Aim for FivePrecalculus

Modeling a non-periodic context

A cooling cup of tea

  • 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 cup of hot tea is set on a table in a room whose temperature stays at 22 degrees Celsius (°C). The temperature of the tea is measured at time t = 0 minutes and again at t = 10 minutes.

The temperature of the tea, in °C, can be modeled by the function T given by T(t) = a · bᵗ + 22, where t is the number of minutes since the first measurement and a and b are constants.

Source: Hypothetical scenario

Temperature of the tea

Minutes since first measurement, t010
Temperature of the tea (°C)9060

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 T(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 temperature of the tea, in degrees Celsius per minute, from t = 0 to t = 10 minutes. Show the computations that lead to your answer. (ii) Use the average rate of change found in part (b)(i) to estimate the temperature of the tea, in degrees Celsius, at t = 4 minutes. Show the work that leads to your answer. (iii) For each time t, let Aₜ be the estimate of the tea's temperature found with the average rate of change from part (b)(i). Your estimate in part (b)(ii) is A₄, and A₄ > T(4). Explain why Aₜ > T(t) for every t in the interval 0 < t < 10. Refer to the graph of T and where it lies relative to the estimates Aₜ.

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Part (c)

1 point

The model T assumes that the temperature of the room stays at 22 °C. Based on the model, will the temperature of the tea ever reach 20 °C? Give a reason for your answer based on the behavior of T as t increases.

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