Aim for FivePrecalculus

Modeling a non-periodic context

Podcast listeners

  • Units 1 and 2
  • 6 points
  • About 18 minutes

You can use a calculator on this question, just like on exam day.

A real-world situation is modeled by a polynomial, piecewise, exponential or logarithmic function. You set up and solve equations for the model's constants, find and use average rates of change with units, and explain an assumption or limit of the model. You'll use a graphing calculator. On the exam: Question 2 of 4. Part A of the free-response section: 2 questions in 35 minutes, graphing calculator required.

The question and its sources

A graphing calculator is required for this question. Make sure it is in radian mode. Unless otherwise specified, decimal approximations should be accurate to three places after the decimal point; avoid rounding intermediate values. Unless otherwise specified, the domain of a function f is the set of all real numbers x for which f(x) is a real number. Show the work that leads to your answers where the question asks for it.

The situation

A new podcast tracks the total number of people who have listened to at least one episode since its launch. Tracking began at time t = 0 weeks, when the podcast already had 12 thousand total listeners. At t = 4 weeks, it had 30 thousand total listeners.

The total number of listeners, in thousands, can be modeled by the function L given by L(t) = a · ln(t + 1) + b, where t is the number of weeks since tracking began and a and b are constants.

Source: Hypothetical scenario

Total listeners

Weeks since tracking began, t04
Total listeners (thousands)1230

Source: Hypothetical data

Suggested time: 18 minutes

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Part (a)

2 points

(i) Use the given data to write two equations that can be used to find the values for constants a and b in the expression for L(t). (ii) Find the values for a and b as decimal approximations.

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Part (b)

3 points

(i) Use the given data to find the average rate of change of the total number of listeners, in thousands of listeners per week, from t = 0 to t = 4 weeks. Show the computations that lead to your answer. (ii) Interpret the meaning of your answer from part (b)(i) in the context of the problem. (iii) Without computing it, determine whether the average rate of change of L from t = 4 to t = 8 is greater than or less than your answer from part (b)(i). Give a reason for your answer based on the graph of L.

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Part (c)

1 point

The podcast's producers claim that, according to the model, the total number of listeners will level off at some maximum value. Do you agree or disagree with this claim? Give a reason for your answer based on the model.

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