGF%F%F%%#ftGF%*$%\"sG\"\"#!\"\"" } {TEXT -1 3 " " }{TEXT 365 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eqn4 := subs(\{F=800*slug*ft/s^2, y=0\}, eqn3);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%eqn4G/,$**\"$+)\"\"\"%%slugGF)%#ftGF)%\"sG!\"#F),$ **$\"+#f0u@$!\")F)F+F)F,F-&%\"MG6#F,F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "eqn5 := M[s] = solve(eqn4, M[s]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%%eqn5G/&%\"MG6#%\"sG,$*&$\"+XaZ'[#!\")\"\"\"%%slugG F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 28 "The weight of the satellite " } {XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 45 " feet above the surface of the earth becomes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn6 := subs (eqn5, eqn3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn6G/%\"FG,$*,$\" +(>`-]$\"\")\"\"\"%#ftG\"\"$%\"sG!\"#%%slugGF,,&*&$\"+Ads\"4#F0F,F-F,F ,%\"yGF,F0F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "To write this in terms of the standard unit of force, the pound, we convert:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "eqn7 := subs(slug = pound*s^2/ft, eqn6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn7G/%\"FG,$**$\"+(>`-]$\"\")\"\" \"%#ftG\"\"#%£GF,,&*&$\"+Ads\"4#!\"#F,F-F,F,%\"yGF,F4F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Wo rk := Int(rhs(eqn7), y = 0 .. 200*5280*ft);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG-%$IntG6$,$**$\"+(>`-]$\"\")\"\"\"%#ftG\"\"#%&p oundGF-,&*&$\"+Ads\"4#!\"#F-F.F-F-%\"yGF-F5F-/F6;\"\"!,$*&\"(+g0\"F-F. F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Work := student[cha ngevar](y=u*ft, Work, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG- %$IntG6$,$**$\"+(>`-]$\"\")\"\"\"%#ftGF-%£GF-,&$\"+Ads\"4#!\"#F-% \"uGF-F3F-/F4;\"\"!\"(+g0\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Work = value( Work );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6 $,$**$\"+(>`-]$\"\")\"\"\"%#ftGF,%£GF,,&$\"+Ads\"4#!\"#F,%\"uGF,F 2F,/F3;\"\"!\"(+g0\",$*($\"+xC+U!)!\"\"F,F-F,F.F,F," }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 " \+ \+ " }{TEXT 356 1 " " }{XPPEDIT 256 0 "Omega;" "6#%&OmegaG" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 373 20 "Exercise 3 Page 601" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 369 "A rocket climbs straight up. Its initial total weight, including \+ fuel, is 7000 pounds. Fuel is consumed at the constant rate of 30 p ounds per mile. Taking into account the decrease in weight due to fuel consumption, but disregarding the decrease in weight due to increasin g elevation, approximate the work done in lifting the rocket the firs t 20 miles into space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 374 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "At height " }{XPPEDIT 18 0 "y ;" "6#%\"yG" }{TEXT -1 34 " measured in feet, the load is " } {XPPEDIT 18 0 "7000-30*y/5280;" "6#,&\"%+q\"\"\"*(\"#IF%%\"yGF%\"%!G&! \"\"F*" }{TEXT -1 32 " pounds. Divide the interval " }{XPPEDIT 18 0 "[0, 20(5280)];" "6#7$\"\"!-\"#?6#\"%!G&" }{TEXT -1 8 " into " } {XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 30 " equal subintervals of leng th " }{XPPEDIT 18 0 "Delta*y;" "6#*&%&DeltaG\"\"\"%\"yGF%" }{TEXT -1 18 ". Choose a point " }{XPPEDIT 18 0 "y[j];" "6#&%\"yG6#%\"jG" } {TEXT -1 9 " in the " }{XPPEDIT 18 0 "j^th;" "6#)%\"jG%#thG" }{TEXT -1 43 " subinterval Lifting the load through the " }{XPPEDIT 18 0 "j^ th;" "6#)%\"jG%#thG" }{TEXT -1 30 " subinterval requires about " } {XPPEDIT 18 0 "(7000-30*y[j]/5280)*Delta*y;" "6#*(,&\"%+q\"\"\"*(\"#IF &&%\"yG6#%\"jGF&\"%!G&!\"\"F.F&%&DeltaGF&F*F&" }{TEXT -1 50 " foot- pounds of work. The total work is about " }{XPPEDIT 18 0 "Sum((7000-3 0*y/5280)*Delta*y,j = 1 .. N);" "6#-%$SumG6$*(,&\"%+q\"\"\"*(\"#IF)%\" yGF)\"%!G&!\"\"F.F)%&DeltaGF)F,F)/%\"jG;F)%\"NG" }{TEXT -1 18 " foot- pounds. As " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 60 " increases th e approximation becomes more accurate. Letting " }{XPPEDIT 18 0 "N;" " 6#%\"NG" }{TEXT -1 99 " tend to infinity we obtain the work as the li mit of the sums, which we recognize as Riemann sums:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 " Work = " } {XPPEDIT 18 0 "limit(Sum((7000-30*y/5280)*Delta*y,j = 1 .. N),N = infi nity);" "6#-%&limitG6$-%$SumG6$*(,&\"%+q\"\"\"*(\"#IF,%\"yGF,\"%!G&!\" \"F1F,%&DeltaGF,F/F,/%\"jG;F,%\"NG/F6%)infinityG" }{TEXT -1 7 " = \+ " }{XPPEDIT 18 0 "Int(7000-30*y/5280,y = 0 .. 20*5280);" "6#-%$IntG6$, &\"%+q\"\"\"*(\"#IF(%\"yGF(\"%!G&!\"\"F-/F+;\"\"!*&\"#?F(F,F(" }{TEXT -1 29 " = 707,520,000 foot-pounds." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 157 " \+ \+ " }{TEXT 375 1 " " } {XPPEDIT 256 0 "Omega;" "6#%&OmegaG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 376 9 "Exercise " }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "A rocket climbs straight up. Its initial total weight, including fuel, is 700 0 pounds. The rate at which the " }{TEXT 380 4 "mass" }{TEXT -1 183 " of fuel is burned is constant. At the surface of the earth the burn r ate, stated in pounds, is 30 pounds per mile Taking into account the d ecrease in weight due to fuel consumption " }{TEXT 379 3 "and" } {TEXT -1 131 " the decrease in weight due to increasing elevation, ca lculate the work done in lifting the rocket the first 20 miles into s pace." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 377 8 " Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "M[R];" "6#&%\"MG6#%\"RG" } {TEXT -1 105 " denote the mass of the rocket (including fuel) in term s of slugs. Then the weight in pounds is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "NewtonsLaw1 := Weight = .140 7716776e17*ft^3/s^2*M[R]/(20917257.22*ft+y)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,NewtonsLaw1G/%'WeightG,$*,$\"+wnr29\"\"(\"\"\"%#ftG \"\"$%\"sG!