The height of the cone is 12, the volume of the cone is 100п. Find the generatrix of the cone.

A cone is a geometric body formed by rotating a right-angled triangle around its leg.

The generatrix of the cone is a line segment connecting the vertex with the boundary of its base.

To calculate the generatrix, consider the axial section of the cone. An axial section is a plane passing through the axis of the cone. It has the shape of an isosceles triangle, in which the generator is the lateral side, and the diameter is the base.

Since the height of an isosceles triangle divides it into two right-angled triangles, we will use the Pythagorean theorem to calculate the generatrix:

L ^ 2 = r ^ 2 + h ^ 2.

To do this, calculate the radius of the base. Let’s use the cone volume formula:

V = 1 / 3πr ^ 2h;

r ^ 2 = 3V / πh;

r ^ 2 = 3 100π / 12π = 300π / 12π = 25;

r = √25 = 5 cm.

L ^ 2 = 5 ^ 2 + 12 ^ 2 = 25 + 144 = 169;

L = √169 = 13 cm.

Answer: the generatrix of the cone is 13 cm.