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 \(\frac{dC}{\mathrm{d}x} =-2e^{-0.5x}\).
It is given that the cost incurred when no machine is produced is $7000. Show that \( C =ae^{-0.5x}+b\), where a and b are constants are determined.
For the following parts, use a = 4 and b = 2[3]
Sketch the graph of C against \(x\), stating the equation of the asymptote and the coordinates of the point of intersection with the \(y-axis\).
It is given that the revenue generated, $R, (in thousands) from selling \(x\) machines in hundreds, is given by \($(10-\frac{1}{5}x-\frac{10}{2x+1})\)[2]
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]
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]