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.
Answers
GPT-4o mini
To find the total volume of water needed for both the swimming pool and the hot tub, we will calculate the volume of each separately, and then sum them.
### Step 1: Calculate the volume of the swimming pool.
The swimming pool consists of two sections:
1. The first section is 10 yards long with a depth of 4 feet.
2. The second section is 15 yards long with a depth of 8 feet.
#### Section 1: 10 yards (4 feet deep)
- **Length**: 10 yards = \(10 \times 3\) feet = 30 feet
- **Width**: 20 yards = \(20 \times 3\) feet = 60 feet
- **Depth**: 4 feet
Now calculate the volume:
\[
\text{Volume}_{1} = \text{Length} \times \text{Width} \times \text{Depth} = 30 \text{ feet} \times 60 \text{ feet} \times 4 \text{ feet}
\]
\[
\text{Volume}_{1} = 30 \times 60 \times 4 = 7200 \text{ cubic feet}
\]
#### Section 2: 15 yards (8 feet deep)
- **Length**: 15 yards = \(15 \times 3\) feet = 45 feet
- **Width**: 60 feet (the same as above)
- **Depth**: 8 feet
Calculate the volume:
\[
\text{Volume}_{2} = \text{Length} \times \text{Width} \times \text{Depth} = 45 \text{ feet} \times 60 \text{ feet} \times 8 \text{ feet}
\]
\[
\text{Volume}_{2} = 45 \times 60 \times 8 = 21600 \text{ cubic feet}
\]
#### Total volume of the swimming pool
\[
\text{Volume}_{\text{pool}} = \text{Volume}_{1} + \text{Volume}_{2} = 7200 + 21600 = 28800 \text{ cubic feet}
\]
### Step 2: Calculate the volume of the hot tub.
The hot tub is a cylinder with a diameter of 3 yards and a depth of 4 feet. To find its volume, we need its radius.
- **Diameter**: 3 yards = 3 × 3 feet = 9 feet
- **Radius**: \( \frac{9}{2} = 4.5 \) feet
- **Depth**: 4 feet
Now calculate the volume of the hot tub:
\[
\text{Volume}_{\text{hot tub}} = \pi r^2 h
\]
where \( r = 4.5 \text{ feet} \) and \( h = 4 \text{ feet} \).
\[
\text{Volume}_{\text{hot tub}} = \pi (4.5)^2 (4) \approx 3.14159 \times 20.25 \times 4
\]
\[
\text{Volume}_{\text{hot tub}} \approx 3.14159 \times 81 = 254.64784 \text{ cubic feet}
\]
### Step 3: Total volume of water needed
Finally, add the volumes of the pool and the hot tub:
\[
\text{Total Volume} = \text{Volume}_{\text{pool}} + \text{Volume}_{\text{hot tub}} = 28800 + 254.64784 \approx 29054.64784 \text{ cubic feet}
\]
Rounding this to the nearest thousandth, we have:
\[
\text{Total Volume} \approx 29054.648 \text{ cubic feet}
\]
Thus, the final answer is:
\[
\boxed{29054.648}
\] cubic feet.
### Step 1: Calculate the volume of the swimming pool.
The swimming pool consists of two sections:
1. The first section is 10 yards long with a depth of 4 feet.
2. The second section is 15 yards long with a depth of 8 feet.
#### Section 1: 10 yards (4 feet deep)
- **Length**: 10 yards = \(10 \times 3\) feet = 30 feet
- **Width**: 20 yards = \(20 \times 3\) feet = 60 feet
- **Depth**: 4 feet
Now calculate the volume:
\[
\text{Volume}_{1} = \text{Length} \times \text{Width} \times \text{Depth} = 30 \text{ feet} \times 60 \text{ feet} \times 4 \text{ feet}
\]
\[
\text{Volume}_{1} = 30 \times 60 \times 4 = 7200 \text{ cubic feet}
\]
#### Section 2: 15 yards (8 feet deep)
- **Length**: 15 yards = \(15 \times 3\) feet = 45 feet
- **Width**: 60 feet (the same as above)
- **Depth**: 8 feet
Calculate the volume:
\[
\text{Volume}_{2} = \text{Length} \times \text{Width} \times \text{Depth} = 45 \text{ feet} \times 60 \text{ feet} \times 8 \text{ feet}
\]
\[
\text{Volume}_{2} = 45 \times 60 \times 8 = 21600 \text{ cubic feet}
\]
#### Total volume of the swimming pool
\[
\text{Volume}_{\text{pool}} = \text{Volume}_{1} + \text{Volume}_{2} = 7200 + 21600 = 28800 \text{ cubic feet}
\]
### Step 2: Calculate the volume of the hot tub.
The hot tub is a cylinder with a diameter of 3 yards and a depth of 4 feet. To find its volume, we need its radius.
- **Diameter**: 3 yards = 3 × 3 feet = 9 feet
- **Radius**: \( \frac{9}{2} = 4.5 \) feet
- **Depth**: 4 feet
Now calculate the volume of the hot tub:
\[
\text{Volume}_{\text{hot tub}} = \pi r^2 h
\]
where \( r = 4.5 \text{ feet} \) and \( h = 4 \text{ feet} \).
\[
\text{Volume}_{\text{hot tub}} = \pi (4.5)^2 (4) \approx 3.14159 \times 20.25 \times 4
\]
\[
\text{Volume}_{\text{hot tub}} \approx 3.14159 \times 81 = 254.64784 \text{ cubic feet}
\]
### Step 3: Total volume of water needed
Finally, add the volumes of the pool and the hot tub:
\[
\text{Total Volume} = \text{Volume}_{\text{pool}} + \text{Volume}_{\text{hot tub}} = 28800 + 254.64784 \approx 29054.64784 \text{ cubic feet}
\]
Rounding this to the nearest thousandth, we have:
\[
\text{Total Volume} \approx 29054.648 \text{ cubic feet}
\]
Thus, the final answer is:
\[
\boxed{29054.648}
\] cubic feet.