## Exam-Style Questions on Modelling## Problems 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).

## 2. | IB Standard |

A Big Wheel at an amusement park has a diameter of length 70 metres which rotates at a constant speed. The bottom of the wheel is h metres above the ground. A seat starts at the bottom of the wheel.The wheel completes one revolution in 4 minutes. [The diagram is not to scale]

(a) After 2 minutes, the seat is 76m above the ground. Find h.

(b) After t minutes, the height of the seat above ground is given by \(f(t) =41+ k \cos{ \frac{\pi t}{2}} \) for \(0 \le t \le 16 \).

Find the value of k.

(c) Find when the seat is 40 m above the ground for the third time.

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