\"#&%\"MG6#%\"RGF,,&*&$\"+Ads\"4#F0F,F-F,F,%\"yGF,F0F," }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "We use th is to solve for the initial mass of the fuel/rocket system:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "eq n1 := M[R] = solve(subs(\{y=0*ft, Weight = 7000*slug*ft/s^2\}, Newtons Law1), M[R]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn1G/&%\"MG6#%\"R G,$*&$\"+9gmv@!\"(\"\"\"%%slugGF/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Thus, initially, " }{XPPEDIT 18 0 "M[R ] = 217.5666014*slug;" "6#/&%\"MG6#%\"RG*&-%&FloatG6$\"+9gmv@!\"(\"\" \"%%slugGF." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Now let " }{XPPEDIT 18 0 "M[fuel];" "6#&%\"MG6# %%fuelG" }{TEXT -1 40 " be the mass of fuel burned each mile. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "NewtonsLaw2 := Weight = .140 7716776e17*ft^3/s^2*M[fuel]/(20917257.22*ft+y)^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,NewtonsLaw2G/%'WeightG,$*,$\"+wnr29\"\"(\"\"\"%#ftG \"\"$%\"sG!\"#&%\"MG6#%%fuelGF,,&*&$\"+Ads\"4#F0F,F-F,F,%\"yGF,F0F," } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "eqn2 := M[fuel] = solve(su bs(\{y=0*ft, Weight = 30*slug*ft/s^2\}, NewtonsLaw2), M[fuel]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/&%\"MG6#%%fuelG,$*&$\"+ " 0 "" {MPLTEXT 1 0 45 "eqn3 := M[R] = rhs(eqn1)-rhs (eqn2)*y/5280/ft;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/&%\"MG6# %\"RG,&*&$\"+9gmv@!\"(\"\"\"%%slugGF/F/**$\"+uE'fw\"!#8F/F0F/%\"yGF/%# ftG!\"\"F7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The weight of the fuel/rocket s ystem " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 45 " feet above the s urface of the earth becomes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eq n4 := subs(eqn3, NewtonsLaw1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%e qn4G/%'WeightG,$*,$\"+wnr29\"\"(\"\"\"%#ftG\"\"$%\"sG!\"#,&*&$\"+9gmv@ !\"(F,%%slugGF,F,**$\"+uE'fw\"!#8F,F6F,%\"yGF,F-!\"\"F " 0 "" {MPLTEXT 1 0 40 "eqn5 := subs(slug = pound*s^2/ft, eqn4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn5G/%'WeightG,$*,$\"+wnr29\"\"(\" \"\"%#ftG\"\"$%\"sG!\"#,&**$\"+9gmv@!\"(F,%£GF,F/\"\"#F-!\"\"F,*, $\"+uE'fw\"!#8F,F6F,F/F7F-F0%\"yGF,F8F,,&*&$\"+Ads\"4#F0F,F-F,F,F=F,F0 F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Work := Int(rhs(eqn5), y = 0*ft .. 20*5280*ft);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG-%$IntG6$,$*,$\"+wnr29\"\"(\"\"\"%#ftG\"\" $%\"sG!\"#,&**$\"+9gmv@!\"(F-%£GF-F0\"\"#F.!\"\"F-*,$\"+uE'fw\"!# 8F-F7F-F0F8F.F1%\"yGF-F9F-,&*&$\"+Ads\"4#F1F-F.F-F-F>F-F1F-/F>;\"\"!,$ *&\"'+c5F-F.F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Work := student[changevar](y=u*ft, Work, u);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG-%$IntG6$,$*,$\"+wnr29\"\"(\"\"\"%#ftGF-%£GF-,&$\"+9gm v@!\"(!\"\"*&$\"+uE'fw\"!#8F-%\"uGF-F-F-,&$\"+Ads\"4#!\"#F-F9F-F=F4/F9 ;\"\"!\"'+c5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Work := val ue( Work );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG,$*($\"*r*=Sq\" \"!\"\"\"%#ftGF*%£GF*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 158 " \+ \+ " }{TEXT 378 1 " " } {XPPEDIT 256 0 "Omega;" "6#%&OmegaG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 40 "2. Heavy Cables: Exercises f rom the Text" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 351 20 "Exercise 4 Page 601" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 218 "A man stands at the top of a tall building and pulls a chain up the side of the building. The chain is 50 feet lon g and \nweighs 3 pounds per linear foot. How much work does the man do in pulling the chain to the top?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 352 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 188 "Orient the vertical ax is so that the origin is at the top of the building and the positive d irection is downward. The chain is realized as the interval [0,50]. Si nce the weight density is " }{XPPEDIT 18 0 "3*pounds/ft;" "6#*(\"\"$\" \"\"%'poundsGF%%#ftG!\"\"" }{TEXT -1 50 ", an infinitesmal section of the cable of length " }{XPPEDIT 18 0 "dy*ft;" "6#*&%#dyG\"\"\"%#ftGF% " }{TEXT -1 9 " weighs " }{XPPEDIT 18 0 "3*dy*pounds;" "6#*(\"\"$\"\" \"%#dyGF%%'poundsGF%" }{TEXT -1 21 ". If that section is " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 49 " feet from the top, the work done li fting it is " }{XPPEDIT 18 0 "y*3*dy*ft*pounds;" "6#*,%\"yG\"\"\"\"\" $F%%#dyGF%%#ftGF%%'poundsGF%" }{TEXT -1 18 ". The total work " } {XPPEDIT 18 0 "W;" "6#%\"WG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "W := Int(y*3 *pounds/ft, y=0*ft..50*ft);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"WG- %$IntG6$,$**\"\"$\"\"\"%\"yGF+%'poundsGF+%#ftG!\"\"F+/F,;\"\"!,$*&\"#] F+F.F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "W = value(W);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$**\"\"$\"\"\"%\"yGF*%'po undsGF*%#ftG!\"\"F*/F+;\"\"!,$*&\"#]F*F-F*F*,$*(\"%]PF*F-F*F,F*F*" }}} {PARA 0 "" 0 "" {TEXT -1 134 " \+ \+ " }{TEXT 353 1 " " }{XPPEDIT 256 0 "Omega;" "6 #%&OmegaG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 341 19 "Exercise 6 Page 601" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 186 "A heavy uniform cable is used to \+ lift a 300 pound load from ground level to the top of a 100 foot high \+ building. If the cable weighs 20 pounds per linear foot then how much \+ work is done?