Aim for FivePrecalculus

Modeling a non-periodic context

A fish population

  • Unit 1
  • 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

Biologists track the population of a species of fish in a lake. They take counts at the start of 2018 (time t = 0), 2021 (t = 3) and 2024 (t = 6).

The population, in hundreds of fish, can be modeled by the function P given by P(t) = at² + bt + c, where t is the number of years since the start of 2018 and a, b and c are constants.

Source: Hypothetical scenario

Fish population

Years since start of 2018, t036
Population (hundreds of fish)4.09.411.2

Source: Hypothetical data

Suggested time: 18 minutes

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Part (a)

2 points

(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 P(t). (ii) Find the values for a, b and c.

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Part (b)

3 points

(i) Use the given data to find the average rate of change of the fish population, in hundreds of fish per year, from t = 0 to t = 3. Show the computations that lead to your answer. (ii) Use the average rate of change found in part (b)(i) to estimate the fish population, in hundreds of fish, at t = 2. Show the work that leads to your answer. (iii) Let Eₜ represent the estimate of the population at time t, using the average rate of change from part (b)(i). Is E₂ greater than or less than P(2)? Explain your answer with a reference to the graph of P and its relationship to Eₜ.

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Part (c)

1 point

Use the model P to predict the fish population at the start of 2034 (t = 16). Based on your answer and the context of the problem, explain whether the model is appropriate for making this prediction.

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