Expert: Scotto - 6/5/2008

Question
consider a square piece of paper with sides equals to 1 unit. we label the four vertices as ABCD. now inscribe a quarter circle with radius of 1 unit such that its center is at vertex A. next fold corner labeled as vertex A to touch the circumference of the Quarter circle. we want the fold to create a triangle (the folded part forms a triangle). note that the end points of the crease have to be on side AB and side AD in order to do this. our task is to find the exact area of the smallest and the largest of these triangles created in the above process. use calculus to do this and make sure you explain your logic.

C_______D
|       |
|       |
|_______|
B       A
quarter circle in middle fold A to circle.
i had this probelm on a test and got it incorrect, i'm told that its easier to do it without trig and i cant figure it out.

Answer
Flip the square over so BC is on the right side and call A the origin.  It is known that for any trianlge, the area is bh/2 where b is the base and h is the height.  Note that the base and height can be measured in any direction, so for any triangle in this form, we can take the base to be x and the height to be y.

It is alson known that (x,y) is on a circle, so x^2+y^2=1 =>
y^2=1-x^2.  The are of our square is then x*root(1-x^2).

Using the derivative, the maximum and minimum can be found.  It should be noted that x is on the interval [0,1].