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 342 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 " Orient the vertical axis so that the origin is at the top of the build ing and the positive direction is downward. The work " }{XPPEDIT 18 0 "W[L];" "6#&%\"WG6#%\"LG" }{TEXT -1 49 "lifting the load is (300)(10 0) foot-pounds. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "W[L] := 300*100*ft*pounds;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"WG6#%\" LG,$*(\"&++$\"\"\"%#ftGF+%'poundsGF+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Since the density of the cable is \+ " }{XPPEDIT 18 0 "delta = 20*pounds/ft;" "6#/%&deltaG*(\"#?\"\"\"%'pou ndsGF'%#ftG!\"\"" }{TEXT -1 61 ", the weight of an infinitesimal leng th of cable of length " }{XPPEDIT 18 0 "dy*ft;" "6#*&%#dyG\"\"\"%#ftG F%" }{TEXT -1 6 " is " }{XPPEDIT 18 0 "delta*dy;" "6#*&%&deltaG\"\" \"%#dyGF%" }{TEXT -1 21 ". If that length is " }{XPPEDIT 18 0 "y;" "6 #%\"yG" }{TEXT -1 74 " feet from the top of the building then the wor k done on that length is " }{XPPEDIT 18 0 "y*delta*dy;" "6#*(%\"yG\" \"\"%&deltaGF%%#dyGF%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "The total work " }{XPPEDIT 18 0 "W[C];" "6#&%\"WG6#%\"CG" }{TEXT -1 26 " lifting the cable is " }{XPPEDIT 18 0 "Int(y*delta,y = 0 .. 1 00*ft);" "6#-%$IntG6$*&%\"yG\"\"\"%&deltaGF(/F';\"\"!*&\"$+\"F(%#ftGF( " }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "delta := 20* pounds/ft;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&deltaG,$*(\"#?\"\"\"% 'poundsGF(%#ftG!\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "W [C] := Int(y*delta, y = 0 .. 100*ft);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"WG6#%\"CG-%$IntG6$,$**\"#?\"\"\"%\"yGF.%'poundsGF.%#ftG!\"\"F. /F/;\"\"!,$*&\"$+\"F.F1F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The total work \+ is " }{XPPEDIT 18 0 "W[L]+W[C];" "6#,&&%\"WG6#%\"LG\"\"\"&F%6#%\"CGF( " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "Work = W[L]+value(W[C]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%%WorkG,$*(\"'++8\"\"\"%#ftGF(%'poundsGF(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 " \+ \+ " }{TEXT 343 1 " " }{XPPEDIT 256 0 "Omega;" "6#%&OmegaG " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 344 19 "Exercise 7 Page 601" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 138 "With regard to the preceding exercise, how much work is done i n lifting the load from ground level to a height 30 feet above ground \+ level?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 345 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 12 "The density " }{XPPEDIT 18 0 "delta;" "6#%&deltaG " }{TEXT -1 55 " has the same value assigned in the preceding exercise ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Her e there are three components of the total work. There are three compon ents to the total work. The work " }{XPPEDIT 18 0 "W[L];" "6#&%\"WG6# %\"LG" }{TEXT -1 47 " lifting the load is (300)(30) foot-pounds. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "W[L] := 300*pounds*30*ft;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"WG6#%\"LG,$*(\"%+!*\"\"\"%#ftGF+ %'poundsGF+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "The lowest 70 feet of the cable weighs " }{XPPEDIT 18 0 "70*ft*delta;" "6#*(\"#q\"\"\"%#ftGF%%&deltaGF%" }{TEXT -1 45 " \+ and is lifted 30 feet. Thus, the work " }{XPPEDIT 18 0 "W[Cable_b ottom];" "6#&%\"WG6#%-Cable_bottomG" }{TEXT -1 48 " lifting the lowe st 70 feet of the cable is " }{XPPEDIT 18 0 "70*ft*delta*30*ft;" "6# *,\"#q\"\"\"%#ftGF%%&deltaGF%\"#IF%F&F%" }{TEXT -1 3 " . " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "W[Cab le_bottom] := 70*ft*delta*30*ft; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> &%\"WG6#%-Cable_bottomG,$*(\"&+?%\"\"\"%#ftGF+%'poundsGF+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "Next consider the top 30 feet of cable. Since the d ensity of the cable is " }{XPPEDIT 18 0 "delta = 20*pounds/ft;" "6#/% &deltaG*(\"#?\"\"\"%'poundsGF'%#ftG!\"\"" }{TEXT -1 60 ", the weight o f an infinitesimal length of cable of length " }{XPPEDIT 18 0 "dy*ft; " "6#*&%#dyG\"\"\"%#ftGF%" }{TEXT -1 6 " is " }{XPPEDIT 18 0 "delta* dy;" "6#*&%&deltaG\"\"\"%#dyGF%" }{TEXT -1 21 ". If that length is " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 74 " feet from the top of the building then the work done on that length is " }{XPPEDIT 18 0 "y*de lta*dy;" "6#*(%\"yG\"\"\"%&deltaGF%%#dyGF%" }{TEXT -1 18 ". The total \+ work " }{XPPEDIT 18 0 "W[Cable_top];" "6#&%\"WG6#%*Cable_topG" } {TEXT -1 37 " lifting the top 30 feet of cable is" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(y*delta,y = 0 .. 30*ft);" "6#-%$Int G6$*&%\"yG\"\"\"%&deltaGF(/F';\"\"!*&\"#IF(%#ftGF(" }{TEXT -1 5 " . \n \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "W[Cable_top] := Int(y* delta,y = 0 .. 30*ft);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"WG6#%*C able_topG-%$IntG6$,$**\"#?\"\"\"%\"yGF.%'poundsGF.%#ftG!\"\"F./F/;\"\" !,$*&\"#IF.F1F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "The total work is " } {XPPEDIT 18 0 "W[L]+W[Cable_bottom]+W[Cable_top];" "6#,(&%\"WG6#%\"LG \"\"\"&F%6#%-Cable_bottomGF(&F%6#%*Cable_topGF(" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Work := value(W[L]+W[Cable_bottom]+W[Cable_top]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG,$*(\"&++'\"\"\"%#ftGF(%'poundsGF(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 " \+ \+ " }{TEXT 346 1 " " }{XPPEDIT 256 0 "Omega;" "6#%&OmegaG" }{TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 347 19 "Exercise 8 Page 601" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "With regard to the cable and load of Exercise 6, if the load is lifted to the top of the build ing then how much work is done in lifting it the final (uppermost) 30 \+ feet?