If pyramid with a square base with side length s and a right cone with radius r have equal heights and equal volumes, then which equation must be true?

Options

A. `s=sqrt(pir)`
B. `s=(sqrt(r))/(pi)`
C. `s=pisqrt(r)`
D. `s=rsqrt(pi)`

Answer

To solve the problem, we need to find the relationship between the side length \( s \) of a pyramid with a square base and the radius \( r \) of a right cone, given that they have equal heights and equal volumes.

### Step-by-Step Solution:

1. **Volume of the Pyramid**:
The volume \( V \) of a pyramid with a square base is given by the formula:
\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \]
For a square base with side length \( s \), the area of the base is \( s^2 \). Therefore, the volume of the pyramid becomes:
\[ V_{\text{pyramid}} = \frac{1}{3} s^2 h \]

2. **Volume of the Cone**:
The volume \( V \) of a right cone is given by the formula:
\[ V = \frac{1}{3} \times \pi \times r^2 \times \text{Height} \]
Thus, the volume of the cone is:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \]

3. **Setting Volumes Equal**:
Since the volumes of the pyramid and the cone are equal, we can set their volume equations equal to each other:
\[ \frac{1}{3} s^2 h = \frac{1}{3} \pi r^2 h \]

4. **Canceling Common Terms**:
We can cancel the common terms \( \frac{1}{3} \) and \( h \) (assuming \( h \neq 0 \)):
\[ s^2 = \pi r^2 \]

5. **Finding the Relationship**:
To find the relationship between \( s \) and \( r \), we take the square root of both sides:
\[ s = r \sqrt{\pi} \]

### Final Equation:
Thus, the equation that must be true is:
\[ s = r \sqrt{\pi} \]