To find the volume of the sphere when given the circumference of a cross-section, we can follow these steps carefully:
1. Given Circumference:
The circumference [tex]\( C \)[/tex] of the sphere's cross-section is provided as [tex]\( 12.56 \)[/tex] inches.
2. Find the Diameter:
The formula for the circumference of a circle is given by:
[tex]\[
C = \pi d
\][/tex]
Where [tex]\( \pi \)[/tex] (pi) is [tex]\( 3.14 \)[/tex] and [tex]\( d \)[/tex] is the diameter. We can solve for the diameter [tex]\( d \)[/tex] by dividing the circumference by [tex]\( \pi \)[/tex]:
[tex]\[
d = \frac{C}{\pi} = \frac{12.56}{3.14} = 4.0 \, \text{in}
\][/tex]
3. Find the Radius:
The radius [tex]\( r \)[/tex] of the sphere is half of the diameter. Therefore:
[tex]\[
r = \frac{d}{2} = \frac{4.0}{2} = 2.0 \, \text{in}
\][/tex]
4. Calculate the Volume:
The formula for the volume [tex]\( V \)[/tex] of a sphere is:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
Plugging in the radius [tex]\( r = 2.0 \)[/tex] inches and [tex]\( \pi = 3.14 \)[/tex]:
[tex]\[
V = \frac{4}{3} \times 3.14 \times (2.0)^3
\][/tex]
First, calculate [tex]\( (2.0)^3 \)[/tex]:
[tex]\[
(2.0)^3 = 8
\][/tex]
Then plug this into the volume formula:
[tex]\[
V = \frac{4}{3} \times 3.14 \times 8 = \frac{4}{3} \times 25.12 = \frac{100.48}{3} \approx 33.49333333333333
\][/tex]
5. Round the Volume:
To find the rounded volume to the nearest tenth:
[tex]\[
V \approx 33.5 \, \text{in}^3
\][/tex]
Therefore, the volume of the sphere is closest to:
[tex]\[
\boxed{33.5 \, \text{in}^3}
\][/tex]
So, the correct answer is [tex]\( b. \ 33.5 \, \text{in}^3 \)[/tex].
