To find the dimensions of the cube box that can snugly fit the spherical soap with a surface area of 63.585 in², we first need to determine the radius of the soap.
The formula for the surface area \( A \) of a sphere is:
\[ A = 4 \pi r^2 \]
Where \( r \) is the radius of the sphere. Given:
\[ A = 63.585 , \text{in}^2 \]
Substituting for \( A \) and \( \pi \):
\[ 63.585 = 4 \times 3.14 \times r^2 \]
Now, calculate \( 4 \times 3.14 \):
\[ 4 \times 3.14 = 12.56 \]
Now substitute this value back into the equation:
\[ 63.585 = 12.56 \times r^2 \]
To solve for \( r^2 \):
\[ r^2 = \frac{63.585}{12.56} \approx 5.06 \]
Now take the square root to find \( r \):
\[ r \approx \sqrt{5.06} \approx 2.25 , \text{in} \]
The diameter \( d \) of the sphere is twice the radius:
\[ d = 2r \approx 2 \times 2.25 = 4.5 , \text{in} \]
Since the cube box must fit the diameter of the sphere snugly, the dimensions of the cube box need to be:
\[ 4.5 , \text{in} \times 4.5 , \text{in} \times 4.5 , \text{in} \]
Thus, the correct choice for the dimensions of the cube box is:
4.5 in. × 4.5 in. × 4.5 in.