Ahmad constructs a square-based pyramid and a cube out of newspaper. The two figures have the same height and the same base area. What is the volume of the pyramid if the volume of the cube is 54 cubic units?(1 point)

Responses

36 cubic units
36 cubic units

6 cubic units
6 cubic units

18 cubic units
18 cubic units

90 cubic units

1 answer

To find the volume of the square-based pyramid given that it has the same height and base area as a cube with a volume of 54 cubic units, we can follow these steps:

  1. Calculate the side length of the cube: The volume \( V \) of the cube is given by the formula: \[ V = s^3 \] where \( s \) is the side length. Given that the volume is 54 cubic units, we find \( s \): \[ s^3 = 54 \implies s = \sqrt[3]{54} = 3 \sqrt[3]{2} \quad (\text{approximately } 3.78) \] We will use exact values for further calculations.

  2. Calculate the base area of the cube: The base area \( A \) of the cube (which is a square) is: \[ A = s^2 \] Substituting \( s = \sqrt[3]{54} \): \[ A = (s)^2 = (s^3)^{2/3} = 54^{2/3} \]

  3. Calculate the height of the cube: The height \( h \) of the cube is equal to its side length \( s \): \[ h = s = \sqrt[3]{54} \]

  4. Calculate the volume of the pyramid: The volume \( V_p \) of a square-based pyramid is given by the formula: \[ V_p = \frac{1}{3} A h \] We plug in the values for base area \( A \) and height \( h \): \[ V_p = \frac{1}{3} (54^{2/3})(54^{1/3}) = \frac{1}{3} (54^{2/3 + 1/3}) = \frac{1}{3} (54^{3/3}) = \frac{1}{3} (54) = 18 \text{ cubic units} \]

So, the volume of the pyramid is 18 cubic units.

The correct answer is: 18 cubic units.