A prism and a pyramid have the same base and the same height. Find the ratio of the volumes of the prism and the pyramid.

Options

A. `1:1`
B. `1:3`
C. `3:1`
D. Cannot be determined

Answer

To find the ratio of the volumes of a prism and a pyramid that have the same base and height, we can follow these steps:

### Step 1: Write the formula for the volume of the prism.
The volume \( V \) of a prism is given by the formula:
\[ V_{\text{prism}} = \text{Area of base} \times \text{Height} \]

### Step 2: Write the formula for the volume of the pyramid.
The volume \( V \) of a pyramid is given by the formula:
\[ V_{\text{pyramid}} = \frac{1}{3} \times \text{Area of base} \times \text{Height} \]

### Step 3: Set up the ratio of the volumes.
We want to find the ratio of the volume of the prism to the volume of the pyramid:
\[ \text{Ratio} = \frac{V_{\text{prism}}}{V_{\text{pyramid}}} \]

### Step 4: Substitute the volume formulas into the ratio.
Substituting the formulas from Steps 1 and 2 into the ratio, we get:
\[ \text{Ratio} = \frac{\text{Area of base} \times \text{Height}}{\frac{1}{3} \times \text{Area of base} \times \text{Height}} \]

### Step 5: Simplify the ratio.
In the ratio, the "Area of base" and "Height" terms cancel out:
\[ \text{Ratio} = \frac{1}{\frac{1}{3}} = 3 \]

### Step 6: Express the ratio in a standard form.
The ratio can be expressed as:
\[ \text{Ratio} = 3:1 \]

### Final Answer:
The ratio of the volumes of the prism to the pyramid is \( 3:1 \).

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