5. A rectangle is to be inscribed in a semicircle of radius 10, with one side lying on the diameter of the semicircle. What is the maximum possible area of the rectangle?

A. 5(2)1/2

B. 50

C. 100

D. 60(5)1/2

E. 145

 

Solution: The answer is C

 

The following is a graph of a semicircle of radius 10 and an inscribed rectangle of length L = 2x and width W = (100-x2)1/2:

Let A(x) be the area of the inscribed rectangle in terms of x. Since the area of any rectangle is given by the product of its length and width,

To find the maximum area, choose the maximum value of A(x) evaluated at all the positive real roots of A’(x). Since A’(x) has only one positive real root,

The maximum area is given by,