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

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 $p={\color {NavyBlue}-{\frac {q}{5}}+20}\,.$