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Question
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?(1 point)
Responses
4.5 in. × 4.5 in. × 4.5 in.
4.5 inches times 4.5 inches times 4.5 inches
2.25 in. × 2.25 in. × 2.25 in.
2.25 inches times 2.25 inches times 2.25 inches
1.2 in. × 1.2 in. × 1.2 in.
1.2 inches times 1.2 inches times 1.2 inches
4 in.× 4 in.× 4 in.
To find the dimensions of the cube box that Marcus needs to package his spherical soaps, we first need to find the diameter of the soap, which will be the side length of the cube box since he wants the soap to fit snugly.
The surface area (A) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
where \( r \) is the radius of the sphere.
Given the surface area is 63.585 in² and using 3.14 as the value of pi, we can solve for \( r \).
\[ 63.585 = 4 \times 3.14 \times r^2 \]
\[ 63.585 = 12.56 \times r^2 \]
\[ r^2 = \frac{63.585}{12.56} \]
\[ r^2 = 5.063 \]
Now we take the square root of both sides to find \( r \):
\[ r = \sqrt{5.063} \]
To find the radius, you can either estimate the square root or use a calculator. Since the question doesn't require an exact answer, let's estimate:
\[ r \approx 2.25 \]
Now that we know the radius, the diameter (D) of the sphere is twice the radius:
\[ D = 2r \]
\[ D = 2 \times 2.25 \]
\[ D = 4.5 \text{ inches} \]
So the dimensions of the cube box, which are equal to the diameter of the sphere, would need to be:
\[ 4.5 \text{ in.} \times 4.5 \text{ in.} \times 4.5 \text{ in.} \]
Therefore, the correct option is:
4.5 in. × 4.5 in. × 4.5 in.