To determine the slant height of a right circular cone given its volume and height, we can follow these steps:
1. Identify Given Values:
- Volume of the cone, [tex]\( V = 100π \, \text{cm}^3 \)[/tex]
- Height of the cone, [tex]\( h = 12 \, \text{cm} \)[/tex]
2. Formula for the Volume of a Cone:
The formula for the volume of a right circular cone is:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
where [tex]\( r \)[/tex] is the radius of the base of the cone.
3. Rearrange the Volume Formula to Find the Radius [tex]\( r \)[/tex]:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
Solving for [tex]\( r^2 \)[/tex]:
[tex]\[
100 \pi = \frac{1}{3} \pi r^2 \cdot 12
\][/tex]
[tex]\[
100 \pi = 4 \pi r^2
\][/tex]
Divide both sides by [tex]\( 4 \pi \)[/tex]:
[tex]\[
r^2 = \frac{100 \pi}{4 \pi} = 25
\][/tex]
Taking the square root of both sides:
[tex]\[
r = \sqrt{25} = 5 \, \text{cm}
\][/tex]
4. Determine the Slant Height [tex]\( l \)[/tex] Using the Pythagorean Theorem:
The slant height [tex]\( l \)[/tex] forms the hypotenuse of a right triangle where the other two sides are the radius [tex]\( r \)[/tex] and the height [tex]\( h \)[/tex]. By the Pythagorean theorem:
[tex]\[
l^2 = r^2 + h^2
\][/tex]
Substitute the known values:
[tex]\[
l^2 = 5^2 + 12^2
\][/tex]
[tex]\[
l^2 = 25 + 144
\][/tex]
[tex]\[
l^2 = 169
\][/tex]
Taking the square root of both sides:
[tex]\[
l = \sqrt{169} = 13 \, \text{cm}
\][/tex]
Therefore, the slant height of the cone is [tex]\( 13 \, \text{cm} \)[/tex].
