Marcus is making spherical soaps to sell in his online store. The surface area of a soap is 63.585 in.^2 and he wants to package them into a cube box so that it fits snugly. Using 3.14 as the value of pi, what should be the dimensions of the cube box?

Question

1.2 in. × 1.2 in. × 1.2 in.
4 in.× 4 in.× 4 in.
2.25 in. × 2.25 in. × 2.25 in.
4.5 in. × 4.5 in. × 4.5 in.

Answer 1

To determine the dimensions of the cube box that can snugly fit the spherical soap, we first need to find the radius of the soap using the given surface area. The surface area \( A \) of a sphere is given by the formula:

\[ A = 4 \pi r^2 \]

where \( r \) is the radius. We can rearrange this formula to solve for \( r \):

\[ r^2 = \frac{A}{4 \pi} \]

Given that the surface area \( A \) is 63.585 in² and using \( \pi \approx 3.14 \), we can plug in the values:

\[ r^2 = \frac{63.585}{4 \times 3.14} \]

Calculating the denominator:

\[ 4 \times 3.14 = 12.56 \]

Now we can find \( r^2 \):

\[ r^2 = \frac{63.585}{12.56} \approx 5.06 \]

Now calculate \( r \):

\[ r \approx \sqrt{5.06} \approx 2.25 \text{ in.} \]

Since the diameter \( d \) of the soap (which is twice the radius) will determine the side length of the cube box, we find:

\[ d = 2r \approx 2 \times 2.25 = 4.5 \text{ in.} \]

Now, to find the dimensions of the snug cube box, the side length must be equal to the diameter of the soap. Therefore, the cube box must have dimensions of:

\[ 4.5 \text{ in.} \times 4.5 \text{ in.} \times 4.5 \text{ in.} \]

Among the options provided, the correct dimensions for the cube box are:

\[ \text{4.5 in. × 4.5 in. × 4.5 in.} \]