Given a perimeter of 180, how do you find the length and the width of the rectangle of maximum area?

1 Answer
Sep 24, 2016

Given a perimeter of 180, the length and width of the rectangle with maximum area are 45 and 45.

Explanation:

Let #x=# the length and #y=# the width of the rectangle.
The area of the rectangle #A =xy#

#2x+2y=180# because the perimeter is #180#.

Solve for #y#
#2y=180-2x#
#y=90-x#

Substitute for #y# in the area equation.
#A=x(90-x)#
#A=90x-x^2#

This equation represents a parabola that opens down. The maximum value of the area is at the vertex.

Rewriting the area equation in the form #ax^2+bx+c#
#A=-x^2+90xcolor(white)(aaa)a=-1, b=90, c=0#

The formula for the #x# coordinate of the vertex is
#x=(-b)/(2a)= (-90)/(2*-1)=45#

The maximum area is found at #x=45#
and #y=90-x=90-45=45#

Given a perimeter of 180, the dimensions of the rectangle with maximum area are 45x45.