Science:Math Exam Resources/Courses/MATH104/December 2015/Question 08 (b)/Solution 1

Let $R,C,P$ denote the revenue, cost, and profit functions, respectively. We seek to maximize the profit $P(p)$ over integer prices $p.$

By definition, revenue is the unit price times the quantity demanded, i.e.,

$R(p)=p\cdot q(p).$

Cost is the sum of fixed and variable costs; we are given that

• "the weekly fixed costs will be ${\color {OrangeRed}\$140}$"
• "the variable cost of production will be ${\color {OliveGreen}\$4}$ per toaster"

Note that the cost is a function of the number of units produced, i.e., the weekly demand $q.$ We therefore have

$C(q)={\color {OrangeRed}140}+{\color {OliveGreen}4}q,$

where $q$ is multiplied by 4 since each toaster costs $4 to produce (while the fixed costs remain constant).

Finally, profit is revenue minus cost. From part (a) we know that $q(p)={\color {Orange}-5(p-16)+20}$ ($q$ itself is a function of $p$), so

${\begin{aligned}P(p)&=R(p)-C(q(p))\\&=p\cdot q(p)-[140+4\cdot q(p)]\\&=(p-4)\cdot q(p)-140\\&=(p-4)[{\color {Orange}-5(p-16)+20}]-140\\&=(p-4)(-5p+100)-140\\&=-5p^{2}+120p-540,\end{aligned}}$

which is maximized when $p={\tfrac {-120}{2\cdot -5}}=12.$ Therefore the company should charge $\color {blue}\$12$ for each toaster.