The volume of a solid right pyramid with a square base is [tex]V[/tex] units[tex]\(^3\)[/tex] and the length of the base edge is [tex]y[/tex] units. Which expression represents the height of the pyramid?

A. [tex]\frac{3V}{y^2}[/tex] units

B. [tex](3V - y^2)[/tex] units

C. [tex](V - 3y^2)[/tex] units

D. [tex]\frac{V}{3y^2}[/tex] units



Answer :

Let's consider the problem of finding the height of a right pyramid with a square base, given the volume and the length of the base edge.

### Step-by-Step Solution:

1. Volume Formula for a Pyramid:
The volume [tex]\( V \)[/tex] of a right pyramid with a square base is given by:
[tex]\[ V = \frac{1}{3} \times \text{base\_area} \times \text{height} \][/tex]

2. Area of the Square Base:
Since the base is a square with edge length [tex]\( y \)[/tex], the area of the base ([tex]\(\text{base\_area}\)[/tex]) is:
[tex]\[ \text{base\_area} = y^2 \][/tex]

3. Substitute Base Area into Volume Formula:
Substitute [tex]\( y^2 \)[/tex] for the base area in the volume formula:
[tex]\[ V = \frac{1}{3} \times y^2 \times \text{height} \][/tex]

4. Solve for Height:
We need to solve for the height ([tex]\(\text{height}\)[/tex]). Rearrange the equation to isolate height on one side:
[tex]\[ \text{height} = \frac{3V}{y^2} \][/tex]

Thus, the expression that represents the height of the pyramid is:
[tex]\[ \boxed{\frac{3V}{y^2}} \][/tex]

So, among the given choices, the correct option is:
[tex]\[ \frac{3 V}{y^2} \text{ units} \][/tex]