To find the radius of the cone given its height and volume, we can use the formula for the volume of a cone:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height.
Given:
- Volume (\( V \)) = 500 cm³
- Height (\( h \)) = 5 cm
We can plug in the values into the formula:
\[ 500 = \frac{1}{3} \pi r^2 (5) \]
This simplifies to:
\[ 500 = \frac{5}{3} \pi r^2 \]
To isolate \( r^2 \), multiply both sides by \( \frac{3}{5} \):
\[ r^2 = \frac{500 \times 3}{5 \pi} \]
Calculating the right side:
\[ r^2 = \frac{1500}{5 \pi} \]
\[ r^2 = \frac{300}{\pi} \]
Now, take the square root of both sides to find \( r \):
\[ r = \sqrt{\frac{300}{\pi}} \]
Now, let's evaluate the responses from the options given:
-
The radius is the same as a cylinder with a volume of 500/3 and the same height.
- Not correct since the cone's volume is 500, not 500/3.
-
The radius is the same as a cylinder with the same volume and same height.
- True, the radius of the cone would be different from the cylinder's radius but the condition of having the same volume and height is true.
-
The radius is the same as a cylinder with a volume of 500(3) and same height.
- Not correct since it would imply the cylinder's volume is 1500 which is not accurate.
-
The radius is the same as a cylinder with the same volume and height of 3.
- Not correct; the height comparison is incorrect.
Thus, the most appropriate response is:
The radius is the same as a cylinder with the same volume and same height.