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### 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.

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### **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.

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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?

A swimming pool is 20 ft wide, 40 ft long, 3 ft deep at the shallow end, and 9 feet deep at its deepest point. A cross-section is shown in the figure. (a) Express the volume of the water in the pool as a function of height h of the water above the deepest point. (b) Determine the domain and range of the function found in part (a).

Calculate the volume of water in a swimming pool with varying depth as a function of height using piecewise functions. This problem involves geometry and algebraic concepts suitable for high school students.

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