The height of a cylinder is increasing at a rate of 5 inches per​ second, while the radius is decreasing at a rate of 2 inches per second. If the height is currently 38 ​inches,
and the radius is 16 inches, then find the rate of change in the volume.
ROUND YOUR ANSWER TO ONE DECIMAL
PLACE.
what is the rate of change in the volume

The volume \( V \) of a cylinder is given by the formula:

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

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

To find the rate of change of the volume \( \frac{dV}{dt} \), we'll need to apply the product rule since both \( r \) and \( h \) are functions of time \( t \). Differentiating the volume formula with respect to time:

\[
\frac{dV}{dt} = \frac{d}{dt} \left( \pi r^2 h \right)
\]

Applying the product rule:

\[
\frac{dV}{dt} = \pi \left( 2r \frac{dr}{dt} h + r^2 \frac{dh}{dt} \right)
\]

We are given the following values:
- \( \frac{dh}{dt} = 5 \) inches/second (rate of change of height),
- \( \frac{dr}{dt} = -2 \) inches/second (rate of change of radius, since it’s decreasing),
- \( h = 38 \) inches (current height),
- \( r = 16 \) inches (current radius).

Substituting these values into the formula:

\[
\frac{dV}{dt} = \pi \left( 2(16)(-2)(38) + (16)^2(5) \right)
\]

Now let's calculate the value.

It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can follow these steps to manually calculate the rate of change in the volume:

1. Use the formula \( \frac{dV}{dt} = \pi \left( 2r \frac{dr}{dt} h + r^2 \frac{dh}{dt} \right) \).
2. Substitute \( r = 16 \), \( h = 38 \), \( \frac{dr}{dt} = -2 \), and \( \frac{dh}{dt} = 5 \) into the formula.
3. Calculate both parts inside the parentheses:
   - First term: \( 2(16)(-2)(38) = -2432 \).
   - Second term: \( (16)^2(5) = 1280(5) = 6400 \).
4. Add these terms together: \( -2432 + 6400 = 3968 \).
5. Multiply by \( \pi \): \( 3968 \times \pi \approx 12463.2 \).

So, the rate of change in the volume is approximately \( 12463.2 \) cubic inches per second.

Let me know if you have any questions! Here are some related questions to explore further:

1. What happens to the rate of volume change if the radius increases instead of decreasing?
2. How would the rate of volume change differ if the height's growth rate decreased?
3. Can you find the rate of volume change if the cylinder is a right cone instead?
4. How does the cylinder's surface area change with these rates?
5. What role does calculus play in real-world applications involving rates of change?

**Tip:** When using related rates, always track which dimensions are increasing or decreasing to ensure correct sign usage.

The height of a cylinder is increasing at a rate of 5 inches per second, while the radius is decreasing at a rate of 2 inches per second. If the height is currently 38 inches, and the radius is 16 inches, then find the rate of change in the volume. ROUND YOUR ANSWER TO ONE DECIMAL PLACE.

This problem involves finding the rate of change in the volume of a cylinder when the height is increasing at 5 inches per second and the radius is decreasing at 2 inches per second. The cylinder's current height is 38 inches, and the radius is 16 inches. Using related rates and calculus, this solution explores how the volume changes over time.

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