Mike sets up a company to manufacture and sell cameras. He believes that the cost, $C$ thousand do

Question

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 $$C = 60 - \sqrt{t + 1} - \frac{5}{\sqrt{t + 1}}, \quad \text{for}\hspace{0.5em} 0 \le t \le 10.$$ Without using a calculator, find the coordinates of the turning point on the graph of $C$ against $t$.[3] Sketch the graph of $C$ against $t$, indicating the coordinates of any intersections with the axes.[2] Using your calculator, find the value of $\displaystyle\int\nolimits_{0}^{10} \left( 60 - \sqrt{t + 1} - \frac{5}{\sqrt{t + 1}} \right) \, \mathrm{d}t$. 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. Explain why $k = h$.[1] Hence, find the exact values of $k$ and $h$, given that $R = 8$ when $t = 3$.[2] Find the rate at which $R$ is increasing when $t = 3$.[1] Find the range of values of $t$ for which the revenue exceeds the cost.[1]

Accepted Answer

(a)$(4,60-2\sqrt 5)$ (c)553 (e)$k=h=\frac{8}{e^3-1}$ (f)$\frac{dR}{\mathrm{d}t}=8.42$ (g)$4.89 < t \leq 10$

2024 EJC H1 Q5

2024 EJC H1 Q5