triangle is , where is the half perimeter, , so that , where
Clearly is maximum when is maximum.
.
For a maximum (clearly it will not be a minimum in this case), one must have simultaneously
;
that is,
An immediate solution is .
If we now introduce this condition in the value of , we find
.
For maximum or minimum, , which gives or .
Clearly gives minimum area; gives the maximum, for , which is for and for .
Example (6). Find the dimensions of an ordinary railway coal truck with rectangular ends, so that, for a given volume the area of sides and floor together is as small as possible.