Question

you are the owner of a rectangular orchard adjacent to a straight river. you have 1000 ft of fence that you want to use to enclose it. no fencing is required along the river. if x is the length of a side perpendicular to the river and y is the length of the side parallel to the river, find the values of x and y that will maximize the enclosed area.

Answer 1

The optimal dimensions of the rectangular orchard to maximize the enclosed area with 1000 ft of fencing, with one side along the river, are 250 ft for the width (x) and 500 ft for the length (y).

Step-by-step explanation:

To find the values of x and y that will maximize the enclosed area of a rectangular orchard using only 1000 ft of fencing, and with one side along a river (hence no fence required on that side), we must use optimization techniques.

First, we express the perimeter that requires fencing in terms of x and y: Perimeter = 2x + y. Since we have 1000 ft of fence, we set this equal to 1000, thus: 2x + y = 1000.

Next, we rearrange the equation to solve for y: y = 1000 - 2x.

Now we can express the area A of the orchard as a function of x: A = x * y, which becomes A(x) = x(1000 - 2x) = 1000x - 2x^2.

To maximize the area, we take the derivative of A(x) with respect to x and set it to zero: A'(x) = 1000 - 4x. Setting this equal to zero gives: 1000 - 4x = 0; thus, x = 250 ft.

Substituting back into the equation for y, we find: y = 1000 - 2(250), therefore, y = 500 ft.

Therefore, for maximum area, the length x perpendicular to the river should be 250 ft and the length y parallel to the river should be 500 ft.