Answer:
Volume of cylinder(V) is given by:
[tex]V = \pi r^2h[/tex]
where,
r is the radius and
h is the height of the cylinder.
Volume of cone(V') is given by:
[tex]V'=\frac{1}{3} \pi r'^2h'[/tex]
where, r' is the radius and h' is the height of the cone.
As per the statement:
A cylinder with a radius of 12 cm and a height of 20 cm has the same volume as a cone with a radius of 8 cm
⇒r = 12 cm , h = 20 cm and r' = 8 cm
Since, Volume of cylinder is equal to Volume of cone.
⇒V = V'
then;
[tex]\pi r^2h = \frac{1}{3} \pi r'^2h'[/tex]
⇒[tex]r^2h = \frac{1}{3}r'^2h'[/tex]
Substitute the given values to solve for h' we have;
[tex]12^2 \cdot 20 = \frac{1}{3} \cdot 8^2 \cdot h'[/tex]
⇒[tex]144 \cdot 20 = \frac{1}{3} \cdot 64 \cdot h'[/tex]
Multiply both sides by 3 we have;
⇒[tex]144 \cdot 60=64 \cdot h'[/tex]
⇒[tex]8640=64 \cdot h'[/tex]
Divide both sides by 64 we have;
[tex]135 = h'[/tex]
or
h' = 135 cm
Therefore, the height of the cone is, 135 cm
