Question

At time $t>0$ the volume of a sphere is increasing at a rate proportional to the reciprocal of its radius. At $t=0$ the radius of the sphere is $1$ unit and at $t=15$ the radius is $2$ units. Find the radius of the sphere as a function of time $t$.

• A. $r={(1+t)}^{\frac{1}{4}}$
• B. $r=(1+t{)}^{\frac{1}{2}}$
• C. $r=(1-t{)}^{\frac{1}{4}}$
• D. $r=(1-t{)}^{\frac{1}{2}}$

Answer 1

The correct option is A $r={(1+t)}^{\frac{1}{4}}$

$\frac{dv}{dt}=\frac{k}{r}\Rightarrow 4\pi r^2\frac{dr}{dt}=\frac{k}{r}$
$4\pi r^{3}dr=kdt$
$\pi r^4=kt+c$
at $t=0;r=1$ $\Rightarrow c=\pi$
$r^4=(\frac{kt}{\pi }+1)$
put $r=2;$ $t=15$ $\Rightarrow k=\pi$
$r^4=t+1$
$\therefore r=(1+t{)}^{\frac{1}{4}}$