Posted by Sani on Wednesday, July 4, 2012 at 3:38am.
Find the area of the largest rectangle that can be inscribed in a semicircle of radius "r"?

maths  Reiny, Wednesday, July 4, 2012 at 12:53pm
place your sketch on the xy grid with the centre at (0,0)
let the radius of the semicircle be r, where r is a constant
so the base of the rectangle is 2x, let its height be y
then x^2 + y^2 = r^2
y =(r^2  x^2)^(1/2)
area = 2xy
= 2x(r^2x^2)^(1/2)
d(area)/dx = (2x)(1/2)(r^2  x^2)^(1/2) (2x) + 2(r^2x^2)^(1/2)
= 0 for a max/min of area
2(r^2  x^2)^(1/2) = 2x^2/(r^2x^2)^(1/2)
x^2 = r^2  x^2
2x^2 = r^2
x^2 = r^2/2
x = r/√2 = (√2/2)r
then largest area
= 2x√(r^2  x^2)
= (4√2/2)r (r^2  r^2/2^(1/2)
= 2√2r(r/√2) = 2r^2
or, in a real simple way
suppose we look at the whole circle, the largest "rectangle we can fit inside the circle is a square, where x = y
then the sides of the square are 2x and 2y,
and the area is 4xy, but x=y and x^2 + x^2 = r^2 > x^2 = r^2/2
so the largest area = 4x^2
= 4(r^2/2) = 2r^2