Question

A sphere has a center.

Which of the following is true about the diameter of a sphere?(1 point)

Responses

The diameter is the shortest length on a sphere.
The diameter is the shortest length on a sphere.

The diameter is not a chord.
The diameter is not a chord.

The diameter of a sphere passes through the center of the circle.
The diameter of a sphere passes through the center of the circle.

The diameter is 2.5 times the length of the radius.

The diameter of a sphere passes through the center of the circle.

Use the formula for the surface area of a sphere to find the surface area of a basketball that has a diameter of 9.2 inches. Use 3.14 pi and round your answer to the nearest hundredth, if necessary.(1 point)

Responses

84.64 in.2
84.64 inches squared

1,063.08 in.2
1,063.08 inches squared

66.44 in.2
66.44 inches squared

265.77 in.2

To find the surface area of a sphere, you can use the formula:

\[ \text{Surface Area} = 4\pi r^2 \]

Since you're given the diameter is 9.2 inches, you first need to find the radius \( r \) of the sphere. The radius is half of the diameter:

\[ r = \frac{\text{Diameter}}{2} = \frac{9.2 \text{ inches}}{2} = 4.6 \text{ inches} \]

Now you can plug the radius into the surface area formula using \( \pi \approx 3.14 \):

\[ \text{Surface Area} = 4 \times 3.14 \times (4.6 \text{ inches})^2 \]

\[ \text{Surface Area} = 4 \times 3.14 \times 21.16 \text{ inches}^2 \]

\[ \text{Surface Area} = 12.56 \times 21.16 \text{ inches}^2 \]

\[ \text{Surface Area} \approx 265.77 \text{ inches}^2 \]

So, the surface area of a basketball with a diameter of 9.2 inches is approximately 265.77 square inches, rounded to the nearest hundredth.

Find the surface area of a sphere with a diameter of 30 inches. Leave your answer in terms of pi.(1 point)

Responses

900π in.2
900 pi inches squared

3,600π in.2
3,600 pi inches squared

2,826π in.2
2,826 pi inches squared

225π in.2

To find the surface area of a sphere, you can use the formula:

\[ \text{Surface Area} = 4\pi r^2 \]

Given that the diameter of the sphere is 30 inches, you first need to find the radius \( r \) of the sphere. The radius is half of the diameter:

\[ r = \frac{\text{Diameter}}{2} = \frac{30 \text{ inches}}{2} = 15 \text{ inches} \]

Now you can plug the radius into the surface area formula:

\[ \text{Surface Area} = 4\pi (15 \text{ inches})^2 \]

\[ \text{Surface Area} = 4\pi (225 \text{ inches}^2) \]

\[ \text{Surface Area} = 900\pi \text{ inches}^2 \]

So, the surface area of a sphere with a diameter of 30 inches is \(900\pi\) square inches when you leave the answer in terms of \( \pi \).

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

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

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

2.25 in. × 2.25 in. × 2.25 in.

To find the dimensions of the cube box needed to fit a spherical soap snugly, we need to determine the diameter of the soap. The surface area of a sphere is given by the formula:

\[ \text{Surface Area} = 4\pi r^2 \]

The surface area of the soap is given as 63.585 square inches, and we'll use \( \pi \approx 3.14 \). We want to solve for the radius \( r \) first:

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

Now we solve for \( r^2 \) first:

\[ r^2 = \frac{63.585}{4 \times 3.14} \]

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

\[ r^2 \approx 5.062 \]

Now we find the radius \( r \):

\[ r \approx \sqrt{5.062} \]

\[ r \approx 2.25 \text{ inches} \]

Now that we have the radius, we can find the diameter, which is twice the radius:

\[ \text{Diameter} = 2 \times r \]

\[ \text{Diameter} = 2 \times 2.25 \text{ inches} \]

\[ \text{Diameter} = 4.5 \text{ inches} \]

Since the diameter of the sphere is 4.5 inches, the cube box must have internal dimensions that are at least 4.5 inches on each side to accommodate the sphere.

Therefore, the dimensions for the cube box should be:

\[ 4.5 \text{ in.} \times 4.5 \text{ in.} \times 4.5 \text{ in.} \]

4.5 inches times 4.5 inches times 4.5 inches.