2024 JPJC H1 Q5

John has a proposal to start a business and hires a consultant to help him project the costs and sales over a period of six years before deciding whether to proceed to start the business. The consultant modelled the cost of running the business, \(C\) thousand dollars per year, at any point in time. The model he uses is

\[C = t^3 - 8t^2 + 16t + 30, \quad \text{for}\hspace{0.5em} 0 \le t \le 6\]

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 minimum or a maximum.[4]
2. Calculate the value of \(\displaystyle\int_{0}^{6} t^3 - 8t^2 + 16t + 30 \, \mathrm{d}t\). In the context of the question, what does this value represent?[2]

The consultant also modelled the sales, \(S\) thousand dollars per year, at any point in time using the model

\[S = 3e^{1.5t-5} + 30, \quad \text{for}\hspace{0.5em} 0 \le t \le 6\]

3. Sketch the graph of \(S\) against \(t\), stating the exact coordinates of any intersections with the coordinate axes.[2]
4. Find \(\frac{\mathrm{d}S}{\mathrm{d}t}\) and explain in context what the expression for \(\frac{\mathrm{d}S}{\mathrm{d}t}\) indicates about the sales over the period of six years.[2]
5. Find the exact value of \(\displaystyle\int_{0}^{6} 3e^{1.5t-5} + 30 \, \mathrm{d}t\).[2]
6. Using your answers in (ii) and (v), justify if John should proceed with the proposal to start the business.[1]