Question

1. the volume of a pyramid varies jointly with its height and the area of its base. a pyramid with a height of 16 feet and a base area of 78 square feet has a volume of 416 cubic feet. if a pyramid has a base area of 102 square feet and a volume of 238 cubic feet, find its height.

Explanation:

Step1: Define joint variation formula

Let $V$ = volume, $h$ = height, $B$ = base area. Joint variation: $V = kBh$, where $k$ is the constant of variation.

Step2: Solve for constant $k$

Substitute $V=416$, $h=16$, $B=78$:
$$416 = k \times 78 \times 16$$
Calculate $78 \times 16 = 1248$, so $k = \frac{416}{1248} = \frac{1}{3}$

Step3: Use $k$ to find unknown height

Substitute $V=238$, $B=102$, $k=\frac{1}{3}$ into $V = kBh$:
$$238 = \frac{1}{3} \times 102 \times h$$
Calculate $\frac{1}{3} \times 102 = 34$, so $h = \frac{238}{34}$

Answer:

7 feet