1.27: Figure 1 shows the sector of a circle with centre .

4PM1/2R/June/2023 — Question 3 · 10 marks

1.27 diagram 1
Figure 1 shows the sector OAB\displaystyle OAB of a circle with centre O\displaystyle O.
The radius of the circle is r\displaystyle r cm and the angle AOB\displaystyle AOB is θ\displaystyle \theta radians.
The area of the sector is 675cm2\displaystyle 675 \mathrm{cm}^2
(a) Show that the perimeter of the sector, P\displaystyle P cm, is given by
P=2r+1350rP = 2r + \frac{1350}{r}
(3)
Given that r\displaystyle r can vary,
(b) find, using calculus, the minimum value of P\displaystyle P
Give your answer in the form ab\displaystyle a\sqrt{b} where a\displaystyle a is an integer and b\displaystyle b is a prime number.
(5)
(c) Justify that the value of P\displaystyle P you found in (b) is a minimum.
(2)