optimizing area

Question

Donna has 320 meters of fencing and wishes to form three sides of a rectangular field. The fourth side borders a river and will not need fencing. One of the sides has length x (in meters).

(a) Find a function that gives the area Ax of the field (in square meters) in terms of x.

(b) What side length x gives the maximum area that the field can have? (in meters)

(c) What is the maximum area that the field can have? (in square meters)

Answer 1

the maximum area is always a square

320/4=80

each side would ordinarily be 80 m

since one side is not used, put the 80 m not used with the 80 m on the opposite side to get 160 m

you have a rectangle 80 by 160 by 80 by "the river"

the area is A=80*160=12,800 sq m

if you need algebra work...

P=l+2w and A=lw

320=l+2w and l=320-2w

A=lw

A=(320-2w)w

A=320w-2w^2

-b/2a=-320/-4=80 again