Answer :
Sure, I'll provide a detailed, step-by-step solution to find the volume of a hemisphere with a diameter of 8 feet, rounded to the nearest tenth of a cubic foot.
### Step-by-Step Solution:
1. Given:
- Diameter of the hemisphere [tex]\( d = 8 \)[/tex] ft.
2. Calculate the radius:
- The radius, [tex]\( r \)[/tex], can be found by dividing the diameter by 2.
[tex]\[ r = \frac{d}{2} = \frac{8}{2} = 4 \text{ ft} \][/tex]
3. Formula for the volume of a hemisphere:
- The formula to calculate the volume of a hemisphere is:
[tex]\[ \text{Volume} = \frac{2}{3} \pi r^3 \][/tex]
4. Substitute the radius into the formula:
- Calculate the volume using [tex]\( r = 4 \text{ ft} \)[/tex]:
[tex]\[ \text{Volume} = \frac{2}{3} \pi (4)^3 \][/tex]
- First, calculate [tex]\( 4^3 \)[/tex]:
[tex]\[ 4^3 = 4 \times 4 \times 4 = 64 \][/tex]
- Now, multiply by [tex]\( \pi \)[/tex] and then by [tex]\( \frac{2}{3} \)[/tex]:
[tex]\[ \text{Volume} = \frac{2}{3} \pi \times 64 \][/tex]
- Simplify the constants:
[tex]\[ \text{Volume} = \frac{2}{3} \times 64 \times \pi \][/tex]
5. Calculate the numerical value:
- Using [tex]\( \pi \approx 3.14159 \)[/tex]:
[tex]\[ \text{Volume} = \frac{2}{3} \times 64 \times 3.14159 \][/tex]
- Multiply [tex]\( 64 \times 3.14159 \)[/tex]:
[tex]\[ 64 \times 3.14159 \approx 201.06176 \][/tex]
- Now, multiply by [tex]\( \frac{2}{3} \)[/tex]:
[tex]\[ \text{Volume} = \frac{2}{3} \times 201.06176 \approx 134.04117 \][/tex]
6. Round to the nearest tenth:
- The volume, [tex]\( 134.04117 \)[/tex], rounded to the nearest tenth is:
[tex]\[ 134.0 \][/tex]
### Final Answer:
The volume of the hemisphere is approximately [tex]\( 134.0 \)[/tex] cubic feet.
### Step-by-Step Solution:
1. Given:
- Diameter of the hemisphere [tex]\( d = 8 \)[/tex] ft.
2. Calculate the radius:
- The radius, [tex]\( r \)[/tex], can be found by dividing the diameter by 2.
[tex]\[ r = \frac{d}{2} = \frac{8}{2} = 4 \text{ ft} \][/tex]
3. Formula for the volume of a hemisphere:
- The formula to calculate the volume of a hemisphere is:
[tex]\[ \text{Volume} = \frac{2}{3} \pi r^3 \][/tex]
4. Substitute the radius into the formula:
- Calculate the volume using [tex]\( r = 4 \text{ ft} \)[/tex]:
[tex]\[ \text{Volume} = \frac{2}{3} \pi (4)^3 \][/tex]
- First, calculate [tex]\( 4^3 \)[/tex]:
[tex]\[ 4^3 = 4 \times 4 \times 4 = 64 \][/tex]
- Now, multiply by [tex]\( \pi \)[/tex] and then by [tex]\( \frac{2}{3} \)[/tex]:
[tex]\[ \text{Volume} = \frac{2}{3} \pi \times 64 \][/tex]
- Simplify the constants:
[tex]\[ \text{Volume} = \frac{2}{3} \times 64 \times \pi \][/tex]
5. Calculate the numerical value:
- Using [tex]\( \pi \approx 3.14159 \)[/tex]:
[tex]\[ \text{Volume} = \frac{2}{3} \times 64 \times 3.14159 \][/tex]
- Multiply [tex]\( 64 \times 3.14159 \)[/tex]:
[tex]\[ 64 \times 3.14159 \approx 201.06176 \][/tex]
- Now, multiply by [tex]\( \frac{2}{3} \)[/tex]:
[tex]\[ \text{Volume} = \frac{2}{3} \times 201.06176 \approx 134.04117 \][/tex]
6. Round to the nearest tenth:
- The volume, [tex]\( 134.04117 \)[/tex], rounded to the nearest tenth is:
[tex]\[ 134.0 \][/tex]
### Final Answer:
The volume of the hemisphere is approximately [tex]\( 134.0 \)[/tex] cubic feet.
