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

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

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Modeling with Polynomial & Rational Functions

1
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1 mark

The variable space y is inversely proportional to x. If space y equals 18 when x equals 2, what is the value of space y when x equals 6?

  • 2

  • 6

  • 9

  • 54

Choose your answer

2
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1 mark

The table gives values of space f open parentheses x close parentheses at selected values of x.

space x

0

1

2

3

4

space f open parentheses x close parentheses

2

5

14

35

74

Based on this data, which type of polynomial function is most appropriate to model space f?

  • Linear

  • Quadratic

  • Cubic

  • Quartic

Choose your answer

3
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1 mark

The piecewise-defined function f is given by

space f open parentheses x close parentheses equals open curly brackets table row cell x squared plus 1 comma space space space space space space x less than 1 end cell row cell 4 minus x comma space space space space space space space space x greater or equal than 1 end cell end table close

What is the value of space f open parentheses 0 close parentheses plus f open parentheses 1 close parentheses plus f open parentheses 3 close parentheses ?

  • 4

  • 5

  • 8

  • 13

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4
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1 mark

The depth, in metres, of water at a sensor in a tidal estuary relative to the mean water level is modelled by the function W defined by

W open parentheses t close parentheses equals fraction numerator 84 t cubed minus 950 t squared plus 3556 t minus 4620 over denominator 16 t squared plus 12 t minus 45 end fraction

where t is measured in hours after 6 a.m., for 2 \leq t \leq 9. Based on the model, how many hours did it take for the water depth to increase from 0 metres to 4 metres above the mean level?

  • 7 . 386

  • 5 . 212

  • 3 . 212

  • 2 . 175

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5
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1 mark

A car is traveling on a highway. The total distance, D open parentheses t close parentheses (in miles), the car has traveled t hours after entering the highway is modeled by the linear function D open parentheses t close parentheses equals 65 t plus 5. Which of the following best describes an underlying assumption of this model?

  • The car's distance from the starting point doubles every hour.

  • The car has traveled 5 miles per hour since entering the highway.

  • The car's speed is constant at 65 miles per hour.

  • The car's speed increases by 65 miles per hour each hour.

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6
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1 mark

The quadratic function space f, given by space f open parentheses x close parentheses equals a x squared plus b x plus c, models a set of data. The graph of space f passes through the points open parentheses 0 comma 5 close parentheses, open parentheses 2 comma 9 close parentheses, and open parentheses 4 comma 21 close parentheses. What is the value of a?

  • 0

  • 1

  • 2

  • 5

Choose your answer

7
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1 mark

The profit (in thousands of dollars) earned by a small business t months after launch is modeled by P open parentheses t close parentheses equals negative 0.4 t cubed plus 4.8 t squared minus 12 t plus 15 for 0 less or equal than t less or equal than 8. What is the maximum value of P open parentheses t close parentheses on this interval?

  • 6.442

  • 15.000

  • 21.400

  • 29.958

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8
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1 mark

A parking garage charges a flat fee of $5 for parking up to 1 hour. Beyond 1 hour, an additional $3 per hour is charged at a continuous rate. Let C open parentheses t close parentheses represent the total parking charge, in dollars, for parking t hours, where t greater than 0. Which of the following piecewise functions correctly models C open parentheses t close parentheses ?

  • C open parentheses t close parentheses equals open curly brackets table row cell 5 comma space space space space space 0 less than t less or equal than 1 end cell row cell 3 t comma space space space space t greater than 1 end cell end table close

  • C open parentheses t close parentheses equals open curly brackets table row cell 5 comma space space space space space space space space space space space space 0 less than t less or equal than 1 end cell row cell 3 t plus 2 comma space space space space t greater than 1 end cell end table close

  • C open parentheses t close parentheses equals open curly brackets table row cell 5 comma space space space space space space space space space space space space 0 less than t less or equal than 1 end cell row cell 3 t plus 5 comma space space space space t greater than 1 end cell end table close

  • C open parentheses t close parentheses equals open curly brackets table row cell 5 t comma space space space space space space space space space space 0 less than t less or equal than 1 end cell row cell 3 t plus 2 comma space space space space t greater than 1 end cell end table close

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9
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1 mark

The intensity I (in watts per square meter) of light from a particular source is inversely proportional to the square of the distance d (in meters) from the source. When d equals 1.5 meters, the intensity is 240 watts per square meter. To 3 decimal places, what is the intensity when d equals 4.7 meters?

  • 24 . 445

  • 76 . 596

  • 114 . 894

  • 2 comma 356.267

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10
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1 mark

A model for the squirrel population (in hundreds) in a small park is given by N open parentheses t close parentheses equals negative 0.3 t squared plus 6 t plus 5 for 0 less or equal than t less or equal than 10, where t is the number of years since the park opened. What is the value of t, to 3 decimal places, at which N open parentheses t close parentheses equals 25?

  • 4 . 226

  • 5 . 556

  • 10 . 000

  • 15 . 774

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11
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1 mark

A community garden plot is being designed in the shape of a rectangle. One side will run along an existing wall, so fencing is needed only for the other three sides. The gardener has 40 feet of fencing material.

