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)
Answers
GPT-4o mini
First, we will calculate the volume of the swimming pool and then the volume of the hot tub, and finally sum both volumes to get the total volume of water needed.
### Volume of the Swimming Pool:
The swimming pool has two different depths for different sections:
1. **Shallow end (first 10 yards)**:
- Length = 10 yards
- Width = 20 yards
- Depth = 4 feet
The volume of the shallow end:
\[
V_1 = \text{Length} \times \text{Width} \times \text{Depth}
\]
\[
V_1 = 10 \, \text{yards} \times 20 \, \text{yards} \times 4 \, \text{feet}
\]
First, convert yards to feet (1 yard = 3 feet):
\[
V_1 = (10 \times 3) \, \text{feet} \times (20 \times 3) \, \text{feet} \times 4 \, \text{feet}
\]
\[
= 30 \, \text{feet} \times 60 \, \text{feet} \times 4 \, \text{feet} = 7200 \, \text{cubic feet}
\]
2. **Deep end (remaining 15 yards)**:
- Length = 15 yards
- Width = 20 yards
- Depth = 8 feet
The volume of the deep end:
\[
V_2 = \text{Length} \times \text{Width} \times \text{Depth}
\]
\[
V_2 = 15 \, \text{yards} \times 20 \, \text{yards} \times 8 \, \text{feet}
\]
Again, convert yards to feet:
\[
V_2 = (15 \times 3) \, \text{feet} \times (20 \times 3) \, \text{feet} \times 8 \, \text{feet}
\]
\[
= 45 \, \text{feet} \times 60 \, \text{feet} \times 8 \, \text{feet} = 21600 \, \text{cubic feet}
\]
Now, add both volumes 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 cylindrical with a diameter of 3 yards and a depth of 4 feet.
1. **Calculate the radius**:
\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{3}{2} \, \text{yards} = 1.5 \, \text{yards}
\]
2. **Convert radius to feet**:
\[
\text{Radius in feet} = 1.5 \times 3 = 4.5 \, \text{feet}
\]
3. **Calculate the volume**:
\[
V_{\text{hot tub}} = \pi r^2 h
\]
where \(h\) is the depth (in feet):
\[
V_{\text{hot tub}} = \pi \times (4.5 \, \text{feet})^2 \times 4 \, \text{feet}
\]
\[
= \pi \times 20.25 \times 4 = 81\pi \, \text{cubic feet}
\]
Using \(\pi \approx 3.14159\):
\[
V_{\text{hot tub}} \approx 81 \times 3.14159 \approx 254.469 \, \text{cubic feet}
\]
### Total Volume of Water Needed:
Adding the volume of the pool and hot tub:
\[
V_{\text{total}} = V_{\text{pool}} + V_{\text{hot tub}} = 28800 \, \text{cubic feet} + 254.469 \, \text{cubic feet} \approx 29054.469 \, \text{cubic feet}
\]
**Final Answer**: Rounding to the nearest thousandth:
\[
\boxed{29054.469}
\] cubic feet of water is needed to fill the pool and hot tub.
### Volume of the Swimming Pool:
The swimming pool has two different depths for different sections:
1. **Shallow end (first 10 yards)**:
- Length = 10 yards
- Width = 20 yards
- Depth = 4 feet
The volume of the shallow end:
\[
V_1 = \text{Length} \times \text{Width} \times \text{Depth}
\]
\[
V_1 = 10 \, \text{yards} \times 20 \, \text{yards} \times 4 \, \text{feet}
\]
First, convert yards to feet (1 yard = 3 feet):
\[
V_1 = (10 \times 3) \, \text{feet} \times (20 \times 3) \, \text{feet} \times 4 \, \text{feet}
\]
\[
= 30 \, \text{feet} \times 60 \, \text{feet} \times 4 \, \text{feet} = 7200 \, \text{cubic feet}
\]
2. **Deep end (remaining 15 yards)**:
- Length = 15 yards
- Width = 20 yards
- Depth = 8 feet
The volume of the deep end:
\[
V_2 = \text{Length} \times \text{Width} \times \text{Depth}
\]
\[
V_2 = 15 \, \text{yards} \times 20 \, \text{yards} \times 8 \, \text{feet}
\]
Again, convert yards to feet:
\[
V_2 = (15 \times 3) \, \text{feet} \times (20 \times 3) \, \text{feet} \times 8 \, \text{feet}
\]
\[
= 45 \, \text{feet} \times 60 \, \text{feet} \times 8 \, \text{feet} = 21600 \, \text{cubic feet}
\]
Now, add both volumes 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 cylindrical with a diameter of 3 yards and a depth of 4 feet.
1. **Calculate the radius**:
\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{3}{2} \, \text{yards} = 1.5 \, \text{yards}
\]
2. **Convert radius to feet**:
\[
\text{Radius in feet} = 1.5 \times 3 = 4.5 \, \text{feet}
\]
3. **Calculate the volume**:
\[
V_{\text{hot tub}} = \pi r^2 h
\]
where \(h\) is the depth (in feet):
\[
V_{\text{hot tub}} = \pi \times (4.5 \, \text{feet})^2 \times 4 \, \text{feet}
\]
\[
= \pi \times 20.25 \times 4 = 81\pi \, \text{cubic feet}
\]
Using \(\pi \approx 3.14159\):
\[
V_{\text{hot tub}} \approx 81 \times 3.14159 \approx 254.469 \, \text{cubic feet}
\]
### Total Volume of Water Needed:
Adding the volume of the pool and hot tub:
\[
V_{\text{total}} = V_{\text{pool}} + V_{\text{hot tub}} = 28800 \, \text{cubic feet} + 254.469 \, \text{cubic feet} \approx 29054.469 \, \text{cubic feet}
\]
**Final Answer**: Rounding to the nearest thousandth:
\[
\boxed{29054.469}
\] cubic feet of water is needed to fill the pool and hot tub.