[Note for Mac users] |
Example: Of all rectangles with perimeter 100, which has the greatest area?
Rectangles
Perimeter 100
Maximize Area
Some possibilities include a1 ´ 49 rectangle, a25 ´ 25 rectangle (square), and a30 ´ 20 rectangle.
The width and height are varying, so we draw an x ´ y rectangle and let x be the width, and y the height. Then the area, A, isA = xy and the perimiter isP = 2x + 2y = 100 .
Maximize area: A = xy
We know2x + 2y = 100 , sox + y = 50 , soy = 50 - x , henceA = x(50-x) . Now A is a function of x alone.
We needx > 0 andy > 0 . The latter means50 - x > 0 , orx < 50 . Thus x must be in the interval [0,50].
A'(x) = 50 - 2x = 0 if and only ifx = 25 .
The derivative is always defined, in this case.
A(25) = 25(50-25) = 25 x 25 = 625
A(0) = 0(50-0) = 0
A(50) = 50(50-50) = 0
The maximum of 625 occurs at x = 25.
In this case, we want the dimensions of the rectangle. We have the
x value, so we still need y. Since