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?
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.