Growth of a Circular Stains

Given

• $\mbox{A circular stain grows}$
• $\mbox{Such that the rate of increase of the radius}$
• $\mbox{Is inversely proportional to the square of the radius.} $
• $\mbox{Another stain area grows quicker. Its equation  is:}  $
• $\dfrac{ds}{dt} = \dfrac{2 e^{2t} }{\sqrt{s}}   $

Questions

• $\mbox{What is equation for the change in area  of  the stain over time?} $
• $\mbox{Show} \;   \dfrac{da}{dt} \varpropto \dfrac{1}{\sqrt{a}}   $
• $\mbox{For second stain, for s = 81 at t=0}$
• $\mbox{Find the time when s = 100}$

Rationale

• $\dfrac{dr}{dt} = k \dfrac{1}{r^2} $
• $\dfrac{da}{dt} = \dfrac{da}{dr} * \dfrac{dr}{dt} $
• $ a = \pi r^2 $
• $\dfrac{da}{dr} = 2 \pi r $
• $\dfrac{da}{dt} = ( 2 \pi r ) * (k \dfrac{1}{r^2} )$
• $\dfrac{da}{dt} = \dfrac{2 \pi k}{r}  $
• $\mbox{And da/dt in terms of area is}$
• $a = \pi r^2 $
• $r^2 = \dfrac{a}{\pi} $
• $r = \sqrt{ \dfrac{a}{\pi}} $
• $\mbox{Substituting for r in previous solution}$
• $\dfrac{da}{dt} = \dfrac{2 \pi k}{ \sqrt{ \dfrac{a}{\pi}}} $
• $\dfrac{da}{dt} = \dfrac{2 \pi^{(\dfrac{3}{2})} k} {\sqrt{a}}$
• $\dfrac{da}{dt} \varpropto \dfrac{1}{\sqrt{a}} $
• $\dfrac{ds}{dt} = \dfrac{2 e^{2t} }{\sqrt{s}}   $
• $s^{\dfrac{1}{2}}  ds = 2 e^{2t} dt $
• $\int{s^{\dfrac{1}{2}}  ds} = \int{2 e^{2t} dt} $
• $\dfrac{s^{\dfrac{3}{2}}}{\dfrac{3}{2}}  = e^{2t}  + C$
• $\dfrac{2}{3} s^{\dfrac{3}{2}}  = e^{2t} + C $
• $s^{\dfrac{3}{2}}  = \dfrac{3}{2} e^{2t} + K $
• $\mbox{Solve for K using s=81 at t=0}    $
• $81^{\dfrac{3}{2}}  = \dfrac{3}{2} e^{2 * 0} + K $
• $9^3 = \dfrac{3}{2} e^0  +  K $
• $729 = \dfrac{3}{2}  +  K  $
• $K = 727.5 $
• $\mbox{Solve for t when s is 100} $
• $s^{\dfrac{3}{2}}  = \dfrac{3}{2} e^{2t} + 727.5 $
• $100^{\dfrac{3}{2}}  = \dfrac{3}{2} e^{2t} + 727.5 $
• $10^3  = \dfrac{3}{2} e^{2t} + 727.5 $
• $1000  = \dfrac{3}{2} e^{2t} + 727.5 $ 
• $1000 - 727.5  = \dfrac{3}{2} e^{2t}  = 272.5 $  
• $\dfrac{2}{3} 272.5 = e^{2t}  $  
• $181.6666667 = e^{2t}  $
• $\ln{181.6666667} = 2t  $ 
• $t = \dfrac{\ln{181.6666667}}{2} = 2.601086753 \approx 2.6$

Source

• www.m4ths.com