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.

Would you like more details on any specific step?

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.