AI Tutor
If it costs 500tosetupascreenprinterplus500 to set up a screen printer plus 5 per shirt, printing one shirt costs an eye-watering 505.Butprint1,000shirts,andeachonecostsjust505. But print 1,000 shirts, and each one costs just 5.50.
A rational function is any function formed by dividing one polynomial by another: f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)}, where Q(x)β‰ 0Q(x) \neq 0.
In the real world, rational functions model situations where a fixed resource or quantity is distributed across a changing scale, such as unit costs, chemical concentrations, and shared work rates.
How do fixed setup costs change the price of every single item you make?

Average Cost Models

Total production cost is C(x)=F+vxC(x) = F + v x, where FF is the one-time fixed cost (like machinery) and vv is the variable cost per item (like materials).
To find the average cost per unit Cˉ(x)\bar{C}(x), we divide the total cost by the number of units xx:
Cˉ(x)=C(x)x=F+vxx=Fx+v\bar{C}(x) = \frac{C(x)}{x} = \frac{F + vx}{x} = \frac{F}{x} + v
As xβ†’βˆžx \to \infty, the term Fx\frac{F}{x} shrinks to 00. This means the horizontal asymptote y=vy = v represents the theoretical lowest possible average cost per unit.
Let's solve a production target: suppose a company has fixed costs of 1,200andavariablecostof1,200 and a variable cost of 4 per item. How many items must they produce to bring the average cost down to $6 per item?
Set CΛ‰(x)=6\bar{C}(x) = 6 and solve for xx: 1200+4xx=6β€…β€ŠβŸΉβ€…β€Š1200+4x=6xβ€…β€ŠβŸΉβ€…β€Š1200=2xβ€…β€ŠβŸΉβ€…β€Šx=600\frac{1200 + 4x}{x} = 6 \implies 1200 + 4x = 6x \implies 1200 = 2x \implies x = 600