Calculus 1- Extreme Value of functions

Question

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

Answer 1

Center the rectangle's base at the origin inside the semicircle.

Height = y and Base = 2x

A = base × height = 2xy

The equation for the circle in the first quadrant is:

x²+y² = r² = 1

y = √(1-x²)

A = 2x√(1-x²)

dA/dx = 2√(1-x²) -2x²/(2√(1-x²))

Set dA/dx to zero to optimize A

2√(1-x²) -2x²/(2√(1-x²)) = (2(1-x²)+2x²)/(2√(1-x²)) = 0

2-4x² = 0

x = 1/√2

y = √(1-1/2) = 1/√2

Base = 2x = 2/√2 and height = 1/√2