To find the volume of a rectangular pyramid, you can use the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
In this case, since it is a square pyramid, the base area can be calculated as:
\[ \text{Base Area} = \text{side length}^2 = 16 , \text{miles} \times 16 , \text{miles} = 256 , \text{mi}^2 \]
The perpendicular height of the pyramid is given as \(8\) miles.
Now, substitute the values into the volume formula:
\[ V = \frac{1}{3} \times 256 , \text{mi}^2 \times 8 , \text{mi} \]
Calculating this step-by-step:
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Multiply the base area by the height: \[ 256 \times 8 = 2048 , \text{mi}^3 \]
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Now, calculate the volume: \[ V = \frac{1}{3} \times 2048 = \frac{2048}{3} \approx 682.67 , \text{mi}^3 \]
Therefore, the volume of the rectangular pyramid is approximately:
\[ \boxed{682.67} , \text{mi}^3 \]