2024 TMJC H1 Q5

A startup company that develops and sells high-efficiency electric bicycles predicts that the profit, \(P\) (in hundreds of dollars) generated from selling one newly launched electric bicycle is related to its production cost, \(x\) (in hundreds of dollars) by the equation\[P = 10 \ln(x+1) - 2x, x \geq 0.\]

1. Sketch the graph of \(P\) against \(x\), stating the coordinates of the axial intercepts and the maximum value of \(P\).[3]
2. The company’s financial officer claims that it is still profitable to continue production when the production cost is \(1500\). Justify whether you agree with the officer’s claim.[2]
3. Find the value of \(\frac{\mathrm{d}P}{\mathrm{d}x}\) at \(x = 2\).[1]

The cost function, \(W\) (in hundreds of dollars), incurred in producing \(q\) hundred units of electric bicycles is modelled by the equation\[W = aq^2 + bq + c, q > 0\] where \(a\), \(b\), and \(c\) are constants.

It is given that \(W = 5\) when \(q = 2\), \(W = 17\) when \(q = 3\), and \(W = 37\) when \(q = 4\).

4. Find the values of \(a\), \(b\) and \(c\).[4]
5. With the values of \(a\), \(b\) and \(c\) found in part (iv), use differentiation to find the minimum value of \(W\) (you do not need to justify that the value is a minimum). Give an interpretation of this value in the context of the question.[4]