A manufacturer produces pain relieving tablets - Edexcel - A-Level Maths Pure - Question 9 - 2012 - Paper 3

The surface area A of a cylinder consists of the areas of its two circular bases and the lateral surface area. The formula is:

$A = 2πx^2 + 2πxh$

Substituting h using our result from part (a):

$A = 2πx^2 + 2πx\left(\frac{60}{πx^2}\right)$

Simplifying:

$A = 2πx^2 + \frac{120}{x}$

Thus,

$A = 2πx^2 + \frac{120}{x}$

To find the minimum surface area, we differentiate A with respect to x:

$\frac{dA}{dx} = 4πx - \frac{120}{x^2}$

Setting the derivative equal to zero for optimization:

$4πx - \frac{120}{x^2} = 0$

Rearranging gives:

$4πx^3 = 120$

This simplifies to:

$x^3 = \frac{120}{4π}$

Thus,

$x = \sqrt[3]{\frac{30}{π}}$

Substituting the value of x back into the surface area formula:

First calculate:

$x ≈ 2.12$

Then substitute this back into:

$A = 2π(2.12)^2 + \frac{120}{2.12}$

Calculating gives:

$A ≈ 85.34$,

Thus, rounding to the nearest integer, the minimum value of A is:

$A ≈ 85$.

To confirm that A = 85 is a minimum, we check the second derivative:

$\frac{d^2A}{dx^2} = 4π + \frac{240}{x^3}$

This derivative is positive for all x > 0, implying that A has a local minimum at the calculated value. Thus, it can be stated that:

The minimum value of A occurs at x = 2.12 leading to A = 85.