To find the radius of a cone given its volume and height, you can 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 cone's base, [tex]\( h \)[/tex] is the height of the cone, and [tex]\( \pi \)[/tex] is a constant (approximately 3.14159).
Given:
- Volume [tex]\( V = 735\pi \)[/tex] cubic millimeters
- Height [tex]\( h = 5 \)[/tex] millimeters
1. Start with the volume formula:
[tex]\[ 735\pi = \frac{1}{3} \pi r^2 \times 5 \][/tex]
2. Isolate [tex]\( r^2 \)[/tex] by first multiplying both sides of the equation by 3 to eliminate the fraction:
[tex]\[ 3 \times 735\pi = \pi r^2 \times 5 \][/tex]
[tex]\[ 2205\pi = 5\pi r^2 \][/tex]
3. Divide both sides by [tex]\( 5\pi \)[/tex] to solve for [tex]\( r^2 \)[/tex]:
[tex]\[ \frac{2205\pi}{5\pi} = r^2 \][/tex]
[tex]\[ \frac{2205}{5} = r^2 \][/tex]
[tex]\[ 441 = r^2 \][/tex]
4. Take the square root of both sides to find [tex]\( r \)[/tex]:
[tex]\[ r = \sqrt{441} \][/tex]
[tex]\[ r = 21 \][/tex]
Thus, the radius of the cone is 21 millimeters.
