To find the radius of a right cone when its volume and height are given, we use the formula for the volume of a cone:
\(\[ V = \frac{1}{3} \pi r^2 h \]\)
Where:
- \(\( V \)\) is the volume of the cone
- \(\( r \)\) is the radius of the base of the cone
- \(\( h \)\) is the height of the cone
- \(\( \pi \)\) is a constant (approximately 3.14159)
Given:
- The volume \(\( V \)\) is 700 cubic units
- The height \(\( h \)\) is 21 units
We need to solve for the radius \(\( r \)\).
1. Start by rearranging the volume formula to solve for \(\( r^2 \)\):
\(\[ r^2 = \frac{3V}{\pi h} \]\)
2. Plug the given values into the equation:
\(\[ r^2 = \frac{3 \times 700}{\pi \times 21} \]\)
4. Calculate the numerical value for \(\( r^2 \)\):
\(\[ r^2 \approx \frac{2100}{65.9734} \approx 31.831 \]\)
So, we find that:
\(\[ r^2 \approx 31.831 \]\)
5. Finally, take the square root of \(\( r^2 \)\) to find the radius \(\( r \)\):
\(\[ r \approx \sqrt{31.831} \approx 5.642 \]\)
Therefore, the radius of the cone is approximately 5.642 units.
