SOLUTION: Find the dimension of a rectangle that will yield the maximum area of its perimeter is 54 meters. What is the maximum area?

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Question 1205762: Find the dimension of a rectangle that will yield the maximum area of its perimeter is 54 meters. What is the maximum area?
Found 3 solutions by Edwin McCravy, Alan3354, ikleyn:
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!


Solve the first equation for L (Note: you could have solved it for W)

2L=54-2W
L=27-W

Substitute in 

A=%2827-W%29W

Use the product rule to differentiate with respect to W

dA%2FdW=%2827-W%29%281%29%2BW%28-1%29

dA%2FdW=27-W-W

dA%2FdW=27-2W

Se the derivative = 0, find maxima and minima

27-2W=0
27=2W
27%2F2=W

Substitute in

L=27-W
L=27-27%2F2
L=27%2Aexpr%282%2F2%29-27%2F2
L=54%2F2-27%2F2
L=27%2F2

Whaddyaknow? The length and width are both the same.
27/2 meters = 13.5 meters each. What kind of rectangle 
is that?

A=LW
A=expr%2827%2F2%29expr%2827%2F2%29
A=729%2F4=182.25
The maximum area is 182.25 square meters.

Edwin

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Find the dimension of a rectangle that will yield the maximum area of its perimeter is 54 meters. What is the maximum area?
----------------
Max area is a square
---> 13.5*13.5 = 182.25 sq neters
================
A = L*W
P = 2L + 2W = 54
L+W = 27
A = L*(27-L) = 27L - L^2
d/dL(27L - L^2) = 27 - 2L = 0 ------- 1st derivative = 0
2L = 27
L = 13.5 meters

Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find the dimension of a rectangle that will yield the maximum area
highlight%28cross%28of%29%29 if its perimeter is 54 meters. What is the maximum area?
~~~~~~~~~~~~~~~~~~~~~


At given perimeter, a rectangle having maximum area is a square with the side length
equal to one fourth  (1/4)  of the given perimeter.


It is a classic problem on finding optimal dimensions.

This problem was solved  MANY  TIMES  in this forum.

Therefore,  I created lessons at this site,  explaining the Algebra solution in all details.

The lessons are under these links
    - A rectangle with a given perimeter which has the maximal area is a square
    - A farmer planning to fence a rectangular garden to enclose the maximal area

Read these lessons attentively.
Consider them as your  TEMPLATE.

By the way,  in these lessons,  you will find many useful links to accompanied lessons.
Do not miss them.

Consider my lessons as your textbook,  handbook,  tutorial and  (free of charge)  home teacher.


In your case,  the maximum area is provided by a square with the side length of  54%2F4 = 131%2F2 = 13.5 meters.

The maximum area in this case is   13.5*13.5 = 182.25 square meters.