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

MATH104 December 2015

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Question 08 (c)

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. (c) Find the break-even points for the company. Note: Please simplify your answer such that it is given by an integer.

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If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall that break-even points occur when the profit is zero. Typically, we express such points as demand ($q$) values for which the profit is zero.

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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. Continuing from part (b), we find the roots of the profit function $P(p)=-5p^{2}+120p-540:$ ${\begin{aligned}0&=-5p^{2}+120p-540\\&=-p^{2}+24p-108\\&=-p^{2}+6p+18p-108\\&=-p(p-6)+18(p-6)\\&=(18-p)(p-6).\end{aligned}}$ Hence break-even points occur when $p={\color {OliveGreen}6}$ and $p={\color {PineGreen}18}.$ Typically, break-even points are expressed as demand ($q$) values for which the profit is zero, so we use our result $q(p)=-5(p-16)+20$ from part (a) to convert these to $q=-5({\color {OliveGreen}6}-16)+20={\color {blue}70}$ and $q=-5({\color {PineGreen}18}-16)+20={\color {blue}10}.$