You are assigned a task to a rectangular cubicle with four w

You are assigned a task to a rectangular cubicle with four walls. The cubicle will be designed as in the following diagram, where the wall on the top of the diagram is exactly half as long as the wall on the bottom of the diagram. A door will be placed for the other half of the wall in order to enclose the cubicle, but no wall will be used there. If you have 250 feet of cubicle wall, what is the largest you can make the area of the cubicle? The largest area is square feet. If necessary, round to two decimal places.

Solution

Let length of wall be l

Let width of wall be w

Perimeter of wall = l +2w +l/2 = 3l/2 +2w

3l/2 + 2w = 250

3l/2 = (250 -2w)

l = 2(250 -2w)/3

Area = l*w = 2w(250 -2w)/3 = 500w/3 - 4w^2/3

Maximum are occurs at vertex : w= -b/2a = - (500/3 /2(-4/3)) = 500/8 = 62.5 ft

Area = 500*62.5/3 - 4*62.5^2/3

= 5208.33 square feet