Question

A farmer wishes to fence off the maximum area possible to make a rectangular field. He has 150 meters of fencing. One side of the land borders a river. Find the maximum area of the field.

Answer 1

Answer: Maximum Area of the field is 5625m²

Step-by-step explanation: If one side is a river, there is no need to fence this side. So, calling the sides of the retangule x and y, we know that the perimeter of the fence is the sum of all sides that will have the fence, so,  2x+y = 150.

We want maximum area, and area is xy

A = xy

2x + y = 150

y = 150 - 2x

A = x(150 - 2x)

To find the maximum area, we need de vertex of this parabola.

x(150 - 2x) = 0

150x - 2x² = 0

75x - x² = 0

a=-1 b=75 c=0

For the vertex: Vy = -Δ/4a

Δ = b² - 4ac = 75² - 4.-1.0 = 75² = 5625

Vy = -Δ/4a = -5625/4.(-1) = 5625