To calculate the capacity of the trench in the shape of an upside-down rectangular pyramid (a frustum of a pyramid), we can use the formula for the volume of a rectangular pyramid:
\[ V = \frac{1}{3} \cdot B \cdot h \]
where \( B \) is the area of the base and \( h \) is the height of the pyramid.
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Calculate the area of the top opening (base):
The dimensions of the opening are: \[ \text{Width} = 30 \text{ ft} \] \[ \text{Length} = 250 \text{ ft} \]
Thus the area of the base \( B \) is: \[ B = \text{Width} \times \text{Length} = 30 \text{ ft} \times 250 \text{ ft} = 7500 \text{ ft}^2 \]
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Identify the height of the pyramid:
The deepest part of the trench is: \[ h = 15 \text{ ft} \]
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Calculate the volume of the trench:
\[ V = \frac{1}{3} \cdot B \cdot h = \frac{1}{3} \cdot 7500 \text{ ft}^2 \cdot 15 \text{ ft} \] \[ V = \frac{1}{3} \cdot 112500 \text{ ft}^3 = 37500 \text{ ft}^3 \]
Therefore, the capacity for the trench is \( 37,500 , \text{ft}^3 \).
The correct response is 37,500 ft.³.