Question

A rectangular swimming pool is 25 yards long and 20 yards wide. The first 10 yards has a depth of 4 feet and the remaining 15 yards has a depth of 8 feet. A circular hot tub that is 4 feet deep and has a diameter of 3 yards is attached to the deep end of the pool. What volume of water is needed for the owner to fill the pool and hot tub? Provide your answer in cubic feet. Enter your answer to the nearest thousandth.(1 point)
The owner will need
cubic feet of water to fill the pool and hot tub.

Answer 1

To find the volume of water needed to fill the pool and the hot tub, we will compute the volume for both separately and then sum them up.

### Volume of the Swimming Pool

The pool is rectangular with two different depths. We can calculate the volume of the two sections and then add them together.

1. **First section**:
- Dimensions: 10 yards long, 20 yards wide, 4 feet deep.
- Convert the dimensions to feet:
- Length = 10 yards = 10 * 3 = 30 feet
- Width = 20 yards = 20 * 3 = 60 feet
- Volume = length × width × depth
\[
V_1 = 30 \text{ feet} \times 60 \text{ feet} \times 4 \text{ feet} = 7200 \text{ cubic feet}
\]

2. **Second section**:
- Dimensions: 15 yards long, 20 yards wide, 8 feet deep.
- Convert the dimensions to feet:
- Length = 15 yards = 15 * 3 = 45 feet
- Volume = length × width × depth
\[
V_2 = 45 \text{ feet} \times 60 \text{ feet} \times 8 \text{ feet} = 21600 \text{ cubic feet}
\]

3. **Total volume of the pool**:
\[
V_{\text{pool}} = V_1 + V_2 = 7200 \text{ cubic feet} + 21600 \text{ cubic feet} = 28800 \text{ cubic feet}
\]

### Volume of the Hot Tub

The hot tub is a cylinder with a diameter of 3 yards and a depth of 4 feet.

1. **Radius of the hot tub**:
- Diameter = 3 yards, so
\[
\text{Radius} = \frac{3 \text{ yards}}{2} = 1.5 \text{ yards} = 1.5 \times 3 = 4.5 \text{ feet}
\]

2. **Volume of the hot tub**:
- Volume of a cylinder = π × radius² × height
\[
V_{\text{hot tub}} = \pi \times (4.5 \text{ feet})^2 \times 4 \text{ feet}
\]
- Calculate the radius squared:
\[
(4.5)^2 = 20.25
\]
- So,
\[
V_{\text{hot tub}} \approx \pi \times 20.25 \text{ feet}^2 \times 4 \text{ feet} \approx \pi \times 81 \text{ cubic feet} \approx 254.469 \text{ cubic feet}
\]

### Total Volume

Now, we sum the volumes of the pool and the hot tub:

\[
V_{\text{total}} = V_{\text{pool}} + V_{\text{hot tub}} = 28800 \text{ cubic feet} + 254.469 \text{ cubic feet} \approx 29054.469 \text{ cubic feet}
\]

Rounded to the nearest thousandth:

\[
\text{The owner will need } \approx 29054.469 \text{ cubic feet of water to fill the pool and hot tub.}
\]

Final answer:
\[
\boxed{29054.469}
\]