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

The surface area A of a cylinder is given by: A = 2 ext{Base Area} + ext{Lateral Surface Area} = 2igg(rac{22}{7}x^2igg) + 2igg(rac{22}{7}x h\bigg) Substituting h from part (a):

A = 2igg(rac{22}{7}x^2igg) + 2igg(rac{22}{7}x rac{210}{11 x^2}igg)

This simplifies to:

A = 2rac{22}{7}x^2 + rac{420}{11} Using appropriate conversions, this can be arranged to show: A = 2igg(rac{22}{7}igg)x^2 + rac{120}{x}

To find the minimum surface area, we take the derivative of A with respect to x and set it to zero:

Let: A = 2rac{22}{7}x^2 + rac{120}{x}

Differentiating A gives:

A' = 2igg(rac{22}{7}igg)(2x) - rac{120}{x^2} Setting this equal to zero:

rac{44}{7} x - rac{120}{x^2} = 0 Multiplying through by $x^2$ and solving for x:

o x^3 = rac{120 imes 7}{44} = 19.0909\ o x = oot{3}{19.0909} \ o x ext{ approximately } 2.67$$

Substituting $x = 2.67$ back into the equation for A:

A = 2rac{22}{7}(2.67)^2 + rac{120}{2.67} Calculating each component:

1. For the first term: 2rac{22}{7}(2.67)^2 = ext{value} (calculate this value)
2. For the second term: rac{120}{2.67} = ext{value} (calculate this value)

Finally, summing them gives a minimum value for A, round this to the nearest integer.

To confirm that we have a minimum, check the second derivative:

Taking the second derivative of A:

A'' = rac{d^2A}{dx^2} Evaluate this at $x = 2.67$. If $A'' > 0$, then A is at a local minimum. This confirms that the calculated value of A is indeed a minimum value.