To solve for the volume of a sphere with a given diameter, we need to follow a systematic approach using the formula for the volume of a sphere and then round our result to the nearest hundredth. Here's a step-by-step solution:
1. Understand the formula for the volume of a sphere:
The volume [tex]\( V \)[/tex] of a sphere is calculated using the formula:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
where [tex]\( r \)[/tex] is the radius of the sphere.
2. Calculate the radius:
The diameter of the sphere is given as 25 cm. The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[
r = \frac{25}{2} = 12.5 \text{ cm}
\][/tex]
3. Approximate π:
The problem specifies to approximate [tex]\(\pi\)[/tex] as 3.14.
4. Plug the radius and π into the formula:
Substitute [tex]\( r = 12.5 \)[/tex] cm and [tex]\(\pi = 3.14\)[/tex] into the volume formula:
[tex]\[
V = \frac{4}{3} \cdot 3.14 \cdot (12.5)^3
\][/tex]
5. Calculate [tex]\( (12.5)^3 \)[/tex]:
[tex]\[
(12.5)^3 = 12.5 \times 12.5 \times 12.5 = 1953.125
\][/tex]
6. Calculate the volume:
Now, multiply the values:
[tex]\[
V = \frac{4}{3} \cdot 3.14 \cdot 1953.125
\][/tex]
[tex]\[
V = \frac{4}{3} \cdot 6122.8125
\][/tex]
[tex]\[
V = 4 \cdot 1530.703125
\][/tex]
[tex]\[
V = 6122.8125
\][/tex]
7. Round the result to the nearest hundredth:
[tex]\[
V \approx 8177.08 \text{ cm}^3
\][/tex]
Therefore, the volume of the sphere, rounded to the nearest hundredth, is approximately 8177.08 cm[tex]\(^3\)[/tex].
Considering the choices provided:
A. [tex]$1,471.88 \text{ cm}^3$[/tex]
B. [tex]$1,962.50 \text{ cm}^3$[/tex]
C. [tex]$2,616.67 \text{ cm}^3$[/tex]
D. [tex]$8,177.08 \text{ cm}^3$[/tex]
The correct answer is:
D. [tex]$8,177.08 \text{ cm}^3$[/tex]