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 348 0 "" }}{PARA 0 "" 0 "" {TEXT 350 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "The work " }{XPPEDIT 18 0 "W[L];" "6# &%\"WG6#%\"LG" }{TEXT -1 50 " lifting the load is (300)(30) foot-p ounds. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "W[L] := 300*30 *ft*pounds;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"WG6#%\"LG,$*(\"%+! *\"\"\"%#ftGF+%'poundsGF+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Since the density of the cable is " }{XPPEDIT 18 0 "delta = 20*pounds/ft;" "6#/%&deltaG*(\"#?\"\"\"%'poundsGF'%#ftG! \"\"" }{TEXT -1 61 ", the weight of an infinitesimal length of cable \+ of length " }{XPPEDIT 18 0 "dy*ft;" "6#*&%#dyG\"\"\"%#ftGF%" }{TEXT -1 6 " is " }{XPPEDIT 18 0 "delta*dy;" "6#*&%&deltaG\"\"\"%#dyGF%" } {TEXT -1 21 ". If that length is " }{XPPEDIT 18 0 "y;" "6#%\"yG" } {TEXT -1 74 " feet from the top of the building then the work done on that length is " }{XPPEDIT 18 0 "y*delta*dy;" "6#*(%\"yG\"\"\"%&delt aGF%%#dyGF%" }{TEXT -1 18 ". The total work " }{XPPEDIT 18 0 "W[C];" "6#&%\"WG6#%\"CG" }{TEXT -1 22 " lifting the cable is" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(y*delta,y = 0 .. 30*ft);" "6#- %$IntG6$*&%\"yG\"\"\"%&deltaGF(/F';\"\"!*&\"#IF(%#ftGF(" }{TEXT -1 22 " . The total work is " }{XPPEDIT 18 0 "W[L]+W[C];" "6#,&&%\"WG6#%\"L G\"\"\"&F%6#%\"CGF(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "W[C] := Int(y*delta, y = 0 .. 30*ft);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"WG6#%\"CG-%$IntG6$,$**\"#?\"\"\"%\"yGF.%'poundsGF. %#ftG!\"\"F./F/;\"\"!,$*&\"#IF.F1F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Work = W[L]+value(W[C]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*(\"&++'\"\"\"%#ftGF'%'poundsGF'F',$*(\"&+!=F'F(F'F) F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 " \+ \+ " }{TEXT 349 1 " " }{XPPEDIT 256 0 "Omega;" "6#%&OmegaG" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 35 "3. Springs: Exercises from the Text" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 319 19 "Exercise 9 Page 601" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "If a spring with spring constant 8 pounds/inch is stretched \+ 7 inches beyond its equilibrium position, then how much work is done ?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 320 8 "Solu tion" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 117 "In order to use the standard unit foot-pound for work, we convert the measurements in inches to measurements in feet." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "k := 8*pound/inch:\nk := subs(inch = ft/12, k);\nF := x -> k*x;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG,$*(\"#'*\"\"\"%£GF(%#ftG! \"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operat orG%&arrowGF(*&%\"kG\"\"\"9$F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Work := Int(F(x), x = 0 .. 7/12*ft);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG-%$IntG6$,$**\"#'*\"\"\"%£GF+%#ftG!\" \"%\"xGF+F+/F/;\"\"!,$*(\"\"(F+\"#7F.F-F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Work = value(Work);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$**\"#'*\"\"\"%£GF*%#ftG!\"\"%\"xGF*F*/F.;\"\"!, $*(\"\"(F*\"#7F-F,F*F*,$**\"#\\F*\"\"$F-F+F*F,F*F*" }}}{PARA 0 "" 0 " " {TEXT -1 135 " \+ \+ " }{TEXT 321 1 " " }{XPPEDIT 256 0 "Omega;" "6#%&OmegaG" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 322 21 "Exercise 10, page 601" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 175 "A spring is stretched 2 inches beyond its equilibrium position. If the force required to maintain it in its stretched position is 60 pounds then h ow much work has been done?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 323 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "In order to use the stan dard unit foot-pound for work, we convert the measurements in inches t o measurements in feet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "F := x -> k*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"FGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&%\"kG\"\"\"9$F.F(F(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eqn := 60*pounds = F(2*inch) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,$*&\"#g\"\"\"%'poundsGF) F),$*(\"\"#F)%\"kGF)%%inchGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqn := subs(inch = ft/12, eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,$*&\"#g\"\"\"%'poundsGF)F),$*(\"\"'!\"\"%\"kGF)%#ftGF)F )" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "k := solve(eqn, k);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG,$*(\"$g$\"\"\"%'poundsGF(%#ftG !\"\"F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Work := Int(F(x) , x = 0..2/12*ft);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG-%$IntG6 $,$**\"$g$\"\"\"%'poundsGF+%#ftG!\"\"%\"xGF+F+/F/;\"\"!,$*&\"\"'F.F-F+ F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Work = value(Work);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$**\"$g$\"\"\"%'poundsGF* %#ftG!\"\"%\"xGF*F*/F.;\"\"!,$*&\"\"'F-F,F*F*,$*(\"\"&F*F+F*F,F*F*" }} }{PARA 0 "" 0 "" {TEXT -1 135 " \+ \+ " }{TEXT 324 1 " " }{XPPEDIT 256 0 "Omega;" "6#%&OmegaG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 325 21 "Exercise 1 1, page 602" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 158 "A force of 280 pounds compresses a spring 4 inches from \+ its natural length of 16 inches. How much work is done compressing it a further 4 inches?