2024 RVHS H1 Q5

ABC Health Product Company is embarking on a \(6\)-year plan for the production and sale of a new health product for young adults. Mr Lim, the company’s finance manager, models the cost of production, \(C\) thousand dollars per year, at time \(t\) years into the plan using the equation \(C=t^3-9t^2+15t+55\), for \(0 \le t \le 6\).

1. Use differentiation to find the values of \(t\) which give stationary points on the graph of \(C\) against \(t\). For each point, justify whether it is a maximum or minimum.[5]
2. Sketch the graph of \(C\) against \(t\), giving the coordinates of the end points and any intersections with the axes.[2]

Mr Lim also projects the revenue from sales of the new health product, \(R\) thousand dollars per year, at time \(t\) years using the equation \(R=36\ln(t+1)\), for \(0 \le t \le 6\).

3. Sketch the graph of \(R\) against \(t\) for \(0 \le t \le 6\), indicating the coordinates of the end points.[1]
4. Find the value of \(\displaystyle\int_0^6 36\ln(t+1) \, \mathrm{d}t\), correct to \(3\) decimal places and provide an interpretation of this value in the context of the question.[2]
5. Hence, determine with reason whether the company should embark on this \(6\)-year plan for the new health product.[2]