Question

A rectangle of perimeter $24$ inches is rotated about one of its sides so as to form a cylinder. To attain the maximum volume of the cylinder what would be the dimensions of the rectangle (in inches)

• A. $8,4$
• B. $4,4$
• C. $8,8$
• D. None of these

Answer 1

The correct option is A

$8,4$

Explanation for the correct option:

Step 1. Find the dimensions of the rectangle:

Given $x+y=12$

$\Rightarrow$ $y=12-x$

As we know,

Volume of the cylinder, $\mathit{V}\mathbf{}\mathbf{=}\mathbf{}\mathit{\pi }{\mathit{x}}^{\mathbf{2}}\mathbf{y}$

Put the value of $y$

$\Rightarrow$$V=\pi x^2\left(12-\mathrm{x}\right)$

$\Rightarrow$$V=12{\mathrm{\pi x}}^2-{\mathrm{\pi x}}^3$

Step 2. Differentiate it with respect to $x$

$\frac{dV}{dx}=24{\mathrm{\pi x}}^{}-3{\mathrm{\pi x}}^2$

Now, If $\frac{dV}{dx}=0,x=0,8$

$\Rightarrow x=8$

$\Rightarrow \frac{d^{2}V}{d{x}^2}=24\mathrm{\pi }-6\mathrm{\pi x}$

$\Rightarrow \frac{d^{2}V}{d{x}^2}<0$ at $x=8$

$V$ is maximum if $x=8$

$\therefore$For maximum volume $\mathit{x}\mathbf{}\mathbf{=}\mathbf{}\mathbf{8}$ and $\mathit{y}\mathbf{}\mathbf{=}\mathbf{}\mathbf{4}$

Hence, Option ‘A’ is Correct.