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?
$\frac{dr}{dt}=kr$

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

$\ln{r}=kt+C$

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

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

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

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.
$V=\frac{4}{3}\pi r^3$

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

r=10, dr=.02

Find dV