A startup company that develops and sells high-efficiency electric bicycles predicts that the profit

Question

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$$P = 10 \ln(x+1) - 2x, x \geq 0.$$ Sketch the graph of $P$ against $x$, stating the coordinates of the axial intercepts and the maximum value of $P$.[3] 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] 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 $q = 4$. Find the values of $a$, $b$ and $c$.[4] 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]

Accepted Answer

(iii)1.33 (iv)a=4, b=-8, c=5 (v)minimum value of W = 1

2024 TMJC H1 Q5

2024 TMJC H1 Q5