To determine the dimensions of the cube box that Marcus should use to package the spherical soaps, we need to first find the radius of the soap from its surface area and then calculate the side length of the cube box that will fit the sphere snugly.
The formula for the surface area \( A \) of a sphere is:
\[ A = 4 \pi r^2 \]
Given that the surface area \( A \) is 63.585 in², we can use this to find the radius \( r \):
- Substitute the value of \( A \) and \( \pi \):
\[ 63.585 = 4 \times 3.14 \times r^2 \]
- Solve for \( r^2 \):
\[ r^2 = \frac{63.585}{4 \times 3.14} = \frac{63.585}{12.56} \approx 5.06 \]
- 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:
\[ d = 2r \approx 2 \times 2.25 \approx 4.5 \text{ in} \]
Since the sphere needs to fit snugly into a cube box, the side length of the cube should be equal to the diameter of the sphere. The dimensions of the cube box should therefore be:
\[ 4.5 \text{ in.} \times 4.5 \text{ in.} \times 4.5 \text{ in.} \]
So the answer is:
4.5 in. × 4.5 in. × 4.5 in.
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