1.29: Optimising the perimeter of a badge

0606/12/M/J/16 — Question 10 · 9 marks

The diagram shows a badge, made of thin sheet metal, consisting of two semi-circular pieces, centres

B

and

C

, each of radius

x

cm. They are attached to each other by a rectangular piece of thin sheet metal,

ABCD

, such that

AB

and

CD

are the radii of the semi-circular pieces and

AD = BC = y

cm.

(i)

Given that the area of the badge is

20\text{ cm}^{2}

, show that the perimeter,

P

cm, of the badge is given by

P = 2x + \dfrac{40}{x}

[4]

(ii)

Given that

x

can vary, find the minimum value of

P

, justifying that this value is a minimum.

[5]