Question

How to find the instantaneous rate of change of the volume of a cylinder with respect to its radius for a cylinder with a height of 3 feet and a radius of 4 feet?

Answer 1

To find the instantaneous rate of change of the volume of a cylinder with respect to its radius, we start by using the formula for the volume \( V \) of a cylinder, which is given by:

\[
V = \pi r^2 h
\]

where:
- \( V \) is the volume,
- \( r \) is the radius,
- \( h \) is the height.

In this case, the height \( h \) is 3 feet. Thus, the volume can be rewritten as:

\[
V = \pi r^2 \cdot 3 = 3\pi r^2
\]

Next, we need to find the derivative of \( V \) with respect to \( r \) to get the instantaneous rate of change:

\[
\frac{dV}{dr} = \frac{d}{dr}(3\pi r^2)
\]

Using the power rule for differentiation, we calculate:

\[
\frac{dV}{dr} = 3\pi \cdot 2r = 6\pi r
\]

Now, we need to evaluate this derivative at the specific radius \( r = 4 \) feet:

\[
\frac{dV}{dr} \bigg|_{r=4} = 6\pi \cdot 4 = 24\pi
\]

Therefore, the instantaneous rate of change of the volume of the cylinder with respect to its radius when the radius is 4 feet is:

\[
24\pi \text{ cubic feet per foot}
\]

This means that for each additional foot increase in the radius, the volume of the cylinder increases by \( 24\pi \) cubic feet.