SOLUTION: Three sides of a trapezoid are each 10 cm long. How long is the 4th side, when the area of the trapezoid has the greatest value?

It is clear, that under given conditions, the trapezoid is isosceles with the shorter base of 10 cm long.


See my sketch below.  The upper base is 10 cm;  the lower base is 10+2x cm long.

                  10
         +------------------+
        /|                   \
       / |                    \
  10  /  |                     \  10
     /   |                      \
    /    |                       \
   +------------------------------+

   <- x ->


The mid-line is   = 10+x  cm long.

The height is  .


The area of the trapezoid  is   A(x) = .


To find the maximum of A(x), take the derivative

     =  - 

     =  - 


Equate it to zero

     -  = 0.


Simplify and find x

     = 

     = (10+x)*x

     = 

     = 0

     = 0

    (x-5)*(x+10) = 0


Of two roots,  x= 5 and x= -10, take the positive one, which gives the maximum to the trapezoid's area.


ANSWER.  The area of trapezoid is maximum when the 4-th side is 10+2*5 = 20 cm long.