Question

A rectangular bird sanctuary is being created with one side along a straight riverbank. The remaining three sides are to be enclosed with a protective fence. If there are 28 km of fence available, find the dimension of the rectangle to maximize the area of the sanctuary.

Answer 1

Answer:

  The rectangle is 7 km by 14 km. The 14 km dimension is parallel to the river.

Step-by-step explanation:

Let x represent the length of fence parallel to the river. The remaining fence is divided into two equal pieces for the ends of the enclosure. Then (28 -x)/2 will be the length of the side of the rectangle perpendicular to the river.

The total area of the enclosure is the product of length and width:

  Area = (x)(28-x)/2

This expression describes a parabola opening downward with zeros at x=0 and x=28. The vertex (maximum) is halfway between those zeros, so is at ...

  x = (0 +28)/2 = 14

Area is maximized when the dimension parallel to the river is 14 km and the ends of the enclosure are 7 km.