Q.Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.

Concept: Optimization Word Problem — maximize curved surface area of a cylinder inscribed in a cone using similar triangles and differentiation.

Let the cone have base radius $r$ and height $h$. Let the cylinder have radius $x$ and height $y$. From similar triangles $\triangle QEC \sim \triangle AOC$:

Curved surface area of cylinder: $S = 2\pi x y = 2\pi x \cdot \frac{h(r - x)}{r} = \frac{2\pi h}{r} (rx - x^2)$.

Differentiate $S$ w.r.t. $x$ and set to zero: …

This is a classic optimisation problem where a geometric constraint (the cylinder must fit inside the cone) links the two variables — radius and height — so we can write everything in terms of one variable. The curved surface area of a cylinder is $2\pi \times \text{radius} \times \text{height}$, so once height is expressed in terms of radius, we have a single-variable function to maximise.

Let the cone have base radius $r$ and height $h$. Let the inscribed cylinder have radius $x$ and height $y$. The cylinder sits with its base on the cone’s base and its top rim touching the cone’s sloping surface.

1. Relate $y$ to $x$ using similar triangles. Look at a vertical cross-section through the axis. The cone appears as a triangle with base $2r$ and height $h$. The cylinder appears as a rectangle of width $2x$ and height $y$, with its top corners touching the sloping sides of the triangle. Consider the smaller triangle formed by the cone’s apex, the top edge of the cylinder, and the axis. Its height is $h - y$ and its base radius is $x$. This triangle is similar to the whole cone triangle (height $h$, base radius $r$). Hence:

Solving for $y$:

2. Write the curved surface area $S$ as a function of $x$. The curved surface area of a cylinder is $2\pi x y$. Substituting $y$:

The domain is $0 < x < r$ (the cylinder must be inside the cone, not degenerate).

3. Maximise $S(x)$ using differentiation. Differentiate with respect to $x$:

Set $S'(x) = 0$: …

Method: Using Similar Triangles to Reduce a Solid-Geometry Optimization to One Variable

The standard technique whenever a smaller solid is inscribed inside a larger one (like a cylinder inside a cone) and the two share a linear geometric constraint.

Steps

Step 1: Draw the axial cross-section and identify the similar triangles

Slicing the solid through its axis turns the cone into a triangle and the inscribed cylinder into a rectangle whose top corners touch the triangle's slanted sides — this creates a smaller triangle (from the apex to the cylinder's rim) that is similar to the whole cone's triangle.

Step 2: Write the proportion from similar triangles

Solve this for the dependent variable (the cylinder's height) in terms of the independent one (its radius).

Step 3: Substitute into the quantity being optimised …

Common Mistakes

Mistake 1: Treating the cylinder's height as independent of its radius

Why it's wrong: the cylinder must fit exactly inside the cone, so its height is forced by its radius through the similar-triangles relation $y=\frac{h}{r}(r-x)$ — a student who tries to maximise $2\pi xy$ by differentiating with respect to both $x$ and $y$ separately, or by holding $y$ fixed, breaks this geometric constraint. Correct approach: always substitute the constraint first so the objective becomes a function of one variable before differentiating.

Mistake 2: Maximising the total surface area instead of the curved surface area …