Modeling a non-periodic context

A reservoir in the dry season

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

A city's reservoir is measured during the 10 months after the start of a dry season. The volume of water in the reservoir at the start of the dry season (t = 0) is 50 million cubic meters. The volume is measured again at t = 2 months and t = 6 months.

The volume of water in the reservoir, in millions of cubic meters, can be modeled by the function R given by R(t) = 0.2t³ + bt² + ct + 50, where t is the number of months since the start of the dry season, 0 ≤ t ≤ 10, and b and c are constants.

Source: Hypothetical scenario

Volume of water in the reservoir

Months since start of dry season, t	0	2	6
Volume (millions of cubic meters)	50	57.6	39.2

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 b and c in the expression for R(t). (ii) Find the values for b and c.

Your answer to part (a)

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Part (b)

3 points

(i) Use the given data to find the average rate of change of the volume of water in the reservoir, in millions of cubic meters per month, from t = 2 to t = 6 months. 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) Use the model R to find all intervals of t in 0 ≤ t ≤ 10 on which the volume of water in the reservoir is increasing. Give the endpoints as decimal approximations.

Your answer to part (b)

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Part (c)

1 point

If the volume of water in the reservoir falls below 30 million cubic meters, the city must restrict water use. Based on the model R, will the city need to restrict water use at any time during 0 ≤ t ≤ 10? Give a reason for your answer using the model.

Your answer to part (c)

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