To determine the dimensions of the cube box that will fit the spherical soap, we first need to find the radius of the sphere from its surface area. The formula for the surface area \(A\) of a sphere is given by:
\[ A = 4\pi r^2 \]
Given the surface area \(A = 63.585 , \text{in}^2\) and using \(\pi \approx 3.14\), we can solve for the radius \(r\).
\[ 63.585 = 4 \times 3.14 \times r^2 \]
Calculating \(4 \times 3.14\):
\[ 4 \times 3.14 = 12.56 \]
Now, we can rewrite the equation:
\[ 63.585 = 12.56 r^2 \]
Next, let’s solve for \(r^2\):
\[ r^2 = \frac{63.585}{12.56} \approx 5.06 \]
Now, taking the square root to find \(r\):
\[ r \approx \sqrt{5.06} \approx 2.25 , \text{in.} \]
The diameter \(d\) of the sphere, which is twice the radius, is:
\[ d = 2r \approx 2 \times 2.25 \approx 4.5 , \text{in.} \]
Since the box needs to fit the whole sphere snugly, the side length of the cube box must be at least equal to the diameter of the sphere.
Thus, the dimensions of the cube box should be \(4.5 , \text{in.} \times 4.5 , \text{in.} \times 4.5 , \text{in.}\)
Final answer: 4.5 in. × 4.5 in. × 4.5 in.