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 326 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "In order to use the standard unit foot-pound for work, we conv ert the measurements in inches to measurements in feet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "F := x -> k*x;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&%\"kG\"\"\"9$F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "eqn := 280*pounds = F(4*inch);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$eqnG/,$*&\"$!G\"\"\"%'poundsGF)F),$*(\"\"%F)%\"kGF)%%inchGF)F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqn := subs(inch = ft/12, eq n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,$*&\"$!G\"\"\"%'pounds GF)F),$*(\"\"$!\"\"%\"kGF)%#ftGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "k := solve(eqn, k);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"kG,$*(\"$S)\"\"\"%'poundsGF(%#ftG!\"\"F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "Work := Int(F(x), x = 4/12*ft..8/12*ft);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG-%$IntG6$,$**\"$S)\"\"\"%'poun dsGF+%#ftG!\"\"%\"xGF+F+/F/;,$*&\"\"$F.F-F+F+,$*(\"\"#F+F4F.F-F+F+" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Work = value(Work);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$**\"$S)\"\"\"%'poundsGF*%# ftG!\"\"%\"xGF*F*/F.;,$*&\"\"$F-F,F*F*,$*(\"\"#F*F3F-F,F*F*,$*(\"$S\"F *F+F*F,F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 135 " \+ \+ " }{TEXT 327 1 " " }{XPPEDIT 256 0 "Omega;" "6#%&OmegaG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 328 21 "E xercise 12, page 602" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 149 "If 30 foot-pounds of work are done in stretching a spr ing 3 inches beyond its natural length, then how much work is done str etching it one inch more?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 329 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "In order to use the standard unit foot-pound for work, we conv ert the measurements in inches to measurements in feet." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "F := x -> k*x;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&%\"kG\"\"\"9$F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "Work := (a,b) -> Int(F(x), x = a.. b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkGf*6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF)-%$IntG 6$-%\"FG6#%\"xG/F3;9$9%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "eqn := value(Work(0*inch, 3*inch)) = 30*ft*pounds;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$eqnG/,$**\"\"*\"\"\"\"\"#!\"\"%\"kGF)%%inchGF *F),$*(\"#IF)%#ftGF)%'poundsGF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqn := subs(inch = ft/12, eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,$*(\"#K!\"\"%\"kG\"\"\"%#ftG\"\"#F+,$*(\"#IF+F ,F+%'poundsGF+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "k := so lve(eqn, k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG,$*(\"$g*\"\"\"% #ftG!\"\"%'poundsGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "w ork := Work(3/12*ft,4/12*ft);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%wo rkG-%$IntG6$,$**\"$g*\"\"\"%#ftG!\"\"%'poundsGF+%\"xGF+F+/F/;,$*&\"\"% F-F,F+F+,$*&\"\"$F-F,F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "work = value(work);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$, $**\"$g*\"\"\"%#ftG!\"\"%'poundsGF*%\"xGF*F*/F.;,$*&\"\"%F,F+F*F*,$*& \"\"$F,F+F*F*,$**\"#qF*F6F,F+F*F-F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 135 " \+ \+ " }{TEXT 330 1 " " }{XPPEDIT 256 0 "Omega;" "6#%&OmegaG" }{TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 331 21 "Exercise 13, page 602" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "A spring is stretched \+ a certain length beyond its natural length. What percentage of the tot al amount of work is expended by the first half of the stretch?" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 332 8 "Solution " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }{XPPEDIT 18 0 "L;" "6#%\"LG" }{TEXT -1 141 " be t he distance the spring is stretched beyond equilibrium. We do not know this value and we do not know the value of the spring constant " } {XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 89 ". Both parameters will can cel when we form the ratio that gives the required percentage." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 14 "F := x -> k*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&%\"kG\"\"\"9$F.F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "percentage := Int(F(x), x = 0..L/2)/Int(F(x), x = 0..L)*100*percent;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+percentageG,$**\"$+\"\"\"\"-%$IntG6$*&%\"kGF(%\"xGF( /F.;\"\"!,$*&\"\"#!\"\"%\"LGF(F(F(-F*6$F,/F.;F1F6F5%(percentGF(F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "value(percentage);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"#D\"\"\"%(percentGF&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 " \+ \+ " }{TEXT 333 1 " " } {XPPEDIT 256 0 "Omega;" "6#%&OmegaG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 334 21 "Exercise 14, page 602" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 177 "If 2/3 ft-lb of work is done in extending a spring at rest 2 inches beyon d its equilibrium position, then how much force is required to maintai n the spring in that position?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 335 8 "Solution" }{TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "In order to use the st andard unit foot-pound for work, we convert the measurements in inches to measurements in feet.." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "rest art;\ninch := ft/12;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%inchG,$*&\" #7!