Modelling with Trigonometric Functions Type II Portfolio

This portfolio will be assessed against all six criteria, A - F. It covers circular and exponential functions, and transformations as well.

1. The maximum height of a Ferris wheel at Funworld in Bangalore is 35 m. The wheel takes 2 minutes to make one revolution. Passengers board the Ferris wheel from a platform 2 m above the ground at the bottom of its rotation.

(a) Draw a simple diagram to represent this information. Explain any simplifying assumptions that you make.

(b) Explain how the height above the ground varies as the Ferris wheel makes one complete revolution. You may use a neatly prepared diagram (neatly and on graph paper) to enhance your explanation, but no calculations are required.

(c) Write a formula in the form of h(t) = asin[b(t + c)] + d to describe how the height above the ground (in m) of a rider (i.e. Mr. S) changes as a function of time, t, in minutes. Explain or justify your equation given some of your discussion points in (a) and (b).

(d) Use graphing technology to prepare a graph of your function and label the key points from your discussion in questions 1a, 1b, 1c.

(d) What height do I reach after 36 seconds on the Ferris wheel.

(e) Explain why my total ride time on the Ferris wheel cannot be 7 minutes.

(f) If I am above a height of 25 m above the ground I can see our hostel, I can see Ross working on this assignment. So if I ride on the Ferris wheel for 12 minutes, for how many seconds can I see Ross?

2. You will continue this investigation by considering other algebraic models for periodic functions, specifically polynomial functions. This investigative question is technology dependent.

(a) Make a predication ==> Do polynomial functions like quadratics, cubics, quartics model periodic data? Explain your prediction.

(b) Prepare a table of values for 10 points in one complete revolution of the Ferris wheel (Use technology to generate this data - i.e. same as when you graphed the function).

(c) Prepare a scatterplot of the 10 key points using WINSTATS or a Graphing Display Calculator (or any other statistics software with which you are familiar.)

(d) Have the software or calculator determine the regression equation and correlation coefficient for a quadratic equation for this data. Comment on how well the quadratic function fits the data. Comment on domain restrictions for the quadratic model.

(e) Repeat question 2(d) for a (i) cubic function, (ii) quartic function, and use WINSTAT to fit higher order polynomials. Comment upon goodness of fit and domain restrictions. What degree of polynomial works best.

(f) For your own interest, include more data (i.e. 3 complete cycles). What happens to the polynomial function=s ability to fit the data?

3. Now we can also use trigonometric functions to model other periodic phenomenon. Consider the case of Mr. Santowski going bungie jumping. I start from the top of a crane, 90 m above the ground and the bungie chord that I am attached to has a maximum extension of 80 m, given my weight. A bungie chord will feature a phenomenon called dampened simple harmonic motion, which simply means that I will not always return to the same maximum height of 90 m in each successive oscillation. To simplify our calculations, let's say that I always return to the same minimum height (i.e. maximum bungie chord extension) in each oscillation and let's assume that my maximum height reduces itself by a factor of one-half of the previous maximum height of the previous oscillation. Each oscillation takes the same length of time to complete, let's say 10 seconds.

(a) Draw a simple diagram to represent this information. Explain any simplifying assumptions that you make.

(b) Explain how the Mr. Santowski's height above the ground varies as I oscillate on the bungie chord, assuming that my maximum height reduces itself by this factor of one half of the previous maximum height. You may use a neatly prepared diagram (neatly and on graph paper) to enhance your explanation, but no calculations are required.

(c) Prepare a table of values showing how my maximum height changes with each oscillation and how then how my height above the ground varies with time.

(d) Determine an exponential equation that models how my maximum height changes as a function of (i) number of oscillations and (ii) time.

(e). Now we need to model a sinusoidal equation for the oscillation, using the given information that the maximum chord length was 80 m, the period of the oscillation was 10 seconds, and I started my jump from the top of a 90 m tower. (Recall your work in Question #1)

(f) Now we want to find a function that models my height above the ground at any given time. Having worked through Questions (a) - (e), determine an equation that shows my height above the ground at any time in my jump.