Trigonometry Project-
Perhaps in previous chapters you have done problems involving how fast a car travels if the tires rotate at 300 rpm's or how to calculate the angular velocity if you know the linear velocity.
The following problem uses the rolling wheel idea but in a different way. You will be asked to trace the path of a speck on the outer edge of a wheel as it rolls down the road. The connection to this chapter is that you will need to find the (x,y) coordinates of the speck using functions involving sin(Θ) and cos(Θ). Historically, this problem has two significant results. It is connected to the solution of how to build a perfect clock (a major breakthrough) and the fastest path from a point A to a point B where B is below A but not directly below A. After you have completed this project ask your teacher for more details on these 2 results.
In order to trace the path of a point (x, y) on the outer edge of a tire we will assume the tire has a radius of 15 inches and rolls toward the right To get started (x, y) will be the point touching the x-axis and θ will be 0. (See the diagram below) Θ stands for how many degrees the tire has rotated. You will do this for one full revolution in 30° increments.
1. Fill in the chart. See diagrams below for help. (Use 1 decimal)
| θ |
0 |
30 |
60 |
90 |
120 |
150 |
180 |
210 |
240 |
270 |
300 |
330 |
360 |
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2. On a sheet of graph paper, plot the points (x, y) in your chart. You should see a smooth curve. Otherwise something is off.
3. Find the lateral position of the speck, x, in terms of Θ.
4. In the diagram, length "a" is behind the center but this won't always be the case. For what angles of Θ is "a" in front of the center and how does your function account for this switch?
5. Graph the function x(Θ) on a reasonable domain.
6. Evaluate and interpret the y-intercept in context for x(θ).
7. Explain why there is no x-intercept in context for x(θ)..
8. Does this function increase, decrease, stay constant or a combination of the above? Why does your answer make sense in context of the problem?
9. Find the vertical position of the speck y, in terms of Θ.
10. In the diagram it shows that length "b" is below the center but this won't always be the case. For what angles of Θ is "b" above the center and how does your function in 9 account for this switch?
11. Graph the function y(θ) on a reasonable domain.
12. Does y(θ) increase, decrease, stay constant or a combination of the above? Why does your answer make sense in context of the problem?
13. Evaluate and interpret the high point of y(θ) in context.