To determine the volume of the pyramid based on the given information about the cube, we can follow these steps:
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Find the height of the cube: The volume of a cube is given by the formula:
\[ V = s^3 \]
where \(s\) is the length of one side of the cube. Given that the volume of the cube is 81 cubic units:
\[ s^3 = 81 \]
Taking the cube root of both sides:
\[ s = \sqrt[3]{81} \]
Since \(81 = 3^4\), we find:
\[ s = 3 \sqrt[3]{3} \]
But we can directly compute \(s\) as follows:
\[ 81 = 3^4 \implies s = 3 \text{ units} \]
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Find the height of the cube: The height of the cube is \(s\), so:
\[ \text{Height} = 3 \text{ units} \]
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Calculate the base area of the pyramid: Since the pyramid has the same base area as the cube, and since the base of a cube is a square, the area of the base (which is also \(s^2\)) is given by:
\[ \text{Base Area} = s^2 = 3^2 = 9 \text{ square units} \]
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Determine the volume of the pyramid: The volume of a pyramid is given by the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Substituting in the base area and height:
\[ V = \frac{1}{3} \times 9 \text{ (square units)} \times 3 \text{ (units)} \]
Simplifying this:
\[ V = \frac{1}{3} \times 27 = 9 \text{ cubic units} \]
Thus, the volume of the pyramid is 9 cubic units.