To find the dimensions of the cube box that can snugly fit the spherical soap, we first need to calculate 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 that the surface area is \( 63.585 , \text{in}^2 \) and using \( \pi \approx 3.14 \), we can set up the equation:
\[ 63.585 = 4 \times 3.14 \times r^2 \]
Now we solve for \( r^2 \):
\[ 63.585 = 12.56r^2 \]
\[ r^2 = \frac{63.585}{12.56} \approx 5.06 \]
Next, we find the radius \( r \):
\[ r \approx \sqrt{5.06} \approx 2.25 , \text{in} \]
Since the diameter \( d \) of the sphere is twice the radius:
\[ d = 2r \approx 2 \times 2.25 \approx 4.5 , \text{in} \]
A cube that fits snugly around the sphere must have each side equal to the diameter of the sphere. Therefore, the dimensions of the cube box should be:
\[ 4.5 , \text{in.} \times 4.5 , \text{in.} \times 4.5 , \text{in.} \]
The correct response is:
4.5 in. × 4.5 in. × 4.5 in.
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