To determine the dimensions of the cube box that can fit the spherical soap, we first need to find the radius of the soap using its surface area and then calculate the diameter of the soap, which will equal the side length of the cube.
The formula for surface area \( S \) of a sphere is given by:
\[ S = 4 \pi r^2 \]
Where \( r \) is the radius of the sphere and \( \pi \approx 3.14 \).
Given that the surface area \( S \) is 63.585 in², we can set up the equation:
\[ 4 \times 3.14 \times r^2 = 63.585 \]
Calculating \( 4 \times 3.14 \):
\[ 4 \times 3.14 = 12.56 \]
Now substituting this back into the equation:
\[ 12.56 r^2 = 63.585 \]
Next, we solve for \( r^2 \):
\[ r^2 = \frac{63.585}{12.56} \approx 5.06 \]
Taking the square root to find the radius \( r \):
\[ r \approx \sqrt{5.06} \approx 2.25 \text{ in.} \]
The diameter \( d \) of the sphere (which will be equal to the side length of the cube box) is:
\[ d = 2r = 2 \times 2.25 \approx 4.5 \text{ in.} \]
Now we can determine the dimensions of the cube box, which will snugly fit the spherical soap:
The dimensions of the cube box should be:
4.5 in. × 4.5 in. × 4.5 in.
Thus, the correct response is:
4.5 in. × 4.5 in. × 4.5 in.