SOLUTION: Rectangle R has varying length l and width w but with a constant perimeter of 4 feet. a)Express the area A as a function of length l Write a list of everything you know about thi

Algebra ->  Rectangles -> SOLUTION: Rectangle R has varying length l and width w but with a constant perimeter of 4 feet. a)Express the area A as a function of length l Write a list of everything you know about thi      Log On


   

Question 894736: Rectangle R has varying length l and width w but with a constant perimeter of 4 feet.
a)Express the area A as a function of length l
Write a list of everything you know about this function.
b)For what values of land w will the area of R be the greatest?
Give an algebraic explanation. Then give a geometric explanation. Be sure to include a graph with relevant points labeled.
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Length is L, width is w.

A=wL, and 2w%2B2L=4, the constant perimeter of 4 feet.
Simplify perimeter equation to w%2BL=2.

Part (a) asks for A(L).
from perimeter equation, w=2-L.
A=%282-L%29L
A=2L-L%5E2
Or if you want,
highlight%28A%28L%29=2L-L%5E2%29, area as a function of length.

The function A(L) is quadratic, and has a maximum value, based on coefficient on L%5E2 being -1.

The maximum value for area will be at L=w, meaning a square shape. This will simply be L=1 and w=1; and the area will be 1 square foot.

Part (b): you asked for an algebraic explanation but I did not give it that way. I simply gave a discussion.

Try this:
Find roots for A(L).
2L-L%5E2=0
L%282-L%29=0
L=0 or L=2.
The maximum value occurs in the exact middle of these roots.