# Thread: if the radius of a sphere is increasing at a rate ...

1. ## if the radius of a sphere is increasing at a rate ...

1) the radius of a sphere is increasing at a rate proportional to its radius. if the radius is 4 initially, and the radius is 10 after 2 seconds, what will the radius be after 3 seconds?

2) use differentials to approximate the change in volume of a sphere when the radius is increased from 10 cm to 10.02 cm.

2. Originally Posted by yoman360
1) the radius of a sphere is increasing at a rate proportional to its radius. if the radius is 4 initially, and the radius is 10 after 2 seconds, what will the radius be after 3 seconds?
$\displaystyle \frac{dr}{dt}=kr$

$\displaystyle \frac{dr}{r}=k\,dt$

$\displaystyle \ln{r}=kt+C$

r(0)=4 so $\displaystyle C=\ln{4}$

r(2)=10 so $\displaystyle k=\frac{1}{2}\ln{\frac{5}{2}}$

$\displaystyle r=4(2.5)^{\frac{t}{2}}$

Now find r(3)

Originally Posted by yoman360
2) use differentials to approximate the change in volume of a sphere when the radius is increased from 10 cm to 10.02 cm.
$\displaystyle V=\frac{4}{3}\pi r^3$

$\displaystyle dV=4\pi r^2\,dr$

r=10, dr=.02

Find dV