Q.Find the dimensions of the rectangle of perimeter $36$ cm which will sweep out a volume as large as possible when revolved about one of its sides. Also find the maximum volume.

Key idea: Revolving the rectangle about one side gives a cylinder — that side is the height, the other side is the radius. Maximise its volume under the perimeter constraint.

Let the sides be $x$ (the axis of revolution) and $y$. Perimeter: $2(x+y)=36\Rightarrow x+y=18$.

Revolving about the side $x$ gives a cylinder of radius $y$ and height $x$:

Differentiate: $V'(y)=\pi(36y-3y^2)=3\pi y(12-y)$. So $V'=0\Rightarrow y=12$ (reject $y=0$). Then $x=18-12=6$. …

Picture the solid

When a rectangle is revolved about one of its sides, that side becomes the axis (the height of a cylinder) and the perpendicular side sweeps out the radius. So the solid is a cylinder with

Step 1 — set up variables and the constraint

Let the rectangle have sides $x$ and $y$, and revolve it about the side of length $x$. The perimeter is fixed:

Revolving about $x$: radius $=y$, height $=x$.

Step 2 — volume as one variable

Step 3 — differentiate and find the critical point

Setting $\dfrac{dV}{dy}=0$ gives $y=0$ (degenerate, zero volume) or $y=12$. So the useful critical value is $y=12$ cm, and then $x=18-12=6$ cm.

Step 4 — confirm it is a maximum …

Method: Optimizing a Solid of Revolution Generated from a Fixed-Perimeter Shape

When a two-dimensional shape with a fixed perimeter is revolved about one of its own sides, that side becomes the axis (height) of the resulting solid, and the perpendicular side becomes the radius — recognising this geometric translation is the first and most important step.

Steps

Step 1: Identify which side becomes the axis and which becomes the radius

Revolving a shape about a side turns that side into the solid's height (or axis length), while the side perpendicular to it sweeps out the radius. Assign variables to both sides of the original shape accordingly.

Step 2: Write the fixed-perimeter constraint and the volume formula for the resulting solid

Use the given perimeter to write a linear relation between the two side-variables. Write the volume of the resulting solid (e.g. $V = \pi r^2 h$ for a cylinder) in terms of both variables.

Step 3: Reduce to one variable, differentiate, and solve …

Common Mistakes

Mistake 1: Swapping which side becomes the radius and which becomes the height

When a rectangle is revolved about one of its sides, that side becomes the cylinder's axis (its height), and the other side sweeps out the radius. Why it's wrong: revolving about the side of length $x$ but then writing $V=\pi x^2y$ (treating $x$ as the radius instead of the height) gives a completely different, wrong volume function and a different critical point. Correct approach: explicitly state which side is the axis before writing $V=\pi(\text{radius})^2(\text{height})$, and keep that assignment consistent through differentiation.

Mistake 2: Not checking that $y=0$ is a degenerate, rejected critical point …