Question 1183445
<br>
p=2b+2h=48
b+h=24<br>
h = 24-b<br>
Area = bh = b(24-b)<br><pre>
  b   h   Area
 --------------
  6   18   108
  8   16   128
  12  12   144
  16   8   128</pre>
"Conjecture": The area appears to be maximum when the length and width are equal.<br>
In fact this is the case; and it is easily proven.<br>
Let the base be 12+x and the height be 12-x; that makes base plus height 24, so the perimeter is 48.<br>
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.<br>