What value of x will result in a rectangle with maximum area among all these rectangles?

Question

Consider a rectangle, with a fixed perimeter of 200 units. Let  dimensions of the rectangle be represented by x and 100 - x. By changing  the value of x we can generate a lot of  different rectangles, each with a constant perimeter of 200 units (For example x = 30, 100 - x = 70, or x = 60 & 100 - x = 40,, etc.). 

 

What value of x will result in a rectangle with maximum area among all these rectangles?

 

Answer 1

using algebra...

P=2l+2w

200=2l+2w

100=l+w

100-w=l

A=lw

A=(100-w)(w)

A=100w-w^2

this is a parabola with a maximum point (h,k)

h=-b/2a

h=-100/(-2)

h=50

x=50 units

 

using arithmetic... 

the rectangle with the greatest area will always be a square

200/4=50 units

Answer 2

The area is x(100 - x) = 100x - x^2.

There are 2 standard ways to solve this problem.

Without using calculus, one can solve this by completing the square.

 100x - x^2 = -(x - 50)^2 + 50^2

 

This has maximum value when the square has minimum value (note the negative sign), which occurs when x - 50 = 0, or x = 50.  The maximum value is 50^2 = 2500.

 

Using calculus, one takes the derivative of 100x - x^2 and set it equal to 0; 100 - 2x = 0; x = 50

The second derivative is -2 is negative, so this is a maximum;  You can then plug in x = 50 and find the maximum function value of 2500.