2024 TJC H1 Q5

Jane starts up a company that produces and sells a particular type of machine. The total cost incurred, \(C\), in thousands, to produce \(x\) machines, in hundreds, is related by \(\frac{dC}{\mathrm{d}x} =-2e^{-0.5x}\).

1. It is given that the cost incurred when no machine is produced is 7000 dollars. Show that \( C =ae^{-0.5x}+b\), where a and b are constants to be determined.
   
   For parts (b)–(d), use the different prescribed cost model \(C=4e^{-0.5x}+2\).[3]
2. Sketch the graph of C against \(x\), stating the equation of the asymptote and the coordinates of the point of intersection with the \(C\)-axis.
   
   It is given that the revenue generated, \(R\), (in thousands) from selling \(x\) machines in hundreds, is given by \(R=10-\frac{1}{5}x-\frac{10}{2x+1}\)[2]
3. Use a non-calculator method to find the number of machines that she should sell to attain the maximum revenue, justifying that this value gives the maximum.[5]
4. By forming a suitable inequality and adding a suitable graph on the sketch in (b), find the maximum number of machines that Jane should produce to sell so she will make a profit.[3]