Aim for FivePrecalculus

Modeling a non-periodic context

Medicine in the bloodstream

  • 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 patient receives a medicine through an IV drip that starts at time t = 0 hours and stops at t = 2 hours. While the drip is running, the amount of medicine in the patient's bloodstream increases at a constant rate. After the drip stops, the amount decreases. Measurements show 50 milligrams (mg) of medicine in the bloodstream at t = 2 and 21.6 mg at t = 5.

The amount of medicine in the bloodstream, in mg, is modeled by the piecewise function M given by M(t) = kt for 0 ≤ t < 2, and M(t) = a · bᵗ for t ≥ 2, where t is the number of hours since the drip started and k, a and b are positive constants.

Source: Hypothetical scenario

Medicine measurements

Hours since drip started, t25
Medicine in bloodstream (mg)5021.6

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 M(t) when t ≥ 2. (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 amount of medicine in the bloodstream, in milligrams per hour, from t = 2 to t = 5 hours. 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) Is the amount of medicine in the bloodstream decreasing faster at t = 2.5 hours or at t = 4.5 hours? Give a reason for your answer based on the graph of M for t ≥ 2.

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Part (c)

1 point

The model assumes that the amount of medicine in the bloodstream does not jump suddenly at the moment the drip stops. Use this assumption to find the value of k. Explain your reasoning.

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