Let P=(x,y) be the point in quadrant 1 that is a vertex of the rectangle and is on the semicircle of radius r=8

Question

what is the perimeter of the rectangle as a function of x

Answer 1

Since x,y can be related to the radius, 8, using the pythagorean theorem, we can quantify y and x as follows:

a^2 + b^2 = c^2

turns into

y^2 + x^2 = 8^2

From there, we solve for y.

y = sqrt(64 - x^2)

Our perimeter will equal the sum of x and y multiplied by four, since x and y fall only in the first quadrant.

P = 4y + 4x

Plug in the found value for y

P = 4*sqrt(64 - x^2) + 4x