What is the maximum area of a rectangular yard that can be enclosed with 120m of fencing used on three sides?

Question

(1 mark for using correct perimeter formula, 1 mark for finding correct width, 1 mark for finding correct length, 1 mark for finding correct area, 0.5 mark for including a 'therefore' statement, 0.5 mark for including correct units)

Answer 1

Let x = one of the two parallel sides and y = the remaining side.

Perimeter, p = 2x + y = 120, y = 120-2x

Area, A = xy = 120x-2x² m²

The area equation is an inverted parabola. The vertex of the parabola gives the maximum area.

The vertex x value = -b/(2a) where b = 120 and a = -2

x = -120/(-2a) = 120/4 = 30 m

y = 120 - 2x = 120 -60 = 60 m

Therefore, the maximum area = 30(60) = 1800 m²