Cylinder in a cone A right circular cylinder is placed inside a cone of radius R and height H so that the base of the cylinder lies on the base of the cone.


b. Find the dimensions of the cylinder with maximum lateral surface area (area of the curved surface).

Verified step by step guidance

1. Define the variables: Let the radius of the cylinder be r and the height be h. The lateral surface area of the cylinder is given by the formula A = 2πrh.
2. Establish the relationship between the dimensions of the cone and the cylinder. Use similar triangles to express h in terms of r, R, and H. The relationship can be derived as h = H(1 - r/R).
3. Substitute the expression for h into the lateral surface area formula to express A solely in terms of r: A = 2πr(H(1 - r/R)).
4. Differentiate the area function A with respect to r to find the critical points: dA/dr = 2π(H(1 - r/R) - Hr/R).
5. Set the derivative dA/dr equal to zero to find the value of r that maximizes the lateral surface area, and then solve for r.