The volume of a pyramid varies jointly with the base area of the pyramid and its height. The volume of one pyramid is 24

cubic inches when its base area is 3
square inches and its height is 24
inches. What is the volume of a pyramid with a base area of 10
square inches and a height of 9
inches?

The volume of the pyramid is blank cubic inches.

1 answer

The volume \( V \) of a pyramid can be expressed by the formula:

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

Given that the volume varies jointly with the base area and height, we can also write:

\[ V = k \times \text{Base Area} \times \text{Height} \]

where \( k \) is a constant.

We know that the volume of one pyramid is 24 cubic inches when the base area is 3 square inches and the height is 24 inches. From this information, we can calculate the constant \( k \):

\[ 24 = k \times 3 \times 24 \]

Solving for \( k \):

\[ 24 = 72k \] \[ k = \frac{24}{72} = \frac{1}{3} \]

Now, we can use this value of \( k \) to find the volume of the pyramid with a base area of 10 square inches and a height of 9 inches.

Using the formula:

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

Substituting the values:

\[ V = \frac{1}{3} \times 10 \times 9 \] \[ V = \frac{1}{3} \times 90 \] \[ V = 30 \]

Thus, the volume of the pyramid is \( \boxed{30} \) cubic inches.