To find the radius of the cone, 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 \( V = 500 , \text{cm}^3 \)
- Height \( h = 5 , \text{cm} \)
We can plug in the values we know:
\[ 500 = \frac{1}{3} \pi r^2 (5) \]
Next, we can solve for \( r^2 \):
\[ 500 = \frac{5}{3} \pi r^2 \]
Multiplying both sides by \( \frac{3}{5} \):
\[ \frac{3}{5} \times 500 = \pi r^2 \]
Calculating the left side:
\[ 300 = \pi r^2 \]
Now, we can solve for \( r^2 \):
\[ r^2 = \frac{300}{\pi} \]
Taking the square root to find \( r \):
\[ r = \sqrt{\frac{300}{\pi}} \]
Now we calculate \( r \):
Using \( \pi \approx 3.14 \):
\[ r \approx \sqrt{\frac{300}{3.14}} \approx \sqrt{95.5} \approx 9.77 , \text{cm} \]
Thus, the radius of the cone is approximately \( 9.77 , \text{cm} \).
Answer:
The radius is approximately \( 9.77 , \text{cm} \).