Suppose that you have 360 feet of rope

Question

Suppose that you have 360 feet of rope and want to use it to make a rectangle. What dimensions should you make your rectangle if you want to enclose the maximum possible area?

Answer 1

L is the length

w is the width

 

P = 360 = 2L + 2w 

      360 = 2 ( L + w) <--- alternative formula for perimeters factors out the 2

      180 = L + w  <--- divides both sides by 2

      180 - w = L

 

A = L * w =  (180 - w)w

 

    Area function in terms of width:

   Area A(w) = (180 -w ) w

 

  The max occurs at hte AVERAGE of the solutions.

 The solutions are (180 - w)w = 0

                             180 - w  = 0  or w=0

                             w=180 or w=0

 

THe average of the solutions is (180+0)/2 = 180/2 = 90

 

The max area occurs with dimensions width=90 and length L=180 - 90 = 90

 The max area is 90*90 = 8100

 Note that is it actually a square, not a rectangle