AP®MathsCollege BoardPrecalculusExam QuestionsPolynomial & Rational FunctionsModeling with Polynomial & Rational Functions

Modeling with Polynomial & Rational Functions (College Board AP® Precalculus): Exam Questions

11 mins3 questions
1a
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2 marks

A musician released a new song on a streaming service. A streaming service is an online entertainment source that allows users to play music on their computers and mobile devices.

Several months later, the musician began using an app (at time t equals 0) that counts the total number of plays for the song since its release. A “play” is a single stream of the song on the streaming service. The table gives the total number of plays, in thousands, for selected times t months after the musician began using the app. At t equals 0, the total number of plays was 25 thousand. At t equals 2, the total number of plays was 30 thousand. At t equals 4, the total number of plays was 34 thousand.

Months after the musician began using the app

0

2

4

Total number of plays for the song since its release (thousands)

25

30

34

The total number of plays, in thousands, for the song since its release can be modeled by the function D given by D open parentheses t close parentheses equals a t squared plus b t plus c, where D open parentheses t close parentheses is the total number of plays, in thousands, for the song since its release, and t is the number of months after the musician began using the app.

(i) Use the given data to write three equations that can be used to find the values for constants a, b, and c in the expression for D open parentheses t close parentheses.

(ii) Find the values for a, b, and c as decimal approximations.

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1b
Sme Calculator
3 marks

(i) Use the given data to find the average rate of change of the total number of plays for the song, in thousands per month, from t equals 0 to t equals 4 months. Express your answer as a decimal approximation. Show the computations that lead to your answer.

(ii) Use the average rate of change found in part B (i) to estimate the total number of plays for the song, in thousands, for t equals 1.5 months. Show the work that leads to your answer.

(iii) Let A subscript t represent the estimate of the total number of plays for the song, in thousands, using the average rate of change found in part B (i). For A subscript 1.5 end subscript found in part B (ii), it can be shown that A subscript 1.5 end subscript less than D open parentheses 1.5 close parentheses.
Explain why, in general, A subscript t less than D open parentheses t close parentheses for all t, where 0 less than t less than 4. Your explanation should include a reference to the graph of D and its relationship to A subscript t.

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1c
1 mark

The quadratic function model D has exactly one absolute minimum or one absolute maximum. That minimum or maximum can be used to determine a domain restriction for D. Based on the context of the problem, explain how that minimum or maximum can be used to determine a boundary for the domain of D.

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2
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3 marks

On the initial day of sales (t equals 0) for a new video game, there were 40 thousand units of the game sold that day. Ninety-one days later (t equals 91), there were 76 thousand units of the game sold that day.

The number of units of the video game sold on a given day can be modeled by the function G given by G open parentheses t close parentheses equals 40 plus 7.961 ln open parentheses t plus 1 close parentheses, where G open parentheses t close parentheses is the number of units sold, in thousands, on day t since the initial day of sales.

(i) Use the given data to find the average rate of change of the number of units of the video game sold, in thousands per day, from t equals 0 to t equals 91 days. Express your answer as a decimal approximation. Show the computations that lead to your answer.

(ii) Use the average rate of change found in (i) to estimate the number of units of the video game sold, in thousands, on day t equals 50. Show the work that leads to your answer.

(iii) Let A subscript t represent the estimate of the number of units of the video game sold, in thousands, using the average rate of change found in (i). For A subscript 50, found in (ii), it can be shown that A subscript 50 less than G open parentheses 50 close parentheses. Explain why, in general, A subscript t less than G open parentheses t close parentheses for all t, where 0 less than t less than 91.

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3
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2 marks

An electric utility company begins a phased installation of smart meters. At time t equals 0, the company begins tracking the total number of smart meters installed, in thousands.

The table gives the total number of smart meters installed, in thousands, for selected times t months after tracking began.

Months after tracking began, t

0

2

Total number of smart meters installed (thousands)

3.655

4.375

(i) Use the given data to find the average rate of change of the total number of smart meters installed, in thousands per month, from t equals 0 to t equals 2 months. Express your answer as a decimal approximation. Show the computations that lead to your answer.

(ii) Use the average rate of change found in part B (i) to estimate the total number of smart meters installed, in thousands,for t equals 1.8 months. Show the work that leads to your answer.

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