Methods2023 Exam 1 · Q9Tech-free6 marks

Walking tracks and a maximum-area theme park

The shapes of two walking tracks are shown below.

Track 1 is described by the function $f(x) = a - x(x - 2)^2$ .

Track 2 is defined by the function $g(x) = 12x + bx^2$ .

Given that $f(0) = 12$ and $g(1) = 9$ , verify that $a = 12$ and $b = -3$ .

1 mark

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Verify that $f(x)$ and $g(x)$ both have a turning point at $P$ .

Give the co-ordinates of $P$ .

2 marks

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A theme park is planned whose boundaries will form the triangle $\Delta OAB$ where $O$ is the origin, $A$ is at $(k, 0)$ and $B$ is at $\left(k, g(k)\right)$ , as shown below, where $k \in (0, 4)$ .

Find the maximum possible area of the theme park, in km $^2$ .

The same two tracks as before, with a triangle OAB drawn using dashed lines: O is the origin, A is a point (k, 0) on the x-axis, and B is directly above A on track 2 (the dashed parabola), with a right angle marked at A.

3 marks

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