Marcus is making spherical soaps to sell in his online store. The surface area of soap is 63.585 in.² and he wants to package them into a cube box so that it fits snuggly. Using using 3.14 as the value of pi, what should be the dimensions of the cube box?

Answer 1

To find the dimensions of the cube box, we first need to determine the volume of the spherical soap.

The formula for the surface area of a sphere is 4πr^2, where r is the radius of the sphere. In this case, the surface area is given as 63.585 in.².

So, we have:
4πr^2 = 63.585
r^2 = 63.585 / (4π)
r^2 = 5.0717211
r ≈ √5.0717211
r ≈ 2.253 inches

Now, we need to find the volume of the spherical soap:
V = (4/3)πr^3
V = (4/3)π(2.253)^3
V ≈ 54.24 in.³

Since the cube box should snugly fit the spherical soap, the volume of the cube box should be equal to the volume of the spherical soap:
Volume of cube = 54.24 in.³

Let the side length of the cube be x inches. Therefore, the volume of the cube can be calculated as:
V = x^3
x^3 = 54.24
x ≈ ∛54.24
x ≈ 3.738 inches

Therefore, the dimensions of the cube box should be approximately 3.738 inches on each side.