Given the problem, we need to find the rate at which the height of a conical sand pile is changing when the pile is 15 feet high. Here’s how to solve it step by step:

Step 1: Set up the relationships

• The volume $V$ of a cone is given by: $V = \frac{1}{3} \pi r^2 h$
• The diameter of the base is three times the altitude (height), which means: $d = 3h$ Since the diameter $d$ is twice the radius $r$: $r = \frac{d}{2} = \frac{3h}{2}$

Step 2: Substitute the radius into the volume formula

Substitute $r = \frac{3h}{2}$ into the volume formula: $V = \frac{1}{3} \pi \left( \frac{3h}{2} \right)^2 h$ $V = \frac{1}{3} \pi \frac{9h^2}{4} h$ $V = \frac{3\pi}{4} h^3$

Step 3: Differentiate with respect to time

Now, differentiate $V$ with respect to time $t$: $\frac{dV}{dt} = \frac{3\pi}{4} \cdot 3h^2 \frac{dh}{dt}$ $\frac{dV}{dt} = \frac{9\pi}{4} h^2 \frac{dh}{dt}$

Step 4: Solve for $\frac{dh}{dt}$

Given:

• The rate at which sand is falling, $\frac{dV}{dt} = 10$ cubic feet per minute.
• The height of the pile, $h = 15$ feet.

Substitute these values into the differentiated equation: $10 = \frac{9\pi}{4} (15)^2 \frac{dh}{dt}$ $10 = \frac{9\pi}{4} \times 225 \times \frac{dh}{dt}$ $10 = \frac{2025\pi}{4} \frac{dh}{dt}$ $\frac{dh}{dt} = \frac{10 \times 4}{2025\pi}$ $\frac{dh}{dt} = \frac{40}{2025\pi}$

Step 5: Simplify the expression

$\frac{dh}{dt} = \frac{40}{2025\pi} \approx 0.0063 \text{ feet per minute}$

Therefore, the rate at which the height of the pile is changing when the pile is 15 feet high is approximately $0.0063$ feet per minute.

Would you like more details or have any questions about this solution?

Relative Questions:

1. How does changing the rate at which sand falls affect the rate of height increase?
2. What happens if the diameter was proportional to a different multiple of the height?
3. How would the problem change if the volume formula of a cone was different?
4. What is the significance of differentiating with respect to time in related rates problems?
5. How can this method be applied to different shapes or volumes?

Tip:

When working with related rates, always identify how variables are related to each other and ensure you differentiate with respect to time carefully, applying the chain rule where necessary.