\"\"%#ftG\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "F : = x -> k*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)o peratorG%&arrowGF(*&%\"kG\"\"\"9$F.F(F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "W := (a,b) -> Int(F(x), x=a..b);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"WGf*6$%\"aG%\"bG6\"6$%)operatorG%&arrowGF)-%$IntG 6$-%\"FG6#%\"xG/F3;9$9%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "eqn := 2/3*ft*pound = W(0*inch,2*inch);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,$**\"\"#\"\"\"\"\"$!\"\"%#ftGF)%£GF)F)-%$ IntG6$*&%\"kGF)%\"xGF)/F3;\"\"!,$*&\"\"'F+F,F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "eqn := map(value, eqn);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqnG/,$**\"\"#\"\"\"\"\"$!\"\"%#ftGF)%£GF)F),$* (\"#sF+%\"kGF)F,F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "k : = solve(eqn, k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG,$*(\"#[\"\" \"%#ftG!\"\"%£GF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "F(2*inch);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\")\"\"\"%£G F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 133 " \+ \+ " }{TEXT 336 1 " " }{XPPEDIT 256 0 "Omega; " "6#%&OmegaG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 338 21 "Exercise 31, p age 603" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The magnitude of a force that stretches a spring is given by " } {XPPEDIT 18 0 "F(x) = k*x;" "6#/-%\"FG6#%\"xG*&%\"kG\"\"\"F'F*" } {TEXT -1 43 " when the spring is extended a distance " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 42 " beyond its equilibrium position. \+ Let " }{XPPEDIT 18 0 "W(x);" "6#-%\"WG6#%\"xG" }{TEXT -1 60 " denot e the work done in stretching the spring a distance " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 105 " from its equilibrium position. Describ e the curve that results from plotting the parametric equations " } {XPPEDIT 18 0 "F = F(x);" "6#/%\"FG-F$6#%\"xG" }{TEXT -1 6 " , " } {XPPEDIT 18 0 "W = W(x);" "6#/%\"WG-F$6#%\"xG" }{TEXT -1 10 " in the \+ " }{XPPEDIT 18 0 "FW;" "6#%#FWG" }{TEXT -1 7 "-plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 339 8 "Exercise" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The spring is " }{TEXT 340 9 "stretched" }{TEXT -1 6 " so " }{XPPEDIT 18 0 "x;" "6#% \"xG" }{TEXT -1 8 " and " }{XPPEDIT 18 0 "F = k*x;" "6#/%\"FG*&%\"k G\"\"\"%\"xGF'" }{TEXT -1 16 " are positive." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "F := x->k*x;\neqn1 := F = F( x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FGf*6#%\"xG6\"6$%)operatorG %&arrowGF(*&%\"kG\"\"\"9$F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%eqn1G/%\"FG*&%\"kG\"\"\"%\"xGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "eqn2 := x = solve(eqn1, x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn2G/%\"xG*&%\"FG\"\"\"%\"kG!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "eqn3 := W = int(F(xi), xi=0..x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn3G/%\"WG,$*(\"\"#!\"\"%\"kG\"\" \"%\"xGF)F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "eqn4 := subs (eqn2, eqn3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqn4G/%\"WG,$*(\" \"#!\"\"%\"kGF*%\"FGF)\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 48 "The parametric curve is the right half \+ (since " }{XPPEDIT 18 0 "0 <= F;" "6#1\"\"!%\"FG" }{TEXT -1 51 ") of a parabola with vertex at the origin of the " }{XPPEDIT 18 0 "FW;" "6 #%#FWG" }{TEXT -1 7 "-plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 132 " \+ \+ " }{TEXT 337 1 " " }{XPPEDIT 256 0 "Omega;" "6#%&Omega G" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 60 "4. Pumping a Flui d from a Reservoir: Exercises from the Text" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 20 "Exercise 15 page 602" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 291 "A swimming poo l full of water has square base of side 15 feet. It is 10 feet dee p and is being pumped dry. The pump floats on the surface of the water and pumps the water to the top of the pool, at which point the water \+ runs off. How much work does the pump perform in emptying the pool?" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 287 8 "Solution " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "A hor izontal slice of water that is " }{XPPEDIT 18 0 "dy;" "6#%#dyG" } {TEXT -1 26 " feet thick has weight " }{XPPEDIT 18 0 "62.428*15^2*d y;" "6#*(-%&FloatG6$\"&GC'!\"$\"\"\"*$\"#:\"\"#F)%#dyGF)" }{TEXT 290 1 "." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "If the slice is \+ " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 78 " feet from the top, the n the work done pumping it to the top of the pool is " }{XPPEDIT 18 0 "y*62.428*15^2*dy;" "6#**%\"yG\"\"\"-%&FloatG6$\"&GC'!\"$F%\"#:\"\"# %#dyGF%" }{TEXT 289 1 "." }{TEXT -1 24 " \nThe total work is " } {XPPEDIT 18 0 "Int(y*62.428*15^2,y = 0 .. 10);" "6#-%$IntG6$*(%\"yG\" \"\"-%&FloatG6$\"&GC'!\"$F(\"#:\"\"#/F';\"\"!\"#5" }{TEXT 288 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Work := Int(y*62.428*15^2,y \+ = 0 .. 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG-%$IntG6$,$*&$ \")+j/9!\"$\"\"\"%\"yGF-F-/F.;\"\"!\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Work = value(Work);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%$IntG6$,$*&$\")+j/9!\"$\"\"\"%\"yGF,F,/F-;\"\"!\"#5$\"':BqF0" }}} {PARA 0 "" 0 "" {TEXT -1 129 " \+ \+ " }{TEXT 291 1 " " }{XPPEDIT 292 0 "Omega;" "6#%&Om egaG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 20 "Exercise 16 page 602" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 360 "A swimming pool full o f water has rectangular base with side lengths of 16 and 30 feet. It \+ is 10 feet deep but not completely filled: the depth of the water it \+ contains is 8 feet. A pump floats on the surface of the water and pump s the water to the top of the pool, at which point the water runs off. How much work does the pump perform in emptying the pool?