Question

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 10 square inches and a height of 9 inches?
the volume of the pyramid is ______ cubic inches.
the solution is

Explanation:

Step1: Write the joint - variation formula

The volume \(V\) of a pyramid varies jointly with the base area \(B\) and its height \(h\). The joint - variation formula is \(V = kBh\), where \(k\) is the constant of variation.
When \(V = 24\) cubic inches, \(B = 24\) square inches, and \(h = 3\) inches. Substitute these values into the formula: \(24=k\times24\times3\).

Step2: Solve for \(k\)

From \(24 = k\times24\times3\), we can solve for \(k\) as follows:
First, simplify the right - hand side: \(k\times24\times3=72k\). Then, \(k=\frac{24}{72}=\frac{1}{3}\).

Step3: Use the formula to find the new volume

Now, we want to find \(V\) when \(B = 9\) square inches and \(h = 9\) inches. Using the formula \(V=\frac{1}{3}Bh\) (since \(k = \frac{1}{3}\)).
Substitute \(B = 9\) and \(h = 9\) into the formula: \(V=\frac{1}{3}\times9\times9\).
First, \(\frac{1}{3}\times9 = 3\). Then, \(3\times9=27\).

Answer:

27