What is the maximum possible area of the corral?

Question

The back of Dantes property is a creek. Dante would like to enclose a rectangular area using the creek as one side and fencing for the other three sides to create a corral. If there is 420 feet of fencing available what is the maximum possible area of the corral?

Answer 1

using arithmetic...

a square always gives you the maximum area

420/4=105

you want a square 105 feet by 105 feet

however, you are using the creek as one side so take the 105 feet that you don't need because of the creek and add it to the opposite side to get 210 feet

you have a rectangle 105 ft by 210 ft by 105 ft by "the creek"

area=l*w

A=105*210

A=22,050 square feet is the maximum area

 

using algebra...

P=2w+l

420=2w+l

420-2w=l

A=lw

A=(420-2w)(w)

A=420w-2w²

0=420w-2w²

this is a parabola that opens downward so the vertex (h,k) is the maximum point

h=-b/2a

h=-420/-4

h=105 feet

you don't need 105 feet by the creek so add this 105 feet to the opposite 105 feet to get 210 feet

A=105*210

A=22,050 square feet is the maximum area and also k from (h,k) is equal to 22,050

so (h,k)=(105, 22,050)

notice that the 105 by 210 rectangle is really two 105 by 105 squares side by side