Question

topic: rational formulas and variation
progress:
the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer.
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 24 square inches and its height is 3 inches. what is the volume of a pyramid with a base area of 15 square inches and a height of 7 inches?
the volume of the pyramid is ______ cubic inches.
the solution is

Explanation:

Step1: Define the joint variation formula

The volume \( V \) of a pyramid varies jointly with the base area \( B \) and height \( h \), so the formula is \( V = kBh \), where \( k \) is the constant of variation.

Step2: Find the constant \( k \)

We know \( V = 24 \), \( B = 24 \), and \( h = 3 \). Substitute these values into the formula:
\( 24 = k \times 24 \times 3 \)
First, calculate \( 24 \times 3 = 72 \), so the equation becomes \( 24 = 72k \).
Solve for \( k \) by dividing both sides by 72: \( k = \frac{24}{72} = \frac{1}{3} \).

Step3: Calculate the new volume

Now we have \( k = \frac{1}{3} \), \( B = 15 \), and \( h = 7 \). Use the formula \( V = kBh \):
\( V = \frac{1}{3} \times 15 \times 7 \)
First, calculate \( \frac{1}{3} \times 15 = 5 \), then multiply by 7: \( 5 \times 7 = 35 \).

Answer:

35