A sphere is inscribed inside a cube. The sphere touches each face of the cube at one point. The length of each edge of the cube is 2 inches. We need to find: a. The surface area of the cube. b. The surface area of the sphere (rounded to the nearest hundredth). c. The ratio of the surface area of the cube to the surface area of the sphere (rounded to the nearest hundredth).

1. Problem Description

A sphere is inscribed inside a cube. The sphere touches each face of the cube at one point. The length of each edge of the cube is 2 inches. We need to find:

a. The surface area of the cube.

b. The surface area of the sphere (rounded to the nearest hundredth).

c. The ratio of the surface area of the cube to the surface area of the sphere (rounded to the nearest hundredth).

2. Solution Steps

a. Surface area of the cube:

A cube has 6 faces. Each face is a square. The area of one face is the side length squared. Since the side length is 2 inches, the area of one face is $2^2 = 4$ square inches.

The surface area of the cube is 6 times the area of one face.

$Surface Area_{cube} = 6 \times (side)^2$

$Surface Area_{cube} = 6 \times 2^2 = 6 \times 4 = 24$ square inches.

b. Surface area of the sphere:

Since the sphere fits snugly inside the cube and touches each face at one point, the diameter of the sphere is equal to the side length of the cube, which is 2 inches. Therefore, the radius of the sphere is $r = \frac{2}{2} = 1$ inch.

The formula for the surface area of a sphere is:

$Surface Area_{sphere} = 4 \pi r^2$

Substituting $r = 1$ inch, we have:

$Surface Area_{sphere} = 4 \pi (1)^2 = 4 \pi$

Using the approximation $\pi \approx 3.14159$, we get:

$Surface Area_{sphere} \approx 4 \times 3.14159 = 12.56636$

Rounding to the nearest hundredth, we get 12.57 square inches.

c. Ratio of the surface area of the cube to the surface area of the sphere:

We need to find the ratio $\frac{Surface Area_{cube}}{Surface Area_{sphere}}$.

$Ratio = \frac{24}{4 \pi} = \frac{6}{\pi}$

Using the approximation $\pi \approx 3.14159$, we get:

$Ratio \approx \frac{6}{3.14159} \approx 1.909859$

Rounding to the nearest hundredth, we get 1.

9

1.

3. Final Answer

a. 24 $in^2$

b. 12.57 $in^2$

c. 1.91