Straight River Question

Question

A farmer has 200 metres of fence and wants to enclose a rectangular field next to a straight river, using the river as one side of the enclosure. Since the river provides a natural boundary, fencing is only needed on the remaining three sides (two shorter sides and one longer side parallel to the river). What size will maximize the enclosed area? Also, what is the maximum area that can be enclosed?@Block Blast

Answer 1

Let x represent the two sides that are perpendicular to the river. Let y represent the length of the side parallel to the river. The amount of fencing is then 2x + y. Since we know that the fencing available is 200 meters, we have this constraint on x and y:

2x + y = 200

Solve that equation for y to get y in terms of x:

y = 200 - 2x

The area of the rectangle is A=LW or in this case A = xy. Since we have an expression for y in terms of x we can represent the area depending on x only:

A = x(200 - 2x)

A = -2x² + 200x

This is a quadratic function so its graph is a parabola. Since its leading coefficient (-2) is negative the parabola opens downward. That means the coordinates of its vertex give the required information. To find the coordinates of the vertex there are a few methods available. One method is to realize that the x coordinate of the vertex is located along its axis of symmetry and that the equation of the axis of symmetry is given by x = -b/2a where a = (-2) and b = (200).

x = -200/2(-2) = -200/-4 = 50

When x = 50, y = 200 - 2(50) = 100 meters

A(50) = -2(50²) + 200(50) = -5000 + 10000 = 5000 m²

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