Question

Prove that the radius of the right circular cylinder of maximum curved surface inscribed in a cone is half of the radius of the cone.

Hint:

To solve optimization problems involving inscribed shapes, use geometric relations and optimization techniques such as differentiation to maximize the area or volume.

Answer 1

Step 1: Consider a cone with a base radius \( r \) and height \( h \). A cylinder inscribed in the cone has a radius \( r_1 \) and height \( h_1 \). 

Step 2: The volume of the cylinder is maximized when its surface area is maximized. The curved surface area of the cylinder is given by: \[ A = 2 \pi r_1 h_1 \] where \( r_1 \) and \( h_1 \) depend on the geometry of the cone.

 Step 3: Using the geometric relations between the cone's dimensions and the cylinder's dimensions, you can show that the radius of the cylinder at maximum surface area is \( r_1 = \frac{r}{2} \). Thus, the radius of the cylinder of maximum curved surface is half the radius of the cone.