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

MATH104 December 2015

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Question 08 (a)

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. (a) Find the demand function linking $p$ and $q$.

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
The relationship between $p$ and $q$ is linear. Which quantities given in the question relate price and demand?

Hint 2
Recall that the 'slope-point form' of the equation of a line is $y-y_{0}=m(x-x_{0}),$ where $(x_{0},y_{0})$ is a point on the line and $m$ is its slope/rate of change. How might you adapt this for the problem above?

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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. The two key pieces of information required for this question are: "when the unit price is $\${\color {OrangeRed}16}$ per toaster, then the weekly demand is ${\color {OliveGreen}20}$ toasters" "for every $\${\color {WildStrawberry}2}$ decrease in the unit price, the weekly demand increases by ${\color {PineGreen}10}$ toasters" from which it is clear that the unit price $p$ and the weekly demand $q$ are linearly related. Using the 'slope-point' form of the equation of a line and considering $q$ as a function of $p$, we can write $q-q_{0}=m(p-p_{0}),$ where $(p_{0},q_{0})$ is a 'coordinate' consisting of a price and the demand at that price, and $m$ is the slope of the line in the $pq$-plane. Hence we can take $(p_{0},q_{0})=({\color {OrangeRed}16},{\color {OliveGreen}20})$ and $m={\frac {\color {PineGreen}10}{-{\color {WildStrawberry}2}}}$ (when $p$ decreases by 2, $q$ increases by 10). Thus ${\begin{aligned}q-{\color {OliveGreen}20}&={\frac {\color {PineGreen}10}{-{\color {WildStrawberry}2}}}(p-{\color {OrangeRed}16})\\q&={\color {blue}-5(p-16)+20}\,.\end{aligned}}$ Note: Considering $p$ as a function of $q$ is equally correct and leads to the equivalent answer $p={\color {NavyBlue}-{\frac {q}{5}}+20}\,.$