Question

A cylinder is inscribed in a cone as shown in the figure. The radius of the cone is 3 inches and the height is 7 inches. Express the volume, V, of the cylinder in terms of its height, h. Hint: Ratios for similar right triangles will be helpful. V(h)=_______

Answer 1

The volume of the cylinder, V, in terms of its height, h, inside a cone with a height of 7 inches and radius of 3 inches is V(h) = π(9/49)h³.

Step-by-step explanation:

To express the volume of the cylinder in terms of its height, we need to use the properties of similar right triangles formed by the radius of the cylinder, its height, and the slant height of the cone.

Since the height of the cone is 7 inches and the radius is 3 inches, the ratio of the radius of the cylinder to its height will be the same as the ratio 3:7. If the height of the cylinder is h, then its radius will be (3/7)h.

The volume, V, of the cylinder can be calculated using the formula for the volume of a cylinder, V = πr²h. Substituting the expression for the radius in terms of h we get:

V = π(3/7h)²h = π(9/49h²)h = π(9/49)h³.

So, the volume of the cylinder in terms of its height h is V(h) = π(9/49)h³.