Calculus 1 - Question Concerning Optimization

Question

A rectangle is to be inscribed in a half circle of radius 1 ft. what dimensions will give the largest area?

Answer 1

Let base of rectangle is 2x. Then other side is (1 - x²)^1/2. So, area A(x) = 2x(1 - x²)^1/2.

A'(x) = 2(1 - x²)^1/2 - 2x²/(1 - x²)^1/2 = (2 - 2x² - 2x²)/(1 - x²)^1/2 = (2 - 4x²)/(1 - x²)^1/2;

A'(x) = 0 or A'(x) undefined to get all critical numbers. 2 - 4x² = 0; x² = 1/2; x = ±√2/2; 1 - x² = 0; x = ±1

For us because x> 0 we have x = √2/2 and x = 1.But x = 1 and side of inscribed rectangle can' be 2.

So x = √2/2. Left from x = √2/2 A'(x) > 0 and right from x = √2/2 A'(x) < 0. So, at x = √2/2 we have maximum

Answer: dimenshions of rectangle √2 and √2/2 and max area A_max= 1.