Problem 43 - Essential Calculus

Question

A cone of height \(H\) with a base of radius \(r\) is cut by a plane parallel to and \(h\) units above the base. Find the volume of the solid (frustum of a cone) below the plane.

Step-by-Step Solution

Answer

The volume of the frustum of the cone is given by \(V_{frustum} = 1/3 \pi r² H - 1/3 \pi (r \cdot {h/H})² (H-h)\).

Step 1: Calculate the Radius of the Frustum's Upper Base

The frustum is similar to the whole cone, so the ratio between dimensions is maintained. This gives us the ratio between the radii of the lower and upper bases of the frustum and the whole cone height. Thus, the radius \(r'\) of the upper base of the frustum is: \( r' = r \cdot {h/H} \)

Step 2: Calculate The Volumes Of The Cones

Now calculate the volumes of the whole cone and the small cone cut off by the plane using the formula for the volume of a cone \(V = 1/3 \pi r² H\). The volume of the whole cone is \(V_{whole} = 1/3 \pi r² H\), and the volume of the small cone is \(V_{small} = 1/3 \pi (r')² (H-h)\).

Step 3: Calculate The Volume Of The Frustum

The volume of the frustum is the volume of the whole cone minus the volume of the small cone. Thus: \(V_{frustum} = V_{whole} - V_{small} = 1/3 \pi r² H - 1/3 \pi (r')² (H-h)\). Substituting \(r'\) from Step 1 gives: \(V_{frustum} = 1/3 \pi r² H - 1/3 \pi (r \cdot {h/H})² (H-h)\)