To find the dimensions of the cube box that will snugly fit the spherical soaps, we first need to determine the radius of the spherical soap from its surface area.
The formula for the surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]
where \( r \) is the radius of the sphere and \( \pi \) is approximately 3.14.
Given that the surface area \( A \) is 63.585 in², we can substitute the values into the formula and solve for \( r \):
\[ 63.585 = 4 \cdot 3.14 \cdot r^2 \]
Calculating \( 4 \cdot 3.14 \):
\[ 4 \cdot 3.14 = 12.56 \]
Now we can rewrite the equation:
\[ 63.585 = 12.56 \cdot r^2 \]
Next, we can solve for \( r^2 \) by dividing both sides by 12.56:
\[ r^2 = \frac{63.585}{12.56} \approx 5.06 \]
Now, taking the square root of both sides to find \( r \):
\[ r \approx \sqrt{5.06} \approx 2.25 \text{ inches} \]
The diameter \( d \) of the soap (which would be the dimension that the cube box needs to accommodate) is given by:
\[ d = 2r \approx 2 \cdot 2.25 = 4.5 \text{ inches} \]
Since the soap fits snugly in the cube box, the dimensions of the cube box will be:
\[ 4.5 \text{ in.} \times 4.5 \text{ in.} \times 4.5 \text{ in.} \]
So, the correct response is:
4.5 in. × 4.5 in. × 4.5 in.