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 \(g(x)=12x-bx^2\).
The unit of length is kilometres.
- 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)--- 8 WORK AREA LINES (style=lined) ---
- 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 \((k, g(k))\), as shown below, where \(k \in (0, 4)\).
Find the maximum possible area of the theme park, in km². (3 marks)
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