SOLUTION: Rectangle EFGH's base is b, its height is h, and it has a perimeter of 48 units. Determine the area of the rectangle for b=6, b=8, b=12, and b=16. Based on these results, what conj

Algebra ->  Rectangles -> SOLUTION: Rectangle EFGH's base is b, its height is h, and it has a perimeter of 48 units. Determine the area of the rectangle for b=6, b=8, b=12, and b=16. Based on these results, what conj      Log On


   

Question 1183445: Rectangle EFGH's base is b, its height is h, and it has a perimeter of 48 units. Determine the area of the rectangle for b=6, b=8, b=12, and b=16. Based on these results, what conjecture can you make about maximizing the area of a rectangle with a fixed perimeter?
Answer by greenestamps(13203) About Me  (Show Source):
You can put this solution on YOUR website!


p=2b+2h=48
b+h=24

h = 24-b

Area = bh = b(24-b)
  b   h   Area
 --------------
  6   18   108
  8   16   128
  12  12   144
  16   8   128

"Conjecture": The area appears to be maximum when the length and width are equal.

In fact this is the case; and it is easily proven.

Let the base be 12+x and the height be 12-x; that makes base plus height 24, so the perimeter is 48.

The area is then (12+x)(12-x)=144-x^2; that area is maximum when x=0, making the base 12+0=12 and the height 12-0=12.