To find the volume of the pyramid, we need to first calculate the area of the triangular base.
Since both the prism and pyramid have a triangular base, they have the same base dimensions.
Let's say the base of the prism and pyramid is an equilateral triangle with side length \(s\).
The area of an equilateral triangle can be calculated using the formula \(\frac{\sqrt{3}}{4} \times s^2\).
Given that the volume of the prism is 540 inches cubed, we can calculate the side length of the base using the formula for the volume of a prism, which is \(V = \text{base area} \times \text{height}\).
540 = \(\frac{\sqrt{3}}{4} \times s^2\) \times height
\(\frac{\sqrt{3}}{4} \times s^2\) = 540/height
\(s^2\) = \(\frac{4}{\sqrt{3}}\) \times 540/height
\(s^2\) = \(\frac{2160}{\sqrt{3} \times height}\)
\(s = \sqrt{\frac{2160}{\sqrt{3} \times height}}\)
Now that we have found the side length of the base, we can find the area of the base and the volume of the pyramid.
Area of base = \(\frac{\sqrt{3}}{4} \times s^2\)
Volume of pyramid = \(\frac{1}{3}\) \times base area \times height
Volume of pyramid = \(\frac{1}{3}\) \times \(\frac{\sqrt{3}}{4} \times s^2\) \times height
Substitute the value of \(s\) into the formula to get the volume of the pyramid.
A prism and a pyramid both have this triangular base. If both shapes have the height and the volume of the prism is 540 inches cubed what is the volume of the pyramid?
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