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