We are given that the radius of a cone is increasing at a constant rate, while the volume remains constant. We are tasked with finding the rate of change of the height of the cone when the radius is 77 inches.

Step 1: Volume Equation

The volume $V$ of a cone is given by the equation: $V = \frac{1}{3} \pi r^2 h$ where:

• $V$ is the volume (constant at 507 cubic inches),
• $r$ is the radius,
• $h$ is the height.

We are also told that $\frac{dr}{dt} = 3$ inches per second (rate of change of the radius), and we need to find $\frac{dh}{dt}$ (rate of change of the height) when $r = 77$ inches.

Step 2: Differentiate the Volume Equation Implicitly

Since the volume is constant, we can differentiate the volume equation with respect to time $t$.

$\frac{d}{dt}\left(V = \frac{1}{3} \pi r^2 h\right)$

Using the product rule for differentiation, we get: $0 = \frac{d}{dt} \left( \frac{1}{3} \pi r^2 h \right)$ $0 = \frac{1}{3} \pi \left( 2r h \frac{dr}{dt} + r^2 \frac{dh}{dt} \right)$

Simplifying the equation: $0 = 2r h \frac{dr}{dt} + r^2 \frac{dh}{dt}$

Step 3: Solve for $\frac{dh}{dt}$

Now, plug in the given values:

• $\frac{dr}{dt} = 3$ inches per second,
• $V = 507$ cubic inches, so use this to solve for $h$ at $r = 77$ inches.

From the volume equation: $507 = \frac{1}{3} \pi (77)^2 h$ $507 = \frac{1}{3} \pi (5929) h$ $507 = \frac{5929 \pi}{3} h$ Solving for $h$: $h = \frac{507 \times 3}{5929 \pi}$ $h \approx \frac{1521}{5929 \pi}$ $h \approx \frac{1521}{18619.48} \approx 0.0817 \text{ inches}$

Now substitute this value of $h$, along with $r = 77$ inches and $\frac{dr}{dt} = 3$ inches per second, into the differentiated equation:

$0 = 2(77)(0.0817)(3) + (77)^2 \frac{dh}{dt}$ $0 = 37.635 + 5929 \frac{dh}{dt}$ $5929 \frac{dh}{dt} = -37.635$ $\frac{dh}{dt} = \frac{-37.635}{5929} \approx -0.00634 \text{ inches per second}$

Thus, the rate of change of the height is approximately $-0.006$ inches per second (rounded to three decimal places).

Final Answer:

$\frac{dh}{dt} \approx -0.006 \text{ inches per second}$

Would you like more details on this solution, or is there anything unclear?

Here are a few related questions for further exploration:

1. How does the volume of a cone change when both the radius and height are increasing at different rates?
2. What happens to the rate of change of height if the volume were to change instead of remaining constant?
3. How can you derive the formula for the volume of a cone starting from basic geometry?
4. How would the solution change if the rate of change of the radius were different?
5. Can the rate of change of height become positive under any circumstances for this problem?

Tip: When dealing with related rates problems, always identify which quantities are constant and which are changing, and use the given relationships to differentiate and solve for the unknown rate.