Solution to Problem 9.4

© 2002 by Karl Hahn 		KCT logo (click for Home Page)

Step 1: Draw the diagram. That diagram is shown to the right.

Step 2: Bisect the angle made by the two equal sides.

Step 3: Label the various lengths. You can see that shown as well in the diagram.

Step 4: Write the equation that relates x, s, and P. This is also shown in the diagram. The sum of all the lengths around the triangle must equal P.

Step 5: Write the equation for the triangle's height. Again, that is shown in the diagram, and is derived from the Pythagorean formula

Step 6: Write the area equation. That is one half base times height, and as you can see, the base is 2x.

         1
   A  =   
         2

              _______
 (2x)h  =  x √s2 - x2

Step 7: Substitute for s. If, according to the perimeter equation,  2x + 2s = P,  then  s = P/2 - x.  Substituting for s into the area equation and then doing a little algebra: 

            _______________        _________
   A  =  x √(P/2 - x)2 - x2  =  x √P2/4 - Px

Step 8: Find the derivative of the area equation. From the above, take the derivative with respect to x. Since the expression for A as a function of x is a product, you will have to use the product rule to find the derivative. Remember that P is a constant: 

   dA      _________          Px
       =  √P2/4 - Px  -    _________
   dx                    2√P2/4 - Px

Step 9: Set the derivative to zero and solve for x. Once you replace  dA/dx  with zero, you can multiply through by the radical expression to get 

         P2            Px         P2
   0  =      -  Px  -        =        -
          4             2          4

  3Px
     
   2
Divide out  P/4  and this becomes: 

   0  =  P  -  6x
or 
         P
   x  =   
         6

Here's the diagram again. Can you see how the above equation makes it so that the isosceles triangle that maximizes area is equilateral? Clearly the base is  2x,  which must be one third of P. The other two thirds of P is accounted for by the two remaining sides, each of which are length, s. Hence  s = 2x = P/3.  And finally


          _______      ____________
   h  =  √s2 - x2  =  √P2/9 - P2/36  =


   _
  √3P
     
   6

Return to Midterm Practice Exam

You can email me by clicking this button:

  Use your own emailer     Use form