Problem 6

Suppose a rectangle is inscribed in a 30, 60, 90 degree triangle with one side of the rectangle along the hypotenuse of the triangle. If the length of the hypotenuse is h, what is the maximum possible area of the rectangle in terms of h?

[Problem submitted by Vin Lee, LACC Associate Professor of Mathematics.]

Solution for Problem 6:

Notice that the three smaller triangles are also 30, 60, 90-degree triangles. In all such triangles the ratio of the lengths of the sides is . Using this ratio we find

. So,

The area of the rectangle is .

This is a quadratic equation of the form . Since is negative the graph of A versus w is a downward opening parabola. The maximum possible area is at the vertex whose w coordinate is

. So, , and the maximum area is .