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 279 8 "Solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "A horizontal slice of water that is " }{XPPEDIT 18 0 "dy;" "6#%#dyG" }{TEXT -1 25 " feet thick has weight " }{XPPEDIT 18 0 "62.428*16*30*dy;" "6#**-%&FloatG6$\"&GC'!\"$\"\"\"\"#;F)\"#IF)%# dyGF)" }{TEXT -1 7 " . 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x=:tBW>nYtI:yay=::::::::::xIyA:::::: :ZxMy;::::::::::::::::::::::::::::::::yaHLA;:::::::::::::::::::::::::: ::::::::::bynY:JAc]:Zx=Jy;:b]nK:Jy;by:nK:tB:bynY::tB:W>:nYtI::W>ZH<:tI wA:nKtBwAnK:Jycy:::vYxI:::::::::BZZy=::::::::::::::<:ry::::::::Z:J::ya y=:::::::tIwA::::::::::::::::::::::::::::::::::vYtBW>::::::::::::::::: :::::::::::::::::::ZxMy;:W>ZH<:Jycy:nKtBwAnK:Jycy:ZHLAcyZH<:tIwA:nKtBw AnK:Jycy:nKtBbyZHnYtI:b]nK:tIb]ZxMy;: W^HnYtI:W^H<:tIb]nYtI:b]nK:Jycy::::::vYxI:::::::::BZZy=::::::::: :::::<:ry::::::::Z:J:ry:vY:::::::tIwA::::::::::::::::::::::::::::::::: ::::::nKtB:::::::::::::::::::::::::::::::::::::bynY:tBW>:nYJAcynY:JAc] :Zx=tBwax=:tBW>:nY:tI::W^H<:tI:wA::::::::vYxI:::::::::BZZy=::::::::::: :::<:ry::::::::Z:J:ryvY<:ry::::::ZxMy;:::::::::::::::::::::::::::::::: :::::::::yaHLA;::::::::::::::::::::::::::::::::::::bynY::tBW>:ZxMy;:W> ZH<:Jycy:ZH:<:yay=::::J:CZvYxIbynY::::::::::::::::::::::::::::: ::::::::::::::::xIW^H<::::::::::::::::::::::::::::::::::::Jycy:ZHLA;by ZH::::::::::::::::::::::::::::::::::::ZxMy;:W^H<:Jy;by:ZHLA ;wax=:::::::::::::jysy:::::::::Z:::: ::::::::::::::::::::::::::::::::::nYtI:W^HLy;W>:nYtI::::::::::::::::ya y=:::::::::Jjysy:::::::::::ZxMy; ::::::::::::::::::::::::::::::::::::::::::::rynKtB:::::::::::::::::::: :::::::::::::::::wax=:xIwax=::::::::::::::::jysy:::::::::Z:Zymy;:::::::::::::::wax=::::::::::::::::::::: :::::::::::::::::::::::jyc]nK:::::::::::::::::::::::::::::::::::Jycy:: :::::::::::::::::vYxI:::::::::BZZy=::::::::::::::<:ry:::::::::>JJjyCZZyM:;yA::::::::::::Z::xI:::::::BZ>Jjy;ry::::::::::::::::::::::Zx My;::::::::::::::::::::::::::::::::::::::::::::rynKtB::::::::::::::::: ::::::::::rynYtI::::::::::::::::::::::yay=::::::::JJ:<::::::AB:;JD ::::::::::::jysy:>:<:::::::::::::::::::vYxI:;Z::::::::::::::::::::yay= J:B::::::::::::3:" }}{PARA 0 "" 0 "" {TEXT -1 17 "If the slice is " } {XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 78 " feet from the top, then t he work done pumping it to the top of the pool is " }{XPPEDIT 18 0 "y *62.428*16*30*dy;" "6#*,%\"yG\"\"\"-%&FloatG6$\"&GC'!\"$F%\"#;F%\"#IF% %#dyGF%" }{TEXT 281 1 "." }{TEXT -1 101 " Because the initial water d epth is 8 feet, the slice closest to the top of the pool corresponds t o " }{XPPEDIT 18 0 "y = 2;" "6#/%\"yG\"\"#" }{TEXT 282 1 "." }{TEXT -1 23 " \nThe total work is " }{XPPEDIT 18 0 "Int(y*62.428*16*30,y \+ = 2 .. 10);" "6#-%$IntG6$**%\"yG\"\"\"-%&FloatG6$\"&GC'!\"$F(\"#;F(\"# IF(/F';\"\"#\"#5" }{TEXT 280 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Work := Int(y*62.428*16*30,y = 2 .. 10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG-%$IntG6$,$*&$ \")Sa'*H!\"$\"\"\"%\"yGF-F-/F.;\"\"#\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Work = value(Work);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%$IntG6$,$*&$\")Sa'*H!\"$\"\"\"%\"yGF,F,/F-;\"\"#\"#5$\"+?6MQ9F+" }}}{PARA 0 "" 0 "" {TEXT -1 135 " \+ \+ " }{TEXT 293 1 " " }{XPPEDIT 257 0 "Omega; " "6#%&OmegaG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 20 "Exercise 17 page 602" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "Referring to the swimming pool of the preceding problem, if power is interrupted when half the water has been pumped, then how much work has been done?" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 284 8 "Solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 146 "When half the water , initially 8 feet deep, is pumped, 4 feet of water remain. The horizo ntal slice closest to the top of the pool corresponds to " }{XPPEDIT 18 0 "y = 2;" "6#/%\"yG\"\"#" }{TEXT -1 43 " and the final slice pumpe d corresponds to " }{XPPEDIT 18 0 "y = 6;" "6#/%\"yG\"\"'" }{TEXT -1 8 " (where " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 77 " measures th e ditance in feet to the top of the tank). The total work is " } {XPPEDIT 18 0 "Int(y*62.428*16*30,y = 2 .. 6);" "6#-%$IntG6$**%\"yG\" \"\"-%&FloatG6$\"&GC'!\"$F(\"#;F(\"#IF(/F';\"\"#\"\"'" }{TEXT 285 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Work := Int(y*62.428*16*30,y = 2 .. 6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG-%$IntG6$,$*&$\")Sa'*H!\"$\"\"\"%\"yGF-F-/F.;\" \"#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Work = value(Wo rk);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$*&$\")Sa'*H!\"$\" \"\"%\"yGF,F,/F-;\"\"#\"\"'$\"++/Z%z%!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 134 " \+ \+ " }{TEXT 294 2 " " }{XPPEDIT 256 0 "Omega;" "6#%&OmegaG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 270 20 "Exercis e 18 page 602" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 333 "A semi-cylindrical tank filled with water is 16 feet lon g. It has rectangular horizontal cross-sections and vertical cross-sec tions that are semicircular of radius 6 feet. A pump is used which flo ats on the surface of the tank. It pumps water to the top, at which po int the water runs off. How much work is done in pumping the water?" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 8 "Solution " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Orien t the positive " }{XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT -1 77 "-axis so \+ that it is directed downward with the origin at the top of the tank." