# Exam-Style Question on Modelling

## A mathematics exam-style question with a worked solution that can be revealed gradually

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Question id: 472. This question is similar to one that appeared on an IB AI Standard paper (specimen) for 2021. The use of a calculator is allowed.

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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