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. Understand the problem: We need to find the dimensions of a cylinder with maximum lateral surface area that fits inside a cone with given radius R and height H. The lateral surface area of a cylinder is given by the formula: \( A = 2\pi rh \), where \( r \) is the radius and \( h \) is the height of the cylinder.
2. Set up the relationship between the cone and the cylinder: The cylinder is inscribed in the cone, so its dimensions are constrained by the cone's dimensions. The cone's equation in terms of its height and radius is \( \frac{r}{R} = \frac{H-h}{H} \), where \( r \) and \( h \) are the radius and height of the cylinder, respectively.
3. Express the height of the cylinder in terms of its radius: From the relationship \( \frac{r}{R} = \frac{H-h}{H} \), solve for \( h \) to get \( h = H - \frac{Hr}{R} \).
4. Substitute \( h \) in the lateral surface area formula: Replace \( h \) in \( A = 2\pi rh \) with the expression from the previous step to get \( A = 2\pi r(H - \frac{Hr}{R}) \). Simplify this to \( A = 2\pi rH - \frac{2\pi Hr^2}{R} \).
5. Find the maximum lateral surface area: To maximize \( A \), take the derivative of \( A \) with respect to \( r \), set it to zero, and solve for \( r \). This will give the radius of the cylinder that maximizes the lateral surface area. Use this \( r \) to find the corresponding \( h \) using the expression from step 3.

Key Concepts

Lateral Surface Area of a Cylinder

The lateral surface area of a cylinder is the area of the curved surface that connects the top and bottom bases. It can be calculated using the formula A = 2πrh, where r is the radius of the base and h is the height of the cylinder. Understanding this concept is crucial for determining how to maximize the surface area in the context of the problem.

Optimization in Calculus

Optimization involves finding the maximum or minimum values of a function within a given domain. In this problem, we need to apply techniques such as taking derivatives and setting them to zero to find the dimensions of the cylinder that maximize its lateral surface area. This concept is fundamental in calculus for solving real-world problems involving constraints.

Geometric Relationships in 3D Shapes

Understanding the geometric relationships between the cylinder and the cone is essential for this problem. The dimensions of the cylinder are constrained by the dimensions of the cone, specifically its radius R and height H. This relationship can be expressed using similar triangles, which helps in deriving the equations needed to optimize the lateral surface area.