2024 DHS H1 Q4
2024 DHS H1 Q4
8 marks
A mobile app developer develops a new mobile game “MacCraft”. He will monitor the revenue generated from the sales of his mobile game over a period of \(6\) years, before he decides whether to continue with the sales of the mobile game. The mobile app developer projects that the revenue, \(R\) thousand dollars per year can be modelled by
\[R = 5t + 1 - \mathrm{e}^{2t-10},\]
where \(t\) is the time in years.
- Sketch the graph of \(R\) against \(t\) for \(0 \le t \le 6\), giving the coordinates of the stationary point and end-points. Explain the significance of the \(y\)-intercept in the context of the question.[3]
- Find the exact area of the region bounded by the graph of \(R\), the \(t\)-axis and the lines \(t = 0\) and \(t = 3\). Give an interpretation of the area you have found.[4]
- In the context of the question, explain the significance of the shape of the graph found in part (a).[1]
Solution:
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Answer:(a) Maximum \(\left(5+\frac{1}{2}\ln\frac{5}{2},\frac{47}{2}+\frac{5}{2}\ln\frac{5}{2}\right)\); endpoints \((0,1-\mathrm e^{-10})\), \((6,31-\mathrm e^2)\). The intercept is the initial revenue rate. (b) \(\frac{51}{2}-\frac{1}{2}\mathrm e^{-4}+\frac{1}{2}\mathrm e^{-10}\) thousand dollars: total revenue in the first three years. (c) Annual revenue rises to about year 5.46, then falls.
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