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 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 inches times 2.25 inches times 2.25 inches 4.5 in. × 4.5 in. × 4.5 in. 4.5 inches times 4.5 inches times 4.5 inches 4 in.× 4 in.× 4 in. 4 inches times 4 inches times 4 inches
To find the dimensions of the cube box that will fit the spherical soap snugly, we need to calculate the volume of the soap and then find the cube with a matching volume.
The surface area of a sphere is given by the formula: 4πr², where r is the radius of the sphere.
Given that the surface area of the soap is 63.585 in.², we can set up the equation:
4πr² = 63.585
Divide both sides of the equation by 4π:
r² = 63.585 / (4π)
r² ≈ 5.083
Taking the square root of both sides, we find:
r ≈ √5.083
r ≈ 2.25
The radius of the soap is approximately 2.25 inches.
The volume of a sphere is given by the formula: (4/3)πr³.
Plugging in the value of the radius:
Volume = (4/3)π(2.25)³
= (4/3)π(11.390625)
≈ 47.7125 in.³
To find the side length of the cube, we can calculate the cube root of the volume:
Side length = ∛(47.7125)
≈ 3.6902
Therefore, the dimensions of the cube box should be approximately 3.6902 inches by 3.6902 inches by 3.6902 inches. However, since this is not one of the given options, the closest option is 4.5 inches by 4.5 inches by 4.5 inches.