Find the volume of a hemisphere that has a radius of 8 centimeters to the nearest tenth.
Understand the Problem
The question asks us to find the volume of a hemisphere given its radius. We need to use the formula for the volume of a hemisphere and round the answer to the nearest tenth.
Answer
$1072.3 \text{ cm}^3$
Answer for screen readers
$1072.3 \text{ cm}^3$
Steps to Solve
- Recall the formula for the volume of a sphere
The formula for the volume $V$ of a sphere with radius $r$ is:
$$V = \frac{4}{3} \pi r^3$$
- Determine the formula for the volume of a hemisphere
A hemisphere is half of a sphere. Therefore, to find the volume of a hemisphere, we take half of the volume of a sphere:
$$V_{hemisphere} = \frac{1}{2} \cdot \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3$$
- Substitute the given radius into the formula
We are given that the radiuis $r = 8$ cm. Substitute this value into the hemisphere volume formula:
$$V_{hemisphere} = \frac{2}{3} \pi (8)^3$$
- Calculate the volume
$$V_{hemisphere} = \frac{2}{3} \pi (512) = \frac{1024}{3} \pi$$ $$V_{hemisphere} \approx \frac{1024}{3} \cdot 3.14159 \approx 1072.3302$$
- Round to the nearest tenth
Rounding $1072.3302$ to the nearest tenth gives $1072.3$.
$1072.3 \text{ cm}^3$
More Information
The volume of the hemisphere with radius 8 cm, rounded to the nearest tenth, is approximately $1072.3 \text{ cm}^3$.
Tips
A common mistake is to use the formula for the volume of a sphere instead of the hemisphere, or forgetting to divide the volume of a sphere by 2 to get the volume of the hemisphere. Also, errors in calculation and rounding can occur.
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