Solve the word problem

Question

you have 84 feet of fencing to enclose a rectangular region. Find the dimensions of the rectangle that maximize the enclosed area.

 

to solve  use the quadratic function's minimum or maximum value.  

Answer 1

Let l and w be the length and width of the rectangle.

 

the perimeter is 84 so 2l +2w =84.   solving for w yields

 

w = 42 - l.

 

The area A of the rectangle is l*w.  So A = l(42-l) = 42l - l²

 

The equation for the area is an upside down parabola, so its vertex is represents the maximum area.   Use the technique of completing the square to put the equation in format to spot vertex

 

A + 21² = 21² + 42l -l²

 

A + 441 = -1(l-21)²

 

With the equation in the above form, we can see that the vertex is at (21, 441) so that maximum area is 441 square feet.  The length of the rectangle is 21, and the width is 42-21 = 21.   The rectangle having the largest area is a 21 feet by 21 feet square.