Question

the volume v = \frac{4}{3}\pi r^{3} of a spherical balloon changes with the radius.
a. at what rate (in^{3}/in) does the volume change with respect to the radius when r = 3 in?
b. using the rate from part a, by approximately how much does the volume increase when the radius changes from 3 to 3.5 in?
a. at what rate (in^{3}/in) does the volume change with respect to the radius when r = 3 in?
36\pi in^{3}/in
(type an exact answer in terms of \pi.)
b. using the rate from part a, by approximately how much does the volume increase when the radius changes from 3 to 3.5 in?
\square in^{3}
(use the answer from part a to find this answer. round to two decimal places as needed.)

Explanation:

Step1: Recall the derivative formula

The volume of a sphere is $V=\frac{4}{3}\pi r^{3}$. The derivative of $V$ with respect to $r$ is $V^\prime=\frac{dV}{dr}=4\pi r^{2}$.

Step2: Calculate the rate for part a

When $r = 3$ in, substitute $r = 3$ into $V^\prime$. So $V^\prime(3)=4\pi\times3^{2}=36\pi$ in$^{3}$/in.

Step3: Calculate the volume increase for part b

The approximate change in volume $\Delta V$ is given by $\Delta V\approx\frac{dV}{dr}\Delta r$. Here, $\frac{dV}{dr}=36\pi$ (from part a) and $\Delta r=3.5 - 3=0.5$ in. Then $\Delta V\approx36\pi\times0.5 = 18\pi\approx 56.55$ in$^{3}$.

Answer:

56.55