A piece of plexiglass is in the shape of a semi-circle with radius 2m. Determine the dimensions of the rectangle with the greatest area that can be cut from the piece of plexiglass?

Answer 1

Draw a rectangle in a semicircle, let its base be 2x and its height be y
draw a line from the centre to the vertex of the rectangle.
That should be a right-angled triangle with sides x, y, and 2 and
x^2 + y^2= 4 ---> - √(4-x^2)

Area of rect. = 2xy
= 2x(x4-x^2)^(1/2)

d(Area) = 2x(1/2)(4 - x^2)^(-1/2)(-2x) + 2(4 - x^2)^(1/2)
= 0 for a max of Area

this solves to x = 2/√3, subbing that back gives us
y = √8/√3 or 2√2/3

so the dimensions of largest area is 4/√3 by 2√2/√3