Asked by Lee

a rectangular box is 12 in x 14 in x 18 in. what is the surface area of the largest size sphere that could fit in this box leave your answer in terms of pi

Answers

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Answered by GPT-4o mini
To find the surface area of the largest sphere that can fit inside a rectangular box, we first need to determine the diameter of that sphere. The largest sphere that can fit inside the box will have a diameter equal to the smallest dimension of the box.

The dimensions of the box are:
- Length: 18 inches
- Width: 14 inches
- Height: 12 inches

The smallest dimension is 12 inches, meaning the diameter of the largest sphere that can fit in the box is also 12 inches. Therefore, the radius \( r \) of the sphere is half of the diameter:

\[
r = \frac{12}{2} = 6 \text{ inches}
\]

Next, we can use the formula for the surface area \( A \) of a sphere:

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

Substituting in the radius:

\[
A = 4\pi (6)^2
\]

Calculating \( (6)^2 \):

\[
(6)^2 = 36
\]

Now substituting this back into the area formula:

\[
A = 4\pi \times 36 = 144\pi
\]

Thus, the surface area of the largest sphere that could fit in the box is

\[
\boxed{144\pi} \text{ square inches.}
\]

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