2024 EJC H1 Q5

Mike sets up a company to manufacture and sell cameras. He believes that the cost, \(C\) thousand dollars per year, at any point in time, \(t\) years, is modelled by

\[C = 60 - \sqrt{t + 1} - \frac{5}{\sqrt{t + 1}}, \quad \text{for}\hspace{0.5em} 0 \le t \le 10.\]

1. Without using a calculator, find the coordinates of the turning point on the graph of \(C\) against \(t\).[3]
2. Sketch the graph of \(C\) against \(t\), indicating the coordinates of any intersections with the axes.[2]

3. Using your calculator, find the value of \(\displaystyle\int_{0}^{10} \left( 60 - \sqrt{t + 1} - \frac{5}{\sqrt{t + 1}} \right) \, \mathrm{d}t\). In the context of the question, what does this value represent?[2]

Mike is also interested in modelling his revenue, \(R\) thousand dollars per year, at any point in time, \(t\) years. He believes that

\[R = ke^t - h,\]

where \(k\) and \(h\) are positive constants.

4. Explain why \(k = h\).[1]
5. Hence, find the exact values of \(k\) and \(h\), given that \(R = 8\) when \(t = 3\).[2]
6. Find the rate at which \(R\) is increasing when \(t = 3\).[1]
7. Find the range of values of \(t\) for which the revenue exceeds the cost.[1]