The radius of a right circular cone is increasing at a rate of 1.4 in/s while its height is decreasing at a rate of 2.1 in/s. At what rate is the volume of the cone changing when the radius is 142 in. and the height is 129 in.?

We are given a cone with a radius \( r \) and height \( h \). The radius is increasing at a rate of \( \frac{dr}{dt} = 1.4 \, \text{in/s} \), and the height is decreasing at a rate of \( \frac{dh}{dt} = -2.1 \, \text{in/s} \). We are tasked with finding the rate at which the volume of the cone is changing when \( r = 142 \, \text{in} \) and \( h = 129 \, \text{in} \).

### Step 1: Volume Formula for a Cone

The volume \( V \) of a cone is given by the formula:
\[
V = \frac{1}{3} \pi r^2 h
\]

### Step 2: Differentiate the Volume with Respect to Time

To find the rate of change of the volume with respect to time \( \frac{dV}{dt} \), we apply the product rule and chain rule to differentiate \( V \) with respect to time \( t \):
\[
\frac{dV}{dt} = \frac{1}{3} \pi \left( 2r \frac{dr}{dt} h + r^2 \frac{dh}{dt} \right)
\]

### Step 3: Plug in the Given Values

Now, substitute the values:
- \( r = 142 \, \text{in} \)
- \( h = 129 \, \text{in} \)
- \( \frac{dr}{dt} = 1.4 \, \text{in/s} \)
- \( \frac{dh}{dt} = -2.1 \, \text{in/s} \)

Substituting these into the equation:
\[
\frac{dV}{dt} = \frac{1}{3} \pi \left( 2(142)(1.4)(129) + (142)^2(-2.1) \right)
\]

Let me calculate this.The rate at which the volume of the cone is changing at the given moment is approximately \( 9368.23 \, \text{in}^3/\text{s} \).

This means that the volume is increasing at this rate under the specified conditions.

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How does the volume change if both the radius and height increase simultaneously?
2. How would the formula change if the cone's height remained constant while only the radius varied?
3. How does the cone’s surface area change as the radius and height change?
4. What would happen to the volume rate if both the radius and height were decreasing?
5. Can you derive the formula for the slant height of the cone given the radius and height?

**Tip:** When dealing with related rates problems, always keep track of units and signs to ensure accurate differentiation.

Solve a related rates problem involving the volume of a cone. Given that the radius is increasing at 1.4 in/s and the height is decreasing at 2.1 in/s, find the rate at which the volume is changing when the radius is 142 in. and the height is 129 in. This problem covers related rates, geometry, and differentiation concepts for Grades 11-12.

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