Question

Find the dimensions of the rectangle with maximum area that can be inscribed in a circle of radius 10.

Answer 1

Answer:

Step-by-step explanation:

That will be a square with a diagonal of 20

side length of 20sin45

and area of (20sin45)² = 200 units²

prove it you say?

Area of a rectangle is base times height

A = bh

With a radius of 10, the diagonals of any rectangle inscribed will be 20 units

20² = b² + h²

h = $\sqrt{400 - b^2}$

A = bh

A = b$\sqrt{400 - b^2}$

Area will be maximized when the derivative is set to zero

           dA/db = $\sqrt{400 - b^2}$ - b²/ $\sqrt{400 - b^2}$

                   0 = $\sqrt{400 - b^2}$ - b²/ $\sqrt{400 - b^2}$

b²/$\sqrt{400 - b^2}$ =  $\sqrt{400 - b^2}$

                  b² = 400 - b²

                2b² = 400

                  b² = 200

                  b = $\sqrt{200}$

h = $\sqrt{400 - b^2}$

h = $\sqrt{400 - \sqrt{200}^2 }$

h = $\sqrt{200}$

A = bh

A =  $\sqrt{200}$•$\sqrt{200}$

A = 200 units²