Question

Farmer Ed has 8,000 meters of fencing & wants to enclose a rectangular plot that borders a river. If farmer Ed does not fence the side along the river, what is the largest area that can be enclosed ?

(answer is on pic but I just need a step by step explanation)

Answer 1

largest area that can be enclosed = 8,000,000 m²

Explanation:

Since it is a rectangle plot, the area is expressed as; A = xy, where x is length and y is width.

Because it is next to the river, he only needs to fence three sides, so amount of fencing; F = x + 2y.

Since we know the amount of fencing available is 8000m, we get:

8000 = x + 2y

solving for x, we have;

x = 8000 - 2y

substitute 8000 - 2y for x into the area equation to give;

A = (8000 - 2y)y distribute

A = -2y² + 8000y

Now, due to the negative sign next to 2, this will be a parabola which opens down, meaning that the point of maximum area will be at the vertex,

Thus; y = -b/2a = -8000/[2(-2)] = 2000

x = 8000 - 2(2000) = 4000

A = 4000(2000) = 8,000,000 m²