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Let's solve each part of this problem step-by-step.

### Problem Overview
The swimming pool has dimensions:
- Width = 12 ft
- Length = 15 ft
- Height of water = \( h \) ft (variable)

Let:
- \( V \) be the volume of water in the pool, which varies with \( h \).
- \( t \) be time in minutes.

### Part (a): Relating \( V \) to \( h \) and \(\frac{dV}{dt}\) to \(\frac{dh}{dt}\)

#### Step 1: Equation for \( V \) in terms of \( h \)
Since the pool is rectangular, the volume \( V \) is given by:
\[
V = \text{width} \times \text{length} \times \text{height}
\]
Substituting the dimensions:
\[
V = 12 \times 15 \times h = 180h
\]

#### Step 2: Differentiate \( V = 180h \) with respect to \( t \)
To relate the rates \( \frac{dV}{dt} \) and \( \frac{dh}{dt} \), differentiate both sides with respect to \( t \):
\[
\frac{dV}{dt} = 180 \frac{dh}{dt}
\]
This equation provides the relationship between \( \frac{dV}{dt} \) and \( \frac{dh}{dt} \).

### Part (b): Rate of Volume Increase
We are given that the water level is rising at \( \frac{dh}{dt} = \frac{1}{4} \) ft/min. Using the equation from Part (a):
\[
\frac{dV}{dt} = 180 \times \frac{1}{4} = 45 \text{ ft}^3/\text{min}
\]
Thus, the volume of water is increasing at a rate of 45 ft³/min.

### Part (c): Rate of Water Level Rise
Now, we are given that the pool is filled at a rate of \( \frac{dV}{dt} = 6 \) ft³/min. We need to find \( \frac{dh}{dt} \).

Using the relationship from Part (a):
\[
\frac{dV}{dt} = 180 \frac{dh}{dt}
\]
Substitute \( \frac{dV}{dt} = 6 \):
\[
6 = 180 \frac{dh}{dt}
\]
\[
\frac{dh}{dt} = \frac{6}{180} = \frac{1}{30} \text{ ft/min}
\]
So, the water level is rising at a rate of \( \frac{1}{30} \) ft/min when the pool is filled at 6 ft³/min.

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### Related Questions
1. How would the relationship change if the dimensions of the pool were different?
2. What if the pool were not rectangular but had a different shape? How would that affect \( V \) in terms of \( h \)?
3. How would we solve if the water were being drained instead of filled?
4. What assumptions are we making about the uniformity of depth in the pool?
5. How would we handle a scenario with a variable filling rate?

### Tip
When dealing with rates, always identify the relationship between variables first, then differentiate to link their rates.

A rectangular swimming pool 12 ft wide by 15 ft long and of uniform depth is being filled with water.
(a) If t is elapsed time, h is the height of the water, and V is the volume of the water, find equations relating V to h and dV/dt to dh/dt.
(b) At what rate is the volume of the water increasing if the water level is rising at 1/4 ft/min?
(c) At what rate is the water level rising if the pool is filled at a rate of 6 ft^3/min?

Solve a related rates problem involving a rectangular pool with dimensions 12 ft by 15 ft. Learn how to relate volume to height and calculate the rate of volume increase and water level rise. Suitable for students in Grades 10-12.

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