To find the height of the rectangular pyramid, we first need to understand the relationships between the volume, base area, and height of both the rectangular prism and the pyramid.
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Given Information:
- Base area of the rectangular prism (and pyramid) = \( 25 , \text{in}^2 \)
- Volume of the rectangular prism = \( 125 , \text{in}^3 \)
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Volume of the Rectangular Prism: The volume \( V \) of a rectangular prism can be calculated using the formula: \[ V = \text{Base Area} \times \text{Height} \] Plugging in the known values: \[ 125 = 25 \times \text{Height of the prism} \] To find the height of the prism: \[ \text{Height of the prism} = \frac{125}{25} = 5 , \text{in} \]
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Volume of the Rectangular Pyramid: The volume \( V \) of a rectangular pyramid is given by: \[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \] Using the same base area of \( 25 , \text{in}^2 \) and letting \( h \) be the height of the pyramid, we can express the volume of the pyramid as: \[ V = \frac{1}{3} \times 25 \times h \]
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Setting the Height: Since the pyramid has a height congruent to that of the prism, we set the height of the prism (which is \( 5 , \text{in} \)) as the height of the pyramid: \[ h = 5 , \text{in} \]
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Conclusion: The height of the rectangular pyramid is \( 5 , \text{in} \).
Thus, the height of the pyramid is \(\boxed{5}\) inches.