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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "page 81 problem 19 by Davi
d Blood and Andrew Kelley" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 339 "A s
nowball melts in such a way that the rate of change in its volume is p
roportional to its surface area. If the snowball was initially 4 in. i
n diameter (2 in. in radius) and after 30 min its diameter is 3 in. (3
/2 in. in radius), when will its diameter be 2 in. (radius: 1 in.)? Ma
thematically speaking, when will the snowball disappear?" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The equat
ion for the surface area of a sphere: A=4" }{XPPEDIT 18 0 "PI;" "6#%#P
IG" }{XPPEDIT 18 0 "r^2;" "6#*$%\"rG\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 
43 "The equation for the volume of a sphere: V=" }{XPPEDIT 18 0 "4/3;
" "6#*&\"\"%\"\"\"\"\"$!\"\"" }{XPPEDIT 18 0 "PI;" "6#%#PIG" }
{XPPEDIT 18 0 "r^3;" "6#*$%\"rG\"\"$" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 10 "Therefore " }{XPPEDIT 18 0 "diff(V,r);
" "6#-%%diffG6$%\"VG%\"rG" }{TEXT -1 2 "=A" }}{PARA 0 "" 0 "" {TEXT 
-1 47 "Since the change in V is proportional to A and " }{XPPEDIT 18 
0 "diff(V,r);" "6#-%%diffG6$%\"VG%\"rG" }{TEXT -1 9 "=A, then " }
{XPPEDIT 18 0 "diff(V,t);" "6#-%%diffG6$%\"VG%\"tG" }{TEXT -1 3 "=k(" 
}{XPPEDIT 18 0 "diff(V,r);" "6#-%%diffG6$%\"VG%\"rG" }{TEXT -1 2 ")." 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "dV/dt=k(dV/dr)--dV's cancel-- =
> 1/dt=k/dr => dr/dt=k" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de1:=diff(r(t),t)=k;" }}{PARA 11 "
" 1 "" {TEXT -1 0 "" }{XPPMATH 20 "6#>%$de1G/-%%diffG6$-%\"rG6#%\"tGF,
%\"kG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "Here we solved the integ
ral and set our initial condition:(r(0)=2) in order to solve for C." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "sol1:=unapply(rhs(dsolve(
\{de1,r(0)=2\},r(t))),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1GR
6#%\"tG6\"6$%)operatorG%&arrowGF(,&*&%\"kG\"\"\"9$F/F/\"\"#F/F(F(F(" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Here we solve for k with our sec
ond condition:(r(30)=3/2)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
25 "k:=solve(sol1(30)=3/2,k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"k
G#!\"\"\"#g" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "sol1(t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"tG#!\"\"\"#g\"\"#\"\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Therefore r(t)=-t/60+2; here we u
se sol1 as our radius and set the radius to 1 in order to find time(t)
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(sol1(t)=1,t);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#g" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 117 "Therefore it takes sixty minutes for the snowball's diam
ter to melt to two inches (or its radius to melt to one inch)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "solve(sol1(t)=0,t);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$?\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 57 "It takes 120 minutes for the snowball to melt completely.
" }}}}{MARK "0 0 0" 51 }{VIEWOPTS 1 1 0 1 1 1803 }