Fencing Area Question

Question

You have 200 feet of fencing to enclose a rectangular plot that borders a river. If you do not fence the side alone the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

 

a. Describe the variable in words.

b. Write the area as a function of x to be maximized.

c. Determine the length and width by maximizing the area function. Show all algebraic work.

d. What is the largest area that can be enclosed?

Answer 1

P=2l+2w

because of the river there is only one length

P=l+2w

200=l+2w

l=200-2w

A=lw

A=(200-2w)w

0=200w-2w^2

this is a parabola that opens downward so there is a maximum point, the vertex of the parabola

the point is (h,k) where h is the "maximizing" number and k is the maximum area

using y=a(x-h)^2+k where a<0...

if you expand this h=-b/2a

-b/2a=-200/-4=50

you have a square that is 50 by 50 by 50 by 50

however, you don't need fencing along the river so add that 50 to the opposite side of 50

now you have a rectangle that is 50 by 100 by 50 by "the river"

the maximum area is A=lw, A=50*100, A=5000 square feet