6.1 - 2016 Q9
Diagram not
drawn to scale
Figure 4 shows a plan view of a sheep enclosure.
The enclosure ABCDEA, as shown in Figure 4, consists of a rectangle BCDE joined to an equilateral triangle BFA and a sector FEA of a circle with radius x metres and centre F.
The points B, F and E lie on a straight line with FE = x metres and 10 ≤ x ≤ 25
The enclosure ABCDEA, as shown in Figure 4, consists of a rectangle BCDE joined to an equilateral triangle BFA and a sector FEA of a circle with radius x metres and centre F.
The points B, F and E lie on a straight line with FE = x metres and 10 ≤ x ≤ 25
(a) Find, in m2, the exact area of the sector FEA, giving your answer in terms of x, in its
simplest form. (2)
Given that BC = y metres, where y = 0, and the area of the enclosure is 1000 m2,
(b) show that (3)
Given that BC = y metres, where y = 0, and the area of the enclosure is 1000 m2,
(b) show that (3)
(c) Hence show that the perimeter P metres of the enclosure is given by
(d) Use calculus to find the minimum value of P, giving your answer to the nearest metre. (5)
(e) Justify, by further differentiation, that the value of P you have found is a minimum. (2)
(e) Justify, by further differentiation, that the value of P you have found is a minimum. (2)