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ioda
3 years ago
14

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

Mathematics
1 answer:
Ilia_Sergeevich [38]3 years ago
5 0

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

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