Answered

The circumference of a cross section of a sphere is 12.56 inches. (Remember [tex]C = \pi d[/tex]) Find the volume of the sphere. Use 3.14 for [tex]\pi[/tex] and round to the nearest tenth.

A. [tex]10.7 \, \text{in}^3[/tex]
B. [tex]33.5 \, \text{in}^3[/tex]
C. [tex]85.3 \, \text{in}^3[/tex]



Answer :

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].