To determine the height of a solid right pyramid with a square base, we will follow these steps:
1. Given Data:
- Volume of the pyramid, [tex]\( V = 1 \)[/tex] cubic unit.
- Length of the base edge, [tex]\( y \)[/tex] units.
2. Volume Formula for a Pyramid:
The volume [tex]\( V \)[/tex] of a solid right pyramid with a square base is given by the formula:
[tex]\[
V = \frac{1}{3} \cdot \text{base area} \cdot \text{height}
\][/tex]
3. Base Area for a Square Base:
The area of the square base with edge length [tex]\( y \)[/tex] is:
[tex]\[
\text{Base Area} = y^2
\][/tex]
4. Substituting the Base Area:
Substitute the base area [tex]\( y^2 \)[/tex] into the volume formula:
[tex]\[
V = \frac{1}{3} \cdot y^2 \cdot \text{height}
\][/tex]
5. Solving for Height:
Isolate the height on one side of the equation. Start by multiplying both sides by 3 to clear the fraction:
[tex]\[
3V = y^2 \cdot \text{height}
\][/tex]
Now, solve for the height by dividing both sides by [tex]\( y^2 \)[/tex]:
[tex]\[
\text{height} = \frac{3V}{y^2}
\][/tex]
6. Substitute the Given Volume:
Given that [tex]\( V = 1 \)[/tex], we substitute [tex]\( V \)[/tex] into the formula:
[tex]\[
\text{height} = \frac{3 \cdot 1}{y^2} = \frac{3}{y^2}
\][/tex]
Thus, the correct expression representing the height of the pyramid is:
[tex]\[
\boxed{\frac{3}{y^2}}
\][/tex]
From the given choices, the correct answer is:
[tex]\[
\frac{3 V}{y^2} \text{ units}
\][/tex]
Given our substitution [tex]\( V = 1 \)[/tex], it reduces to:
[tex]\[
\frac{3}{y^2} \text{ units}
\][/tex]
Hence, the expression representing the height of the pyramid is [tex]\( \frac{3V}{y^2} \)[/tex] units.
