Modeling a non-periodic context
A scooter-share program
- 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 city runs a scooter-share program. The number of scooters in the program, in hundreds, is recorded at the start of the program (t = 0 years) and again at t = 3 years.
The number of scooters, in hundreds, can be modeled by the function E given by E(t) = a · bᵗ, where t is the number of years since the program started and a and b are positive constants.
Source: Hypothetical scenario
Scooters in the program
| Years since start, t | 0 | 3 |
|---|---|---|
| Scooters (hundreds) | 12 | 21 |
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 E(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 number of scooters, in hundreds of scooters per year, from t = 0 to t = 3 years. 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) For an exponential model such as E, the output values over equal-length input intervals change by the same factor. Use this property and the values E(0) = 12 and E(3) = 21, but not your values of a and b, to find E(6). Show the work that leads to your answer.
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Part (c)
1 pointAt t = 6 years, the program actually had 31.2 hundred scooters. Find the residual for the model E at t = 6, and explain what the residual tells you about the model's prediction at t = 6.
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