Let's solve the problem step by step.
### Step 1: Determine the diameter of the sphere
Given: The circumference [tex]\( C = 12.56 \)[/tex] inches
The formula for the circumference of a circle is [tex]\( C = \pi d \)[/tex], where [tex]\( \pi \approx 3.14 \)[/tex] and [tex]\( d \)[/tex] is the diameter.
[tex]\[ d = \frac{C}{\pi} \][/tex]
Substituting the given values:
[tex]\[ d = \frac{12.56}{3.14} = 4.0 \text{ inches} \][/tex]
### Step 2: Determine the radius of the sphere
The radius [tex]\( r \)[/tex] is half of the diameter.
[tex]\[ r = \frac{d}{2} \][/tex]
Substituting the value of the diameter:
[tex]\[ r = \frac{4.0}{2} = 2.0 \text{ inches} \][/tex]
### Step 3: Determine the volume of the sphere
The formula for the volume of a sphere is:
[tex]\[ V = \frac{4}{3} \pi r^3 \][/tex]
Substituting the values of [tex]\( \pi \)[/tex] and [tex]\( r \)[/tex]:
[tex]\[ V = \frac{4}{3} \times 3.14 \times (2.0)^3 \][/tex]
[tex]\[ V = \frac{4}{3} \times 3.14 \times 8 \][/tex]
[tex]\[ V = \frac{4}{3} \times 25.12 \][/tex]
[tex]\[ V = 33.49333333333333 \text{ cubic inches} \][/tex]
### Step 4: Round the volume to the nearest tenth
To round to the nearest tenth, we look at the first digit after the decimal point which is 4:
[tex]\[ V \approx 33.5 \text{ cubic inches} \][/tex]
### Conclusion
The volume of the sphere, rounded to the nearest tenth, is:
[tex]\[
\boxed{33.5 \text{ cubic inches}}
\][/tex]
Therefore, the correct answer is:
[tex]\[ b. 33.5 \text{ in}^3 \][/tex]
