To find the volume of the sphere given that the circumference of a cross-section is 12.56 inches, and using 3.14 for pi, we will proceed step-by-step:
1. Find the diameter of the sphere:
- The formula for the circumference [tex]\(C\)[/tex] of a circle in terms of its diameter [tex]\(d\)[/tex] is:
[tex]\[
C = \pi \cdot d
\][/tex]
- We're given that the circumference [tex]\(C = 12.56\)[/tex] inches. Using [tex]\(\pi = 3.14\)[/tex]:
[tex]\[
d = \frac{C}{\pi} = \frac{12.56}{3.14} = 4.0 \text{ inches}
\][/tex]
2. Calculate the radius of the sphere:
- The radius [tex]\(r\)[/tex] is half of the diameter:
[tex]\[
r = \frac{d}{2} = \frac{4.0}{2} = 2.0 \text{ inches}
\][/tex]
3. Calculate the volume of the sphere:
- The formula for the volume [tex]\(V\)[/tex] of a sphere in terms of its radius is:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
- Plugging in the values, [tex]\(r = 2.0\)[/tex] inches and [tex]\(\pi = 3.14\)[/tex]:
[tex]\[
V = \frac{4}{3} \times 3.14 \times (2.0)^3 = \frac{4}{3} \times 3.14 \times 8 = \frac{4}{3} \times 25.12 = 33.49333333333333 \text{ cubic inches}
\][/tex]
4. Round the volume to the nearest tenth:
- The volume 33.49333333333333 cubic inches rounded to the nearest tenth is:
[tex]\[
V \approx 33.5 \text{ cubic inches}
\][/tex]
Thus, the volume of the sphere is [tex]\( 33.5 \)[/tex] cubic inches.
Therefore, the correct option is:
[tex]\[
\boxed{33.5 \text{ in}^3}
\][/tex]
