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Exam-Style Questions on ModellingProblems on Modelling adapted from questions set in previous Mathematics exams. |
1. | IB Applications and Interpretation |
In a fantasy story the power value of a dream catcher varies depending on its length. The power values of various dream catchers are recorded in the following table:
| Length, \(x\) cm | 0 | 10 | 15 |
| Power, \(p\) W | 0 | 12 | 22 |
This information was used to create Model A, where \(p\) is a function of \(x\) , \(x \ge 0\).
Model A: \(p(x) = ax^2 + bx\) , where \(a,b \in \mathbb{Z}\).
When the length is 10 cm, Model A can be represented by the equation 50a + 5b = 6.
(a) Write down a second equation to represent Model A, when the length is 15cm.
(b) Find the values of a and b.
(c) Find the coordinates of the vertex of the graph of \(y = p(x)\).
(d) Using the values in the table and your answer to part (c), sketch the graph of \(y = p(x)\) for \(0 \le x \le 15\) and \(0 \le p \le 22\).
Additional data was used to create Model B, a revised model for the power of a dream catcher.
Model B: \(p(x) = 0.06x^2 + 0.68x\)
(e) Use Model B to calculate an estimate for the power of a dream catcher of length 18cm.
The actual power of a dream catcher of length 18cm is 30 W.
(f) Calculate the percentage error in the estimate in part (e).
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