A farmer has 200m of wire fencing from which to build a rectangular enclosure. He intends to use an existing wall for one of the sides.

Find the dimensions that will result in a maximum enclosed area.

Wall

Length = x metres

Breadth = y metres

To solve this problem, we need to determine the dimensions of the rectangular enclosure that will give the maximum area, using 200 meters of wire fencing for three sides of the rectangle since the fourth side is an existing wall.
Here are the steps to solve the problem:

Understand the Variables:


Let x be the length of the rectangle along the existing wall.

Let y be the breadth of the rectangle, perpendicular to the wall.



Formulate the Perimeter:


Since three sides of the rectangle will use the wire fencing, the expression for the total length of wire used is: x + 2 y = 200

Solve for x :
x = 200 − 2 y



Write the Area Function:


The area A of the rectangle is given by: A = x × y

Substitute the expression for x from the perimeter equation: A = ( 200 − 2 y ) × y = 200 y − 2 y 2



Maximize the Area:


To find the maximum area, we need to take the derivative of the area function with respect to y , and set it to zero: d y d A ​ = 200 − 4 y 200 − 4 y = 0

Solve for y :
4 y = 200 y = 50



Find x :


Substitute y = 50 back into the expression for x :
x = 200 − 2 × 50 = 100


Conclusion:


Therefore, the dimensions that give the maximum enclosed area are a length x = 100 meters along the existing wall and a breadth y = 50 meters.

The maximum area is A = 100 × 50 = 5000 square meters.


By following these steps, we find that the maximum area of the rectangle can be achieved with these dimensions.

Answered by EmmaGraceJohnson | 2025-07-06