Question

this pentagon-based pyramid and this pentagon-based prism have the same volume, and their bases have the same area.

complete the statement about the heights.

the height of the pyramid is the same as / two times as large as / three times as large as / five times as large as the height of the prism.

Explanation:

Identify the given conditions

We are given a pentagon-based pyramid and a pentagon-based prism.
They have the same volume:

$$ V_{\text{pyramid}} = V_{\text{prism}} $$

Their bases have the same area:

$$ B_{\text{pyramid}} = B_{\text{prism}} = B $$

State the volume formulas

The volume of a prism is:

$$ V_{\text{prism}} = B \cdot h_{\text{prism}} $$

The volume of a pyramid is:

$$ V_{\text{pyramid}} = \frac{1}{3} B \cdot h_{\text{pyramid}} $$

Relate the heights

Since the volumes are equal, we equate the two formulas:

$$ \frac{1}{3} B \cdot h_{\text{pyramid}} = B \cdot h_{\text{prism}} $$

Solve for the pyramid height

Divide both sides by the common base area \( B \):

$$ \frac{1}{3} h_{\text{pyramid}} = h_{\text{prism}} $$

Multiply both sides by 3:

$$ h_{\text{pyramid}} = 3 \cdot h_{\text{prism}} $$

Interpret the result

The height of the pyramid is three times as large as the height of the prism.

Answer:

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