To find the surface area of the largest sphere that can fit inside a rectangular box, we first need to determine the diameter of the sphere. The largest sphere that can fit inside the box will have its diameter equal to the smallest dimension of the box.
The dimensions of the box are:
- Length = 18 inches
- Width = 14 inches
- Height = 12 inches
Among these dimensions, the smallest is 12 inches. Thus, the diameter of the largest fitting sphere is 12 inches.
To find the radius \( r \) of the sphere, we divide the diameter by 2: \[ r = \frac{12}{2} = 6 \text{ inches} \]
The surface area \( A \) of a sphere is given by the formula: \[ A = 4\pi r^2 \]
Substituting the radius into the formula: \[ A = 4\pi (6^2) = 4\pi (36) = 144\pi \]
Thus, the surface area of the largest sphere that could fit in the box is: \[ \boxed{144\pi} \]
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