Question

The perimeter of a rectangular metallic sheet is 300 cm. It is rolled along one of its sides to form a cylinder. Find the dimensions of the rectangular sheet so that the volume of the cylinder so formed is maximum.

Hint:

For optimization problems, write the expression for the quantity to be optimized, differentiate it, and use the second derivative test to confirm maxima or minima.

Answer 1

Step 1: Define the variables
Let the length of the rectangle be \( x \) cm and the breadth be \( (150 - x) \) cm (since the perimeter is \( 2(x + \text{breadth}) = 300 \)). 
When the rectangle is rolled along its length, the radius \( r \) and height \( h \) of the cylinder formed are: \[ 2\pi r = x \implies r = \frac{x}{2\pi}, \quad h = (150 - x). \] Step 2: Write the volume of the cylinder
The volume \( V \) of the cylinder is given by: \[ V = \pi r^2 h = \pi \left(\frac{x}{2\pi}\right)^2 (150 - x). \] Simplify: \[ V = \pi \frac{x^2}{4\pi^2} (150 - x) = \frac{x^2}{4\pi} (150 - x). \] Step 3: Differentiate \( V \) with respect to \( x \)
The derivative of \( V \) is: \[ \frac{dV}{dx} = \frac{1}{4\pi} \left(2x(150 - x) - x^2\right). \] Simplify: \[ \frac{dV}{dx} = \frac{1}{4\pi} \left(300x - 3x^2\right). \] Step 4: Set \( \frac{dV}{dx} = 0 \) to find critical points
\[ 300x - 3x^2 = 0 \implies x(300 - 3x) = 0 \implies x = 0 \text{ or } x = 100. \] Step 5: Use the second derivative test to confirm maxima
The second derivative is: \[ \frac{d^2V}{dx^2} = \frac{1}{4\pi} (300 - 6x). \] At \( x = 100 \): \[ \frac{d^2V}{dx^2} = \frac{1}{4\pi} (300 - 600) = \frac{-300}{4\pi} = \frac{-75}{\pi} < 0. \] Hence, \( V \) is maximum when \( x = 100 \). 
Step 6: Calculate the dimensions of the rectangle
When \( x = 100 \), the length is \( 100 \) cm, and the breadth is: \[ 150 - x = 50 \, \text{cm}. \]