4. A spherical container is designed to hold as much volume as the rectangular prism above. Its radius is 3.2 in. Find the surface area of the sphere rounded to the nearest square inch.

Question

*
43 square inches
129 square inches
137 square inches
412 square inches

Answer 1

To solve the problem, we'll start by finding the volume of the rectangular prism and equate it to the volume of the sphere, since the spherical container is designed to hold the same volume as the rectangular prism.

However, the problem statement directly involves calculating the surface area of the sphere with a given radius. So let's proceed with that calculation:

1. **Volume of the Sphere:**

The volume \( V \) of a sphere with radius \( r \) is given by the formula:
\[
V = \frac{4}{3} \pi r^3
\]

Given the radius \( r = 3.2 \text{ in} \), we can calculate the volume:
\[
V = \frac{4}{3} \pi (3.2)^3
\]

Before we go into approximations, let’s get that volume using basic computations:
\[
3.2^3 = 32.768
\]
\[
V = \frac{4}{3} \pi \times 32.768
\]
\[
V \approx \frac{4}{3} \times 3.14159 \times 32.768
\]
\[
V \approx \frac{4}{3} \times 102.987
\]
\[
V \approx 137.316 \, \text{in}^3
\]

2. **Surface Area of the Sphere:**

The surface area \( A \) of a sphere with radius \( r \) is given by the formula:
\[
A = 4 \pi r^2
\]

Substituting \( r = 3.2 \text{ in} \):
\[
A = 4 \pi (3.2)^2
\]

Calculate \( (3.2)^2 \):
\[
(3.2)^2 = 10.24
\]

Now, calculate the surface area:
\[
A \approx 4 \times 3.14159 \times 10.24
\]
\[
A \approx 4 \times 32.169
\]
\[
A \approx 128.676 \, \text{in}^2
\]

Rounded to the nearest square inch, the surface area is:
\[
A \approx 129 \, \text{in}^2
\]

Therefore, the surface area of the sphere rounded to the nearest square inch is \( 129 \) square inches. Thus, the correct answer is:

**129 square inches**