Question

Surface Area of Spheres Quick Check
5 of 55 of 5 Items
Question
Marcus is making spherical soaps to sell in his online store. The surface area of a soap is 63.585 in.2
and he wants to package them into a cube box so that it fits snugly. Using 3.14 as the value of pi, what should be the dimensions of the cube box?(1 point)
Responses

2.25 in. × 2.25 in. × 2.25 in.
2.25 inches times 2.25 inches times 2.25 inches

1.2 in. × 1.2 in. × 1.2 in.
1.2 inches times 1.2 inches times 1.2 inches

4 in.× 4 in.× 4 in.
4 inches times 4 inches times 4 inches

4.5 in. × 4.5 in. × 4.5 in.
4.5 inches times 4.5 inches times 4.5 inches

Answer 1

To determine the dimensions of the cube box that Marcus needs for his spherical soap, we first need to find the radius of the soap based on its surface area. The formula for the surface area of a sphere is given by:

\[
A = 4\pi r^2
\]

Where \( A \) is the surface area and \( r \) is the radius of the sphere.

Given that the surface area \( A \) is \( 63.585 \, \text{in}^2 \) and using \( \pi \approx 3.14 \), we can set up the equation:

\[
63.585 = 4 \times 3.14 \times r^2
\]

Calculating the right-hand side:

\[
63.585 = 12.56 \times r^2
\]

Now we divide both sides by \( 12.56 \):

\[
r^2 = \frac{63.585}{12.56} \approx 5.06
\]

Now, we take the square root to find \( r \):

\[
r \approx \sqrt{5.06} \approx 2.25 \, \text{in.}
\]

The diameter \( d \) of the sphere is \( 2r \):

\[
d = 2 \times 2.25 \approx 4.5 \, \text{in.}
\]

Since the cube box must fit the spherical soap snugly, the side length of the cube box must at least be equal to the diameter of the sphere:

Thus, the dimensions of the cube box should be \( 4.5 \, \text{in.} \times 4.5 \, \text{in.} \times 4.5 \, \text{in.} \).

The correct answer is:

**4.5 in. × 4.5 in. × 4.5 in.**