Mr Lim has just set up a company selling handphones. Over a period of $9$ months, his total profit

Question

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. 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] Sketch the graph of $P$ against $t$ for $0 \le t \le 9$.[1] Using integration, find $\displaystyle\int\nolimits_0^9 -(\mathrm{e}^{0.2t} - 4)^2 + 9 \, \mathrm{d}t$, correct to the nearest integer, showing your workings clearly. Explain, in the context of the question, what this value represent.[4] 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]

Accepted Answer

(i)$t=5\ln 4$ (iii)$50$ (iv)$\frac{\mathrm{d}A}{\mathrm{d}t}=-1746$

2024 ACJC H1 Q6

2024 ACJC H1 Q6