To determine the dimensions of the cube box that can snugly fit the spherical soap, we need to first calculate the radius of the soap from its surface area. The formula for the surface area \( A \) of a sphere is given by:
\[ A = 4 \pi r^2 \]
Where:
- \( A \) is the surface area,
- \( r \) is the radius of the sphere, and
- \( \pi \) is approximately 3.14.
We know the surface area \( A = 63.585 , \text{in}^2 \). Using \(\pi = 3.14\), we can rearrange the formula to solve for \( r \):
\[ 63.585 = 4 \times 3.14 \times r^2 \]
Calculating the right side:
\[ 63.585 = 12.56 \times r^2 \]
Next, divide by 12.56 to isolate \( r^2 \):
\[ r^2 = \frac{63.585}{12.56} \approx 5.06 \]
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 also the side length of the cube that fits the sphere snugly) is given by:
\[ d = 2r \approx 2 \times 2.25 = 4.5 , \text{in} \]
Thus, the dimensions of the cube box should be \( 4.5 , \text{in} \times 4.5 , \text{in} \times 4.5 , \text{in} \).
Final response: 4.5 in. × 4.5 in. × 4.5 in.
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