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

6 cubic units
6 cubic units

90 cubic units
90 cubic units

36 cubic units
36 cubic units

18 cubic units

1 answer

To find the volume of the pyramid, we first need to determine the side length of the cube since we know its volume is 54 cubic units.

The volume \( V \) of a cube is given by the formula:

\[ V = s^3 \]

where \( s \) is the side length of the cube. Setting this equal to 54, we find \( s \):

\[ s^3 = 54 \]

Taking the cube root of both sides gives us:

\[ s = \sqrt[3]{54} \]

Next, since the pyramid has the same base area and height as the cube, we first need to evaluate the base area of the cube. The base area \( A \) of the cube is:

\[ A = s^2 \]

Calculating \( s^2 \):

\[ s^2 = (\sqrt[3]{54})^2 = 54^{2/3} \]

Now, we will compute the height of the cube, which is equal to the side length \( s \) of the cube:

\[ h = s = \sqrt[3]{54} \]

The volume \( V_p \) of a square-based pyramid is given by the formula:

\[ V_p = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]

Substituting for base area (which is \( s^2 \)) and height \( (s) \):

\[ V_p = \frac{1}{3} \times s^2 \times s = \frac{s^3}{3} \]

Substituting \( s^3 = 54 \):

\[ V_p = \frac{54}{3} = 18 \]

Therefore, the volume of the pyramid is:

\[ \boxed{18} \text{ cubic units} \]