Question

A rectangle of perimeter 36 cm is revolved around one of its sides to sweep out a cylinder of maximum volume.

Find the dimensions of the rectangle.

Hint:

Take the radius as x and the height as 18 - x, then maximise V = pi x^2 (18 - x) using derivatives.

Answer 1

Step 1: Set up the variables:
Let the rectangle have length \(l\) cm and breadth \(b\) cm.
The perimeter is 36 cm, so \(2(l+b) = 36\), which gives \(l + b = 18\).
Revolve the rectangle about the side of length \(b\). Then the other side \(l\) becomes the radius, and \(b\) becomes the height of the cylinder.
So \(r = l\) and \(h = b = 18 - l\).

Step 2: Write the volume as a function of one variable:
The volume of a cylinder is \(V = \pi r^{2}h\).
\[ V(l) = \pi l^{2}(18 - l) = \pi\,(18l^{2} - l^{3}), \quad 0 < l < 18 \]

Step 3: Find the critical point:
\[ \frac{dV}{dl} = \pi\,(36l - 3l^{2}) = 3\pi l\,(12 - l) \]
Put \(\dfrac{dV}{dl} = 0\). Since \(l \ne 0\), we get \(l = 12\).

Step 4: Apply the second derivative test:
\[ \frac{d^{2}V}{dl^{2}} = \pi\,(36 - 6l) \]
At \(l = 12\), this is \(\pi(36 - 72) = -36\pi < 0\).
So the volume is maximum at \(l = 12\).

Step 5: Find the dimensions:
The radius is \(l = 12\) cm, so the side that stays as the radius is 12 cm.
The other side is \(b = 18 - 12 = 6\) cm.
The maximum volume is \(\pi (12)^{2}(6) = 864\pi\) cm\(^{3}\).

Final Answer:
The rectangle measures 12 cm by 6 cm. It is revolved about the 6 cm side, so the cylinder has radius 12 cm and height 6 cm.\[ \boxed{12 \text{ cm} \times 6 \text{ cm}} \]