optimization

Question

Jane wants to build a rectangular shaped dog pen. She has 500 feet of fencing and she wants to make the area as large as possible.

a. What should the dimensions be to maximize the area?

b. Could Jane enclose more area if she built a circular dog pen?

Answer 1

The perimeter of the rectangle is:

500 = 2L + 2W

500 - 2L = 2W

 

250 - L = W

 

The area of the rectangle is:

Area = L*W

Area = L*(250-L) = -L² +250L        [Substitute 250-L for W)

 

To find the maximum area, take the derivative wrt L and set it to zero:

d(Area)/dL = -2L + 250 = 0

250 = 2L

 

125 = L

 

Solve for W:

W = 250 - L = 250 - 125 = 125

 

To maximize the  enclosed area, Jane should build a square enclosure 125 feet on a side.  The enclosed area would be 125*125 = 15,625 square feet

 

For a circular dog pen:

Circumference = 2(pi)r = 500

r = 500/2(pi)

 

The area enclosed = (pi)r² = (pi)*500²/4(pi)² = 250000/4(pi) = 19,894.4 square feet

 

Yes, a circular area pen would enclose more area