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: Find the proportionality constant

The volume $V$ of a pyramid varies jointly with the base - area $B$ and its height $h$, so $V = kBh$. When $V = 24$ cubic inches, $B = 24$ square inches and $h = 3$ inches. Substitute these values into the formula: $24=k\times24\times3$. Solve for $k$:
\[

$$\begin{align*} 24&=k\times24\times3\\ k&=\frac{24}{24\times3}\\ k&=\frac{1}{3} \end{align*}$$

\]

Step2: Calculate the volume for new base - area and height

Now we want to find the volume when $B = 10$ square inches and $h = 9$ inches. Use the formula $V = kBh$ with $k=\frac{1}{3}$.
\[

$$\begin{align*} V&=\frac{1}{3}\times10\times9\\ V& = 30 \end{align*}$$

\]

Answer:

30