Question

the volume v = \frac{4}{3}\pi r^{3} of a spherical balloon changes with the radius.
a. at what rate (in³/in) does the volume change with respect to the radius when r = 11 in?
b. using the rate from part a, by approximately how much does the volume increase when the radius changes from 11 to 11.1 in?
a. at what rate (in³/in) does the volume change with respect to the radius when r = 11 in?
484\pi in³/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 11 to 11.1 in?
in³
(use the answer from part a to find this answer. round to two decimal places as needed.)

Explanation:

Step1: Recall the derivative of volume formula

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

Step2: Evaluate the derivative at $r = 11$

Substitute $r = 11$ into $V'$. So $V'(11)=4\pi(11)^{2}=484\pi$ $\text{in}^3/\text{in}$. This answers part a.

Step3: Use linear - approximation formula

The linear - approximation formula is $\Delta V\approx V'(r)\Delta r$. Here, $r = 11$, $\Delta r=11.1 - 11=0.1$. Since $V'(11) = 484\pi$, then $\Delta V\approx484\pi\times0.1$.

Step4: Calculate the approximate volume increase

$\Delta V\approx484\pi\times0.1=48.4\pi\approx151.97$ $\text{in}^3$.

Answer:

a. $484\pi$ $\text{in}^3/\text{in}$
b. $151.97$ $\text{in}^3$