area of rectangle in a circle

Question

Find the area of the largest rectangle that can be inscribed in a circle of radius 4

Answer 1

The rectangle will have a diagonal that is the diameter of the circle.

 

x² + y² = c² = 64

 

A = xy

 

y = √(64 - x²)

 

A = x√(64 - x²)

 

 

The value to maximize A satisfies 0 < x < 8.  At this point you can plot this function and see the solution.

 

 

Here take the derivative and set it to zero.

 

dA/dx = -x²/√(64-x²) + √(64-x²)

 

dA/dx = 0:

 

x² = 64 - x² or x² = 32

 

x = 4√2

 

So, y = 4√2.

 

The rectangle is a square and its area is (4√2)(4√2) = 32.