Modeling a non-periodic context
An invasive pond plant
- 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
An invasive water plant is spreading across the surface of a pond whose total surface area is 2,000 square meters (m²). Biologists measure the area of the pond's surface covered by the plant at time t = 2 weeks and again at t = 7 weeks after they first noticed it.
The area covered by the plant, in m², can be modeled by the function A given by A(t) = a · bᵗ, where t is the number of weeks since the plant was first noticed and a and b are positive constants.
Source: Hypothetical scenario
Area covered by the plant
| Weeks since first noticed, t | 2 | 7 |
|---|---|---|
| Area covered (m²) | 18 | 75 |
Source: Hypothetical data
Suggested time: 18 minutes
Your answers are saved in this browser as you type.
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 A(t). (ii) Find the values for a and b as decimal approximations.
0 / 2,500 characters
Part (b)
3 points(i) Use the given data to find the average rate of change of the area covered by the plant, in square meters per week, from t = 2 to t = 7 weeks. 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) A biologist makes a semi-log plot of the model: for several values of t, she plots the points (t, log₁₀ A(t)) and finds that they lie on a line. Find the slope of this line as a decimal approximation. Show the work that leads to your answer.
0 / 2,500 characters
Part (c)
1 pointThe biologists want to use the model A to predict the area covered by the plant far into the future. Explain why the model A cannot give reasonable predictions for all values of t ≥ 2. Use a value from the model to support your explanation.
0 / 2,500 characters
Checking scoring…
Scoring it yourself shows you the rubric, examples and a model answer. Try writing your answer first.