Question

A rectangle is inscribed in a semicircle that has a radius of 10. Find the function that models the area A of the rectangle in terms of its length y.

Answer 1

in the diagram, the width of the shaded rectangle is 2x and its height is y.
so its area is 2xy.
however, since it is inscribed within a semi-circle, we also know:\[x^2+y^2=10^2\]so:\[x=\sqrt{10^2-y^2}\]substitue this into expression for area and you get:\[A=2xy=2y\sqrt{10^2-y^2}=2y\sqrt{100-y^2}\]