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

MATH104 December 2015

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Question 08 (b)

A company is planning to produce a new electric toaster. After conducting an extensive market survey, the research department provides the following information: when the unit price is $16 per toaster, then the weekly demand is 20 toasters; for every $2 decrease in the unit price, the weekly demand increases by 10 toasters. The financial department estimates that the weekly fixed costs will be $140 and the variable cost of production will be $4 per toaster. Let $p$ be the unit price and let $q$ be weekly demand. (b) What price should the company charge in order to maximize the weekly profit? Note: Please simplify your answer such that it is given by an integer.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Recall that profit is revenue minus cost. What are the revenue and cost functions?

Hint 2
Recall that revenue is the unit price times the quantity demanded (see part (a)).

Hint 3
Recall that cost is the sum of fixed and variable costs, one of which depends on the number of units produced.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. 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.