Describing Change: Rates

Project 3.1: Fee-Refund Schedules



Some students at many colleges and universities enroll in courses and then later withdraw from them. Such students may have part-time status upon withdrawing. Part-time students have begun questioning the fee-refund policy, and a public debate is taking place. The Higher Education Commission has scheduled hearings on the issue. The Board of Trustees has hired your firm as consultants to help them prepare their presentation. Recently, the student senate passed a resolution condemning the current fee-refund schedule. Then, the associate vice president issued a statement claiming that further erosion of the university's ability to retain student fees would reduce course offerings.


  1. Preparing Alternative Plans Examine the current fee-refund schedule for your college or university. Present a graph and formula for the current refund schedule. Critique the refund schedule.

    Create alternative fee-refund schedules that include at least two quadratic plans (one concave up and one concave down), an exponential plan, a logistic plan, a no-refund plan, and a complete-refund plan. (Hint: Linear models have constant first differences. What is true about quadratic and exponential models?) For each plan, present the refund schedule in a table, in a graph, and with an equation. Critique each plan from the students' viewpoint and from that of the administration.

    Select the nonlinear plan that you believe to be the best choice from both the students' and the administration's perspective. Outline the reasons for your choice.

  2. Discussing Rates of Change Estimate the rate of change of your model for withdrawals after 1 week, 3 weeks, and 5 weeks. Interpret the rates of change in this context. How might the rate of change influence the administration's view of the model you chose? Would the administration consider a different model more advantageous? If so, why? Why did you not propose it as your model of choice?


  1. Prepare a written report of your results for the Board of Trustees. Include scatter plots, models, and graphs. Include in an appendix the reasoning that you used to develop each of your models.
  2. Prepare a press release for the college or university to use when it announces the adoption of your plan. The press release should be succinct and should answer the questions Who, What, When, Where, and Why. Include the press release in your report to the Board.
  3. (Optional) Prepare a brief (15-minute) presentation on your work. You will be presenting it to members of the Board of Trustees of your college or university.

Project 3.2: Doubling Time



Doubling time is defined as the time it takes for an investment to double. Doubling time is calculated by using the compound interest formula A 5 Pert or the continuously compounded interest formula A 5 Pert. An approximation to doubling time can be found by dividing 72 by 100r. This approximating technique is known as the Rule of 72.

Dr. C. G. Bilkins, a nationally known financial guru, has been criticized for giving false information about doubling time and the Rule of 72 in seminars. Your team has been hired to provide mathematically correct information for Dr. Bilkins to use in future seminar presentations.


  1. Evaluating Doubling-Time Rules Construct a table of doubling times for interest rates from 2% to 20% (in increments of 2%) when interest is compounded annually, semiannually, quarterly, monthly, and daily. Construct a table of doubling-time approximations for interest rates of 2% through 20% when using the Rule of 72. Devise similar rules for 71, 70, and 69. Then construct tables for these rules. Examine the tables and determine the best approximating rule for interest compounded semiannually, quarterly, monthly, and daily. Justify your choices. For each interest compounding listed above, compare percent errors when using the Rule of 72 and when using the rule you choose. Percent error is 100 3 (estimate 2 true value)/(true value). Comment on when the rules overestimate, when they underestimate, and which is preferable.
  2. Evaluating the Sensitivity of Doubling Time Dr. Bilkins is interested in knowing how sensitive doubling time is to changes in interest rates. Estimate rates of change of doubling times at 2%, 8%, 14%, and 20% when interest is compounded quarterly. Interpret your answers in a way that would be meaningful to Dr. Bilkins, who knows nothing about calculus.


  1. Prepare a written report for Dr. Bilkins in which you discuss your results in Tasks 1 and 2. Be sure to discuss whether Dr. Bilkins should continue to present the Rule of 72 or present other rules that depend on the number of times interest is compounded.
  2. Prepare a document for Dr. Bilkins's speechwriter to insert into the seminar presentation (which Dr. Bilkins reads from a Teleprompter). It should include (1) a brief summary of how to estimate doubling time using an approximation rule and (2) a statement about the error involved in using the approximation. Keep in mind that Dr. Bilkins wishes to avoid further allegations of misinforming seminar participants. Also include a brief statement summarizing the sensitivity of doubling time to fluctuations in interest rates. Include the document in your written report.
  3. (Optional) Prepare a brief (15-minute) presentation of your study. You will be presenting it to Dr. Bilkins and the speechwriter. Your presentation should be only a summary, but you need to be prepared to answer any technical questions that may arise.

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