4.472135954999579*R is the Max perimeter
0.8944272*R is the value for x

note:  the downside of this method is there is no easy way to see that the answer is 2*sqrt(5)*R, as noted in the previous solutions.

def y(a):
    """ enter x value and return the y value for a circle of radius 1
    centered at the origin """
    return (1-a**2)**(1/2)

def perim(a):
    """ enter the x value and return the perimeter of a rectangle
    inscribed in a half circle of radius 1 """
    return (4*a + 2*(y(a)))

n = 10000000
x=0
maxPerim = 0
for i in range(1,n):
    newPerim = perim(i/n)
    if newPerim > maxPerim:
        maxPerim=newPerim
        x = i/n

print(x, maxPerim)