Modeling a non-periodic context
Salt water in a tank
- 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 large tank holds 50 liters (L) of water with 100 grams (g) of salt dissolved in it. Starting at time t = 0 minutes, salt water is pumped into the tank at a constant rate. The salt water entering the tank always has the same concentration. The tank is stirred so the salt is evenly mixed, and no water leaves the tank.
The concentration of salt in the tank, in grams per liter (g/L), can be modeled by the function C given by C(t) = (100 + at)/(50 + bt), where t is the number of minutes since pumping began and a and b are positive constants. The concentration is measured at t = 5 and t = 10 minutes.
Source: Hypothetical scenario
Salt concentration in the tank
| Minutes since pumping began, t | 5 | 10 |
|---|---|---|
| Concentration (g/L) | 4 | 5 |
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 C(t). (ii) Find the values for a and b.
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Part (b)
3 points(i) Use the given data to find the average rate of change of the salt concentration, in grams per liter per minute, from t = 5 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 salt concentration, in grams per liter, at t = 8 minutes. Show the work that leads to your answer. (iii) Let Eₜ represent the estimate of the salt concentration at time t, using the average rate of change from part (b)(i). Is E₈ greater than or less than C(8)? Explain your answer with a reference to the graph of C and its relationship to Eₜ.
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Part (c)
1 pointA student claims that if the tank were large enough to keep filling forever, the salt concentration would eventually rise above 8 grams per liter. Do you agree or disagree with this claim? Give a reason for your answer based on the model.
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