SOLUTION: How to calculate the maximum area of rectangle give the fixed perimeter of 24.

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Question 1153651: How to calculate the maximum area of rectangle give the fixed perimeter of 24.
Found 3 solutions by Boreal, ikleyn, Alan3354:
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website!
length is x and width is 12-x, and that gives half perimeter of 12
area is x(12-x)
maximize their product, which is 12x-x^2
the vertex is at -b/2a or -12/-2 or x=6
12-x=6
This is a square with area 36 sq units
Calculus gets the same answer.

Answer by ikleyn(52786)   (Show Source):
You can put this solution on YOUR website!
.

Maximum area of a rectangle with given perimeter is the area of a SQUARE with this perimeter.

It is the general rule.

See the lessons
    - HOW TO complete the square to find the minimum/maximum of a quadratic function
    - Briefly on finding the minimum/maximum of a quadratic function
    - HOW TO complete the square to find the vertex of a parabola
    - Briefly on finding the vertex of a parabola
    - A rectangle with a given perimeter which has the maximal area is a square
in this site.

So, in your case the perimeter is 24 units.

The square with such a perimeter has the side of 24/4 = 6 units,
and its area is 6*6 = 36 square units.

It is MAXIMAL possible area for rectangles with the perimeter of 24 units.

Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website!
How to calculate the maximum area of rectangle give the fixed perimeter of 24.
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P = 2W + 2L = 24
W + L = 12
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1st try W = L = 6 and 6
Area = 6*6 = 36
Any changes --> Area = (W + c)*(W - c) = W^2 - c^2
Any value of c <> 0 ---> a smaller area.