To determine the height of a solid right pyramid with a square base, we need to start with the formula for its volume. The volume [tex]\( V \)[/tex] of a pyramid can be expressed as:
[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]
In this case, the base of the pyramid is a square with edge length [tex]\( y \)[/tex]. Therefore, the area of the square base is:
[tex]\[ \text{Base Area} = y^2 \][/tex]
Substituting the base area into the volume formula, we get:
[tex]\[ V = \frac{1}{3} \times y^2 \times \text{Height} \][/tex]
Let [tex]\( h \)[/tex] represent the height of the pyramid. The equation then becomes:
[tex]\[ V = \frac{1}{3} \times y^2 \times h \][/tex]
We want to solve for the height [tex]\( h \)[/tex]. To do this, we rearrange the equation to isolate [tex]\( h \)[/tex]:
[tex]\[ h = \frac{3 \times V}{y^2} \][/tex]
Thus, the correct expression for the height of the pyramid is:
[tex]\[ \frac{3V}{y^2} \text{ units} \][/tex]
Among the given options:
1. [tex]\(\frac{3 V}{y^2}\)[/tex] units
2. [tex]\((3 V - y^2)\)[/tex] units
3. [tex]\((V - 3 y^2)\)[/tex] units
4. [tex]\(\frac{V}{3 y^2}\)[/tex] units
The correct expression that represents the height of the pyramid is:
[tex]\[ \frac{3 V}{y^2} \text{ units} \][/tex]
So, the correct choice is:
[tex]\[ \boxed{\frac{3 V}{y^2}} \text{ units} \][/tex]
