Afarmer is planning a rectangular area for her chickens. the area of the rectangle will be 200 square feet. three sides of the rectangle will be formed by fencing, which cost 5$ per foot, the fourth side of the rectangle will be formed by a portion of the barn wall, which requires no fencing. in order to minimize the cost of fencing, how long should the fourth wall be?

Let

x---------> the length side of the rectangular area

y---------> the width side of the rectangular area

we know that

the area of the rectangle is equal to

-----> equation

The perimeter of the rectangle is equal to

but remember that the fourth side of the rectangle will be formed by a portion of the barn wall

so

-----> equation

To minimize the cost we must minimize the perimeter

Substitute the equation in the equation

Using a graph tool

see the attached figure

The minimum of the graph is the point

that means for

the perimeter is a minimum and equal to

Find the value of y

The cost of fencing is equal to

therefore

the answer is

the length side of the the fourth wall will be

check the picture below.

well, the angle at the vertex d is clearly not a right-angle, let's check for the one at the vertex f which does look like,   however if it's indeed, that means ef is perpendicular to df, in which case their slopes will be negative reciprocal of each other.

so the negative reciprocal of the slope of ef is 1, however the slope of df is not 1, is 1/2, so nope, they're not perpendicular lines.