To find the dimensions of the cube box that can snugly fit the spherical soap, we first need to determine the radius of the soap based on its surface area.
The formula for the surface area \( S \) of a sphere is given by:
\[ S = 4\pi r^2 \]
Where \( r \) is the radius of the sphere. We can rearrange this formula to solve for \( r \):
\[ r^2 = \frac{S}{4\pi} \]
Substituting the values we have:
\[ S = 63.585 \text{ in}^2 \] \[ \pi = 3.14 \]
Now we can calculate \( r \):
\[ r^2 = \frac{63.585}{4 \times 3.14} \] \[ r^2 = \frac{63.585}{12.56} \approx 5.06 \]
Taking the square root to find \( r \):
\[ r \approx \sqrt{5.06} \approx 2.25 \text{ in} \]
Since the sphere fits snugly in a cube, the diameter of the sphere will be equal to the side length of the cube. The diameter \( d \) of the sphere is given by:
\[ d = 2r \approx 2 \times 2.25 \approx 4.5 \text{ in} \]
Thus, the dimensions of the cube box should be at least \( 4.5 \text{ in} \times 4.5 \text{ in} \times 4.5 \text{ in} \) to fit the soap snugly.
Among the given choices, the correct dimensions for the cube box would be:
4.5 in x 4.5 in x 4.5 in