To find the radius of a right cone given its volume and height, we 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 base
- [tex]\( h \)[/tex] is the height
- [tex]\( \pi \)[/tex] is a constant (approximately 3.14159)
We are given:
- Volume, [tex]\( V = 2167 \, \text{units}^3 \)[/tex]
- Height, [tex]\( h = 18 \, \text{units} \)[/tex]
We need to solve for the radius [tex]\( r \)[/tex]. First, rearrange the volume formula to solve for [tex]\( r^2 \)[/tex]:
[tex]\[ V = \frac{1}{3} \pi r^2 h \][/tex]
Multiply both sides of the equation by 3 to get rid of the fraction:
[tex]\[ 3V = \pi r^2 h \][/tex]
Substitute the given values [tex]\( V = 2167 \)[/tex] and [tex]\( h = 18 \)[/tex]:
[tex]\[ 3 \times 2167 = \pi r^2 \times 18 \][/tex]
Simplify the left side:
[tex]\[ 6501 = \pi r^2 \times 18 \][/tex]
Divide both sides by [tex]\( 18\pi \)[/tex] to isolate [tex]\( r^2 \)[/tex]:
[tex]\[ r^2 = \frac{6501}{18\pi} \][/tex]
Now, calculate the value:
[tex]\[ r^2 = \frac{6501}{18 \times 3.14159} \][/tex]
First, calculate [tex]\( 18 \times 3.14159 \)[/tex]:
[tex]\[ 18 \times 3.14159 \approx 56.5487 \][/tex]
Then, divide [tex]\( 6501 \)[/tex] by [tex]\( 56.5487 \)[/tex]:
[tex]\[ r^2 = \frac{6501}{56.5487} \approx 115.0 \][/tex]
Finally, to find the radius [tex]\( r \)[/tex], take the square root of both sides:
[tex]\[ r = \sqrt{115.0} \approx 10.72 \, \text{units} \][/tex]
Therefore, the radius of the right cone is approximately [tex]\( 10.72 \)[/tex] units.
