"A rectangle is inscribed in a semicircle of radius 2 cm. Find the largest area of such a rectangle".

Question

There is a diagram, but I think the question makes it clear enough what is going on. I'm having problems finding a relationship that I can work with.

Answer 1

draw a radius from the centre to a vertex of the rectangle
Let the base of the rectangle be 2x, making the triangle base x, and let the height of the rectangle by y

Area of rectangle = 2xy
but x^2 + y^2 = 4
y = √(4 - x^2) or (4-x^2)^(1/2)

A = 2x(4-x^2)^(1/2)

take the derivative using the product rule,
set it equal to zero and solve for x

Answer 2

Thank you! Never occurred to me to position the radius like that.