Question

The owner of the Rancho Grande has 2,956 yd of fencing with which to enclose a rectangular piece of grazing land situated along the straight portion of a river. If fencing is not required along the river, what are the dimensions (in yd) of the largest area he can enclose? A rectangular piece of land has been enclosed along a straight portion of a river. The enclosure is bordered by the river (a long side), another long side of fence, and two short sides of fence.

Answer 1

To maximize the enclosed area with 2,956 yd of fencing along a river, the owner can create a three-sided rectangle with dimensions 739 yd by 1,478 yd.

Step-by-step explanation:

To find the dimensions of a rectangular piece of grazing land that can be enclosed with 2,956 yd of fencing when the land is situated along a river, we can make use of the formula for the perimeter of a rectangle. Since the one side of the rectangle is along the river, the owner will only need to fence three sides: two widths (W) and one length (L), giving us the equation 2W + L = 2,956 yd.

The largest enclosed area can be obtained by creating a shape that is closest to a square since for a given perimeter, the square has the maximum area among rectangles. Here, however, since one side is along the river, we are dealing with optimizing a three-sided enclosure to maximize the area A = W * L. To do this, we can express L as L = 2,956 yd - 2W and then plug this into the area formula:

A = W * (2,956 yd - 2W)

To find the value of W that maximizes the area, we take the derivative of A with respect to W and set it equal to zero:

dA/dW = 2,956 yd - 4W = 0

Solving for W gives us W = 739 yd. Substituting W back into the equation for L gives us L = 1,478 yd. So the dimensions of the largest area he can enclose are 739 yd by 1,478 yd.