Question

A farmer is building a fence to enclose a rectangular area consisting of two separate regions. The four walls and one

additional vertical segment (to separate the regions) are made up of fencing, as shown below.
1
If the farmer has 276 feet of fencing, what are the dimensions of the region which enclose the maximal area?
Select the correct answer helnur

Answer 1

46 feet by 69 feet

Explanation:

If the three equal vertical segments on the left, middle, and right of the region each have length L, and the segments on the top and bottom side of the region have length W, then we see that 3L+2W=276 (the total length of the fencing), so we find that

W=276−3L2

Therefore, in terms of the side length L, the area of the region is

L⋅(276−3L2)=276L−3L22

This is a quadratic function, where L is what we usually think of as x. To maximize the area we should find the x-value of its vertex. Recall that the x-value of the vertex of the parabola ax2+bx+c is given by the formula −b2a. So in this case, we have a=−32 and b=138, so the x-value of the vertex is

−b2a=−(138)2(−3/2)=46

So the optimal length of the left, middle, and right segments is 46, and the length of the top and bottom is

W=276−3(46)2=69