Question

the volume of a solid right pyramid with a square base is \\(v\text{ units}^3\\) and the length of the base edge is \\(y\text{ units}\\).

which expression represents the height of the pyramid?

\\(\bigcirc\quad \frac{3v}{y^2}\text{ units}\\)
\\(\bigcirc\quad (3v - y^2)\text{ units}\\)
\\(\bigcirc\quad (v - 3y^2)\text{ units}\\)
\\(\bigcirc\quad \frac{v}{3y^2}\text{ units}\\)

Explanation:

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<reasoning>

Identify the base area formula

Using the Pyramid Base Geometry knowledge point
\[
B = y^2
\]

State the volume formula

The volume \(V\) of a right pyramid is given by:
\[
V = \frac{1}{3} B h
\]
where \(B\) is the base area and \(h\) is the height.

Substitute the base area

Substitute \(B = y^2\) into the volume formula:
\[
V = \frac{1}{3} y^2 h
\]

Solve for the height

Multiply both sides by 3 and divide by \(y^2\) to isolate \(h\):
\[
3V = y^2 h \implies h = \frac{3V}{y^2}
\]
</reasoning>

<answer>
<mcq-correct>(A) \(\frac{3V}{y^2}\) units</mcq-correct>
<mcq-option>(B) \((3V - y^2)\) units</mcq-option>
<mcq-option>(C) \((V - 3y^2)\) units</mcq-option>
<mcq-option>(D) \(\frac{V}{3y^2}\) units</mcq-option>
</answer>

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Answer:

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</pre_analysis>

<reasoning>

Identify the base area formula

Using the Pyramid Base Geometry knowledge point
\[
B = y^2
\]

State the volume formula

The volume \(V\) of a right pyramid is given by:
\[
V = \frac{1}{3} B h
\]
where \(B\) is the base area and \(h\) is the height.

Substitute the base area

Substitute \(B = y^2\) into the volume formula:
\[
V = \frac{1}{3} y^2 h
\]

Solve for the height

Multiply both sides by 3 and divide by \(y^2\) to isolate \(h\):
\[
3V = y^2 h \implies h = \frac{3V}{y^2}
\]
</reasoning>

<answer>
<mcq-correct>(A) \(\frac{3V}{y^2}\) units</mcq-correct>
<mcq-option>(B) \((3V - y^2)\) units</mcq-option>
<mcq-option>(C) \((V - 3y^2)\) units</mcq-option>
<mcq-option>(D) \(\frac{V}{3y^2}\) units</mcq-option>
</answer>

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