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 " \+ " }{OLE 1 370712 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:J=j:J:ZJJ;j: JyyyyyI<:^:f:n:v:>;vyyyyy=:::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::fyyyyyQt=F=N=V=^=f=n=v=>>AFANAVA^AfAnA;jYJZB:=EFENEVE^EfEnEvE>F:::::::JEf:yyyxIN::;`:Z@[ ::JJPMYHS>AM^Aj;J:@:<:=j[vGUMrvC?MoJ::::::::JCNZ;^:vYxI>:<::::::j`J:j: vCSmlF@[KaFFcmnnHEM:>:::::::oJ;@j:j;B:yayA:<::::::M:CH^EJ:>:=:jRj^^HEm pnCfGEM:>::::::n=?R:yyyyyy:>:<::::::?J:f:>:F:;JyKy;vY::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::F:wyy AbR<:Tn=B:[dJ_UWaETX::::LyVDurJ:::::::::::=:;::::::::::F:D 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::::::vYxI:;Z::::::::::::::::::::yay=J:B::::::::::::3:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "A \"slice\" of water that has thickness " }{XPPEDIT 18 0 "dy;" "6#%#dyG" }{TEXT -1 14 " and that is " }{XPPEDIT 18 0 "y;" "6#% \"yG" }{TEXT -1 43 " feet from the top of the tank has width " } {XPPEDIT 18 0 "2*sqrt(36-y^2);" "6#*&\"\"#\"\"\"-%%sqrtG6#,&\"#OF%*$% \"yGF$!\"\"F%" }{TEXT -1 11 ", volume\n" }{XPPEDIT 18 0 "2*16*sqrt(3 6-y^2)*dy;" "6#**\"\"#\"\"\"\"#;F%-%%sqrtG6#,&\"#OF%*$%\"yGF$!\"\"F%%# dyGF%" }{TEXT -1 15 ", and weight " }{XPPEDIT 18 0 "62.428*2*16*sqrt (36-y^2);" "6#**-%&FloatG6$\"&GC'!\"$\"\"\"\"\"#F)\"#;F)-%%sqrtG6#,&\" #OF)*$%\"yGF*!\"\"F)" }{TEXT -1 1 " " }{TEXT 275 1 "." }{TEXT -1 62 " \+ The work done in pumping this slice to the top of the tank is" }} {PARA 0 "" 0 "" {XPPEDIT 18 0 "2*16*62.428*y*sqrt(36-y^2)*dy;" "6#*.\" \"#\"\"\"\"#;F%-%&FloatG6$\"&GC'!\"$F%%\"yGF%-%%sqrtG6#,&\"#OF%*$F,F$! \"\"F%%#dyGF%" }{TEXT 274 1 "." }{TEXT -1 52 " The total work in pump ing the filled tank dry is " }{XPPEDIT 18 0 "Int(2*16*62.428*y*sqrt(3 6-y^2),y = 0 .. 6);" "6#-%$IntG6$*,\"\"#\"\"\"\"#;F(-%&FloatG6$\"&GC'! \"$F(%\"yGF(-%%sqrtG6#,&\"#OF(*$F/F'!\"\"F(/F/;\"\"!\"\"'" }{TEXT 276 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Work := Int(2*16*62.428*y* sqrt(36-y^2),y = 0 .. 6);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%WorkG- %$IntG6$,$*($\"('p(*>!\"$\"\"\"%\"yGF-,&\"#OF-*$)F.\"\"#F-!\"\"#F-F3F- /F.;\"\"!\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Work = va lue(Work);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$*($\"('p(*>! \"$\"\"\"%\"yGF,,&\"#OF,*$)F-\"\"#F,!\"\"#F,F2F,/F-;\"\"!\"\"'$\"+?6MQ 9!\"%" }}}{PARA 0 "" 0 "" {TEXT -1 135 " \+ \+ " }{TEXT 295 1 " " }{XPPEDIT 256 0 "Omega;" "6#%&OmegaG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 266 20 "Exercise 19 page 602" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 354 "A filled reservoir is in the shape of an inverted cone, with the radius of the base of the con e and the depth at the center both being 100 feet. A pump is used wh ich floats on the surface of the reservoir. It pumps water to the top, at which point the water runs off. How much work is done in pumping t he water in the reservoir to a depth of 50 feet?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 8 "Solution" }}{PARA 0 "" 0 "" {TEXT -1 20 "Orient the positive " }{XPPEDIT 18 0 "y;" "6#%\"yG" } {TEXT -1 76 "-axis so that it is directed downward with the origin at \+ the top of the tank" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 18 " " }}{PARA 0 "" 0 "" {TEXT -1 36 " \+ " }{OLE 1 312864 1 "[xm]Br=WfoRrB:::wk ;nyyI;G:;:j:[Z:F:y;;\\:FBNBVB^BfBnBvB>CFCNC VC^CfCnCvC>DIFINIVI^IfInIvI>J>ELq^H_mvJ::::::::g jR<:T><::;UFMC_KVFKX:A:;R:B:F:AlqfG[maNFO=;::::::::_J; @J@Wlj^HMMufF;J:::::::N=?R:=:AZ:vYxY:B:::: ::>;^jk`:>:=:jRj^^HEmpnCfGEM:>::::::n=?R:yyyyyy:>:<::::::?:EJ:>:F:;JyK y;vY:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::F:wyyAbR<:Tn=B:[dJ_UWaETX::::LyVDurJ:::::: :::::=:;::::::::::F:D:<::::::::::::::::::::::::::::::::::::::::::::::: :::::j:C:?>A>CF:fJnJvJ>KFKNK_>a>c>e>g>i>kN:FMNMVM^MfM 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:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::>r:Ar:myyiy=::::::::::::::::::::::::<:xI::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::;`:mysy::::::::::::::::::::::::Z:Z y=:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::JZCjvY xI::::::::::::::::::::::::B:ry:::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::>RRRRRRRRRRRRRRRRRRRRjy;::;`:mysy::::::::::::: :::::::::::Z:vYxI::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::;`:mysy:Z:MZsyvY:::::::::::::::::J:RxIyA:>RRRxIBjvYxI:::::::::::::: ::::::::::B:yay=:::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::JZCj::::::::::::::::::vYxI:::;B:<:yay=<:;Zymy;; Z:jy;;Zymy;;Z:jysy::::>RRRxIyA::::::::::::::::viywI>:<:yay=xeyxIyE:>:jyr ixuywyytiymy<:;:vIxeyxYyA:::::::::::::::::Bj>RxIyA:>RRRRRRRRRRRR::::::::::::jysy:>:<:::::::::::::::::::vYxI:;Z::::::::::::::::::::y ay=J:B:::::::::::::5:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "The diagram below shows a \"slice\" of water of thickn ess " }{XPPEDIT 18 0 "dy;" "6#%#dyG" }{TEXT -1 12 " at level " } {XPPEDIT 18 0 "y;" "6#%\"yG" }{TEXT 262 1 "." }{TEXT -1 28 " The radiu s of the slice is " }{XPPEDIT 18 0 "100-y;" "6#,&\"$+\"\"\"\"%\"yG!\" \"" }{TEXT 261 1 "." }{TEXT -1 39 " The volume of the slice is theref ore " }{XPPEDIT 18 0 "Pi*(100-y)^2*dy;" "6#*(%#PiG\"\"\"*$,&\"$+\"F%% \"yG!\"\"\"\"#F%%#dyGF%" }{TEXT -1 22 " and the weight is " } {XPPEDIT 18 0 "62.428*Pi*(100-y)^2*dy;" "6#**-%&FloatG6$\"&GC'!\"$\"\" \"%#PiGF),&\"$+\"F)%\"yG!\"\"\"\"#%#dyGF)" }{TEXT 260 1 "." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 " \+ " }{OLE 1 309276 1 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