Marcus is making spherical soaps to sell in his online store. The surface area of a soap is 63.585 in.2
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
1.2 in. × 1.2 in. × 1.2 in.
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.
2.25
in.
×
2.25
in.
×
2.25
in
.
2.25 inches times 2.25 inches times 2.25 inches
4 in.× 4 in.× 4 in.
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
in.
×
4.5
in.
×
4.5
in.
4.5 inches times 4.5 inches times 4.5 inches
why not 4.5 x 4.5 x 4.5?
I apologize for the confusion. You are correct. The dimensions of the cube box should be 4.5 inches by 4.5 inches by 4.5 inches.
But why?
To calculate the dimensions of the cube box, we need to find the side length of the cube.
Since a sphere has a surface area of 63.585 in², we can use the formula for the surface area of a sphere to find the radius, and then convert it to a diameter.
Surface Area of a Sphere = 4πr²
63.585 = 4πr²
Divide both sides by 4π:
r² = 63.585 / (4π) ≈ 5.1
Take the square root of both sides:
r ≈ √5.1 ≈ 2.26
Now we have the radius. To get the diameter, multiply the radius by 2:
d ≈ 2 × 2.26 ≈ 4.52
Since the dimensions of a cube are the same for each side, the cube box should have sides that are approximately 4.52 inches long. Rounded to the nearest tenth, this gives us dimensions of 4.5 inches by 4.5 inches by 4.5 inches.
To find the dimensions of the cube box that would snugly fit a spherical soap with a surface area of 63.585 in^2, we can use the formula for the surface area of a sphere:
Surface Area of Sphere = 4πr^2
Given that the surface area is 63.585 in^2, we can rearrange the formula to solve for the radius:
63.585 = 4πr^2
r^2 = 63.585 / (4π)
r^2 ≈ 5.065
Taking the square root of both sides, we find that the radius is approximately 2.25 inches.
The dimensions of a cube are all the same, so the length, width, and height will be equal. Therefore, we can conclude that the dimensions of the cube box should be 2.25 inches × 2.25 inches × 2.25 inches.
So, the correct answer is option 2: 2.25 in. × 2.25 in. × 2.25 in.