The area of the plot, in square feet, can be modeled by the function A given by

A open parentheses x close parentheses equals x open parentheses 40 minus 2 x close parentheses

where x is the length of the side perpendicular to the wall, in feet, and 0 less than x less than 20.

What is the contextually appropriate range of A?

  • \left(0 , 200\right)

  • \left(0 , 200\right]

  • \left(0 , \infty\right)

  • \left(- \infty , \infty\right)

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12
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1 mark

A telecommunications company's monthly cell phone bill, in dollars, depends on the number of gigabytes (GB) of data used during the month. Customers using less than 5 GB pay a flat $30. Customers using at least 5 GB but less than 20 GB pay a flat $45. Customers using 20 GB or more pay a flat $65.

Let C be the function such that C open parentheses d close parentheses gives the monthly bill, in dollars, for a customer using d GB of data, where d greater or equal than 0.

Which of the following correctly defines C open parentheses d close parentheses?

  • C open parentheses d close parentheses equals open curly brackets table row cell 30 comma space space space 0 less or equal than d less than 5 end cell row cell 45 comma space space space 5 less than d less than 20 end cell row cell 65 comma space space space d greater or equal than 20 end cell end table close

  • C open parentheses d close parentheses equals open curly brackets table row cell 30 comma space space space 0 less or equal than d less than 5 end cell row cell 45 comma space space space 5 less or equal than d less than 20 end cell row cell 65 comma space space space d greater or equal than 20 end cell end table close

  • C open parentheses d close parentheses equals open curly brackets table row cell 30 comma space space space 0 less or equal than d less or equal than 5 end cell row cell 45 comma space space space 5 less or equal than d less or equal than 20 end cell row cell 65 comma space space space d greater or equal than 20 end cell end table close

  • C open parentheses d close parentheses equals open curly brackets table row cell 30 comma space space space space space space space space space space space space space space space space space space space space space space space space 0 less or equal than d less than 5 end cell row cell 30 plus 7 over 3 open parentheses d minus 5 close parentheses comma space space space 5 less or equal than d less than 20 end cell row cell 65 comma space space space space space space space space space space space space space space space space space space space space space space space space d greater or equal than 20 end cell end table close

Choose your answer

13
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1 mark

A scatterplot of a data set in the x y-plane shows a sequence of points that switches between increasing and decreasing three times.

Which of the following polynomial function types best models the data?

  • Linear

  • Quadratic

  • Cubic

  • Quartic

Choose your answer

14
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1 mark

The force, in Newtons, between two electrically charged objects is inversely proportional to the square of the distance, in meters, between them. When the distance between the objects is 2 meters, the force is 25 Newtons.

Based on the model, what is the distance between the objects, in meters, when the force is 4 Newtons?

  • 0.320

  • 5.000

  • 12.500

  • 25.000

Choose your answer

15
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1 mark

A polynomial model for the population of a colony (in thousands) is given by P open parentheses t close parentheses equals negative t squared plus 20 t plus 100 for t greater or equal than 0, where t is measured in years from the start of observation. According to this model, after how many years does the population first reach zero?

  • 5 plus 5 square root of 2

  • 10 plus 5 square root of 2

  • 10 plus 10 square root of 2

  • 20 plus 10 square root of 2

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16
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1 mark

Let S open parentheses t close parentheses represent the total amount of snow, in inches, that has fallen during a snowstorm, where t is the number of hours since the start of the storm. The total snowfall is modeled by the function S open parentheses t close parentheses equals negative 1 fourth t squared plus 3 t for 0 less or equal than t less or equal than T, where T is the latest time at which the model is appropriate. What is the largest value of T for which the model is appropriate in this context?

  • T = 3

  • T = 6

  • T = 9

  • T = 12

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17
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1 mark

The cost per unit of producing x items of a product, in dollars, is modeled by the function C given by C open parentheses x close parentheses equals fraction numerator a over denominator x minus h end fraction for some positive constants a and h. When 60 items are produced, the cost per unit is $50 When 80 items are produced, the cost per unit is $25.

Based on the model, what is the cost per unit, in dollars, when 100 items are produced?

  • 12.500

  • 16.667

  • 20.000

  • 30.000

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18
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1 mark

The height, in meters, of a model rocket t seconds after launch is modeled by the function h given by h open parentheses t close parentheses equals negative 4.9 t squared plus 30 t for 0 less or equal than t less or equal than 6.

Based on the model, for how many seconds is the height of the rocket greater than 30 meters?

  • 1.259

  • 3.605

  • 4.864

  • 6.122

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19
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1 mark

The function space f is defined by space f open parentheses x close parentheses equals 2 x plus a for x less than 3, and by space f open parentheses x close parentheses equals x squared minus 4 x plus b for x \geq 3, where a and b are constants. The two pieces of space f agree in value at x equals 3, and space f open parentheses 0 close parentheses equals negative 1. What is the value of a plus b?

  • -7

  • -1

  • 5

  • 7

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20
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1 mark

A polynomial function space f has the following two properties:

  • space f has exactly 1 distinct real zero.

  • space f has exactly 2 local maxima and exactly 2 local minima.

What is the smallest possible degree of space f?

  • 3

  • 4

  • 5

  • 6

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