Suppose I have three points: S, E, and M. The point S is fixed while E and M move in a circle around S. The point E is a distance 1 meter from S and revolves at a rate of 1 revolution per minute. The point M is a distance of 1.50 meters from S and revolves at a rate of 0.53 revolutions per minute. Both move counterclockwise. 1. Treating S as the origin, draw a picture of this where E is in the first quadrant and M is in the second.
Draw rays from S through E and M. Draw a line connecting E and M. Recall the positive x-axis is considered the 0 angle, so draw this as well. Let α be the angle that the point E is at and β be the angle that the point M is at. Let θ be the exterior angle at E and let D be the distance between E and M.
2. Fill in the labels for the lines and angles.
Often times, one or more values will be function of the others. These formulas are often based on geometry. Often areas, volumes, and surface areas are involved. More tricky ones involve similar triangles and circles.
3. Use the Law of Sines to write θ as a function of D, α, and β
4. Use the Law of Cosines to write D as a function of α and β. When dealing with a related rates problem, itâs always good to keep track of whatâs changing and whatâs fixed.
5. Make a table of all quantities, determining if they are fixed or changing. If they are fixed, plug in their values. If they are changing, include their derivatives. You may notice that not all variables have values. The end of the problem typically involves plugging in values for one or more of these.
6. How is D changing when α is Ï/ 3 and β is Ï?
7. How is θ changing when α is 0 and β is Ï/ 2 ?
8. How is θ changing when α and β are both Ï?