Problem Breakdown:

We are given a swimming pool with the following dimensions:

• Width: 20 feet
• Length: 40 feet
• Depth: 3 feet at the shallow end and 9 feet at the deep end.
• The cross-section is shown as a trapezoid from the shallow to deep end.

The goal is to:

1. Express the volume of the pool as a piecewise function of the height $h$ of the water above the deepest point (9 feet).
2. Determine the domain and range of this piecewise function.

Step 1: Understanding the Geometry

We can divide the pool into two sections:

• Region 1: From $x = 0$ to $x = 12$, where the pool's depth linearly increases from 3 ft to 9 ft (sloped region).
• Region 2: From $x = 12$ to $x = 40$, where the pool is uniformly 9 ft deep.

The volume as a function of $h$ depends on how high the water level is relative to the deepest point.

Step 2: Volume in Region 1 (Sloped section)

In this region, the depth increases linearly from 3 ft to 9 ft. Let’s derive the depth at any point along the slope.

• The equation of the line from the shallow end to the deep end is given by: $d(x) = \frac{6}{12}x + 3 = 0.5x + 3$ where $d(x)$ is the depth at any point $x$ from 0 to 12 ft.

• Now, for a given height $h$, the volume of water in this region will be the area of a trapezoid cross-section multiplied by the width (20 ft) and length (12 ft). This is for $h$ values less than or equal to 6 ft, because this region ends at a depth of 9 ft.

Step 3: Volume in Region 2 (Constant depth section)

For $h > 6$, the water height in this region covers the constant depth part, where the depth is 9 ft throughout. For this section, the volume calculation is straightforward since the depth remains constant at 9 ft.

Let me work out the exact piecewise formula for the volume and determine the domain and range.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, I can outline the process for you manually if you'd like.

Would you like a detailed explanation of the next steps for setting up the piecewise volume function and calculating the domain and range?