I'm having trouble with the following problem and have no idea what to do. I tried drawing a horizontal and vertical line down the middle of the window but got nowhere.
A window is in the shape of a 2 by 3 rectangle surmounted by a semicircle. Describe the boundary of the window as $r = f(\theta)$ in a polar coordinate system whose location and orientation you specify. The $f(\theta)$ likely will need to be defined piecewise.
It’s easy to describe in polar coordinates a circle whose center is at the origin. You can benefit from this, since the location of the origin is up to you.
Next, you just need to write a vertical line and a horizontal line in terms of polar coordinates. A vertical line is the graph of a polar function $f(\theta)$ such that, as $\theta$ rotates, the radius $r=f(\theta)$ corresponding to the angle $\theta$ is pushed out or in just enough that the $x$ coordinate of the point doesn’t change. How do you push out or in just enough? Hint: look at the formulas for converting between rectangular and polar coordinates.