Question 970464: the perimeter of a rectangle is 100 cm. determine the possible measures of one of knowing that the area of rectangle should be minimum 500 cm
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! 
Let us define some variables
 = length of the rectangle, in cm.
 = width of the rectangle, in cm.
 =area of the rectangle in square centimeters.
The way  and were defined,
 , and when  we have that special type of rectangle that we call a square.
From geometry (and real life common sense) we know that
   and  .
From the problem, we know that
  .
From all that we know,
 --->  --->  --->  ,
and plugging the expression  for in ,
we get as a function of :
 <--->  .
That tells us that the area of the rectangle is a quadratic function of .
From a memorized formula,
or from looking a the function transformed using algebra as shown below,
we realize that the function has a maximum for  ,
when  .
To both sides of ,, the farther we go from ,
the smaller the value for .
Of course, we defined , , and to make sense with the "real life" situation of the problem, so they are all positive, for starters.
Also since and ,
 --->  --->  --->  , so the function is defined for  ,
and we get only a piece of the quadratic function,
increasing from   to a maximum for ,
when .
<---> <---> <---> 
If you need an area of 500 square centimeters, you would get it when
 , so you solve that to find the minimum width.
Or, you say that you want  and use the expression
<---> to get
 --->  --->  --->  (because we know that  <---> ) .
Then ,
and  .
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