2024 ACJC H1 Q6

Mr Lim has just set up a company selling handphones. Over a period of \(9\) months, his total profit \(P\) thousand dollars per month can be modelled by

\[P = -(\mathrm{e}^{0.2t} - 4)^2 + 9,\]

where \(t\) is the time in months after he set up his company.

1. Use differentiation to find the value of \(t\) which gives a stationary point on the graph of \(P\) against \(t\). Justify whether this point is a maximum or minimum point.[4]
2. Sketch the graph of \(P\) against \(t\) for \(0 \le t \le 9\).[1]
3. Using integration, find \(\displaystyle\int_0^9 -(\mathrm{e}^{0.2t} - 4)^2 + 9 \, \mathrm{d}t\), correct to the nearest integer, showing your workings clearly. Explain, in the context of the question, what this value represent.[4]

4. Mr Lim has a budget for advertising his handphones for sale. At any time, \(t\) months, measured from the start of his advertising campaign, the remaining budget, \(A\) dollars, is modelled by\[A = \frac{200}{3}(-20t + 75 + \frac{375}{2t + 5}).\] Using differentiation, find \(\frac{\mathrm{d}A}{\mathrm{d}t}\) when \(t = 3\), giving your answer to the nearest integer. Explain, in the context of this question, what this value represent.[3]