Find the largest trapezoid that can be inscribed in a semicircle with the lower base of the trapezoid on the diameter of the semicircle.
Let be the radius of the semicircle, let be the lower edge of the trapezoid, and the upper edge. We recall from geometry that the area of the trapezoid is given by
where and are the lengths of the bases and is the height of the trapezoid. Next, we want to find formulas for and in terms of the radius . Since lies on the diameter of the semicircle we have . For we consider the two right triangles drawn in red in the diagram above. The hypotenuse of each of these is , and the legs have lengths and . Therefore,
Substituting these into the formula for the area of a trapezoid, we get an expression for the area of the trapezoid in terms of and ,
Taking the derivative of this area function with respect to , (noting that is a constant since we are given the semicircle of radius to start with)
To find the critical points we set the derivative equal to zero and solve for ,
We know the area function has a maximum at this point since for the derivative we have
Hence, is increasing when and is decreasing when . Thus, the area has a maximum when .
Finally, we use this value of to solve for in terms of ,
Since we already know we then have the lengths of the upper and lower edges of the trapezoid are: