If the volume of a rectangular prism is 600 in.3

, what is the volume of a rectangular pyramid whose base and height is congruent to the prism?(1 point)
Responses

300 in.3
300 in. cubed

1,200 in.3
1,200 in. cubed

1,800 in.3
1,800 in. cubed

200 in.3

1 answer

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

\[ V = \text{length} \times \text{width} \times \text{height} \]

For a rectangular pyramid, the volume \( V \) is given by the formula:

\[ V = \frac{1}{3} \times \text{base area} \times \text{height} \]

Since the base and height of the pyramid are congruent to those of the prism, the base area of the pyramid is the same as the base area of the prism. Thus, the base area \( A \) can be represented as:

\[ A = \text{length} \times \text{width} \]

So, the volume of the pyramid is:

\[ V_{\text{pyramid}} = \frac{1}{3} \times A \times \text{height} \]

Since the volume of the prism is given as \( 600 , \text{in}^3 \):

\[ 600 = \text{length} \times \text{width} \times \text{height} \]

The pyramid's volume can be expressed as:

\[ V_{\text{pyramid}} = \frac{1}{3} \times (\text{length} \times \text{width}) \times \text{height} \]

Notice that:

\[ \text{length} \times \text{width} \times \text{height} = 600 \]

Thus,

\[ V_{\text{pyramid}} = \frac{1}{3} \times 600 = 200 , \text{in}^3 \]

Therefore, the volume of the rectangular pyramid is:

\[ \boxed{200 , \text{in}^3} \]