To find the volume of the pyramid, we first need to determine the dimensions of the cube.
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Volume of the Cube: Given that the volume of the cube is 81 cubic units, we can find the side length \( s \) of the cube using the formula for the volume of a cube:
\[ V = s^3 \]
Setting this equal to 81:
\[ s^3 = 81 \]
Taking the cube root of both sides:
\[ s = \sqrt[3]{81} = 4.326 (approximately) \]
However, 81 is actually \( 3^4 \), so \( s = 3^{\frac{4}{3}} \).
Alternatively, we can calculate the actual cube root more directly: The cube root of 81 can be computed as:
\[ s = 3^{4/3} = 3^{1.3333} \approx 4.326 \]
OR \( s = 4.326 \)
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Base Area of the Pyramid: Since the cube has the same base area as the pyramid, the base area of the cube (which is \( s^2 \)) can be calculated as follows:
\[ A = s^2 = (4.326)^2 \approx 18.738 \]
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Height of the Pyramid: The height of the pyramid is equal to the side of the cube:
\[ h = s = 4.326 \]
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Volume of the Pyramid: The formula for the volume of a pyramid is given by:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
Plugging in the values we found:
\[ V = \frac{1}{3} \times A \times h \approx \frac{1}{3} \times 18.738 \times 4.326 \]
Now, we calculate this:
\[ V \approx \frac{1}{3} \times 18.738 \times 4.326 \approx \frac{1}{3} \times 80.895 \approx 26.965 \]
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Rounding to the Nearest Whole Number: Rounding 26.965 to the nearest whole number, we get:
\[ \text{Volume of the pyramid} \approx 27 \text{ cubic units} \]
Thus, the volume of the pyramid is approximately 27 cubic units.