When Does Making More Get Cheap Enough?

A bakery pays fixed costs for its oven and rent plus a cost for each loaf. The average cost per loaf, total cost divided by loaves, falls as production grows and levels off near the per-loaf cost. Asking when it drops below a target price is a rational inequality.

How it works

Rational functions and rational inequalities

In one sentence

A rational function is a quotient $P(x)/Q(x)$ of polynomials; $\frac{F + vx}{x}$ has a vertical asymptote at $x = 0$ and a horizontal asymptote $y = v$ . To solve a rational inequality, find where the expression equals the bound or is undefined, then test the sign on each interval; here, for $x > 0$ , $\frac{F+vx}{x} \le p \iff (p - v)x \ge F$ .

General formula

{\htmlData{sym=0}{\bar C(x)}} = \frac{{\htmlData{sym=1}{F}} + {\htmlData{sym=2}{v}}{\htmlData{sym=3}{x}}}{\htmlData{sym=3}{{x}}}, \qquad \frac{{\htmlData{sym=1}{F}} + {\htmlData{sym=2}{v}}{\htmlData{sym=3}{x}}}{\htmlData{sym=3}{{x}}} \le {\htmlData{sym=4}{p}} \iff {\htmlData{sym=3}{x}} \ge \frac{\htmlData{sym=1}{{F}}}{{\htmlData{sym=4}{p}} - {\htmlData{sym=2}{v}}} \quad ({\htmlData{sym=4}{p}} > {\htmlData{sym=2}{v}},\ {\htmlData{sym=3}{x}} > 0)

Each extra loaf shares the fixed costs with all the others, so the share per loaf keeps shrinking, but never below what the flour and labor cost.

What the symbols mean

$\bar C(x)$
Average cost
Total cost divided by the number of loaves.
$F$
Fixed costs
Daily costs that don't depend on output: rent, the oven, insurance.
$v$
Cost per loaf
Flour, energy and labor for each extra loaf.
$x$
Loaves
How many loaves are made per day.
$p$
Target price
The price per loaf the average cost has to beat.

Total cost is the fixed costs plus a cost per loaf, $F + vx$ . Dividing by the number of loaves $x$ gives the average cost $\bar C(x) = \frac{F + vx}{x} = v + \frac{F}{x}$ .
The term $F/x$ shrinks as $x$ grows, so the average falls toward the horizontal asymptote $y = v$ ; the vertical asymptote at $x = 0$ is the cost of making nothing.
To sell at price $p$ without losing money, solve $\frac{F + vx}{x} \le p$ . For $x > 0$ , multiplying by $x$ keeps the inequality's direction: $F + vx \le px$ , so $x \ge \frac{F}{p - v}$ .
If the price is at or below the per-loaf cost ( $p \le v$ ) there's no solution: no volume can rescue a price below what each loaf costs to make.

Learn the math

Rational InequalitiesAlgebra · Lesson 18, Solving Equations and Inequalities Rational FunctionsAlgebra · Lesson 33, Common Graphs