Our final problem has no known exact solution. We want to find the area of the shape formed by,
The inequality has two parameters and . They are quantities of length, so they take on a nonnegative value. Let’s try out some values and see what these parameters do. I have colored the resulting shapes in red. (Note, the scales are different.)
When (move along the diagonal, starting from bottom-left), we get the shape of a nice heart. When we increase while holding fixed, the heart morphs into a circle. On the other hand, when we increase while holding fixed, the heart turns into two petals. Well, I see bunny ears.
Clearly, the shape (i.e. area) of our heart depends on the radius and the ear length . Can you guess the formula for the area ?