Linear models and their limits

Using linear models to compare plans and find break-even points, cost and revenue for a small business, and recognising when a straight-line model stops making sense.

This is where linear relationships earn their keep. Phone plans, gym memberships, car hire, a small side business: each one is a pair of lines, and the question is always the same. Which is better for me, and when does that change?

Comparing two options

Graph both models on the same axes. The lower line is cheaper. Where the lines cross, the two cost the same: that is the break-even point. To find it exactly, set the two rules equal and solve.

Example 1●●

Plan A costs $\$20$ a month plus $\$4$ per GB of data. Plan B costs $\$35$ a month plus $\$1.50$ per GB. Find the break-even point and say which plan suits a heavy user.

$A = 4g + 20$ and $B = 1.5g + 35$

Write each plan as a rule, with g the GB used.

$4g + 20 = 1.5g + 35$

Break even: the costs are equal.

$2.5g = 15$, so $g = 6$

Subtract 1.5g and 20 from both sides.

$A = 4 \times 6 + 20 = 44$. Both cost $\$44$ at $6$ GB.

Substitute into either rule to get the cost.

Under $6$ GB, Plan A is cheaper. Over $6$ GB, Plan B is cheaper, so a heavy user should choose Plan B.

Plan B has the smaller gradient, so it rises more slowly and is lower past the crossing.

InteractiveWhich plan is cheaper?

Try this

Start with the phone plans and check the break-even point from Example 1. Then raise Plan B's monthly fee by $\$5$ and predict which way the crossing moves before you look. Finally, pick the gym and find the smallest number of visits a month at which the membership is worth it.

Cost, revenue and profit

Cost is usually linear: a fixed set-up cost plus a cost for each item.
Revenue (money coming in) is selling price times the number sold: direct variation.
Profit is revenue minus cost. The business breaks even where the lines cross. Below that it makes a loss.

Example 2●●

Arjun prints custom phone cases. He spent $\$120$ on a design licence, each case costs him $\$6$ to make, and he sells them for $\$14$. Find the break-even point and his profit on $40$ cases.

$C = 6n + 120$ and $R = 14n$

Cost: fixed plus per item. Revenue: price times number.

$14n = 6n + 120$, so $8n = 120$ and $n = 15$ cases

Break even where they are equal.

For $40$ cases: $R = 560$, $C = 360$, so the profit is $\$200$.

Profit is revenue minus cost.

When the model stops working

Every model is a simplification. A good answer says where it makes sense.

Domain. Many models only make sense for some inputs. You can't sell $-3$ cases, a tank can't hold less than $0$ L, and a phone plan may cap data at $40$ GB.
Whole numbers. Cases, people and visits come in whole numbers, so the "line" is really a set of dots. A break-even of $3.75$ visits means $4$ visits in practice.
The rate changes. Bulk discounts, surge pricing, a car using more fuel at high speed: real rates aren't always constant.
Extrapolation. A trend that held over the data may not hold far beyond it.

Example 3●●●

A car's fuel tank holds $50$ L. On a highway trip the fuel left is modelled by $F = 50 - 0.075d$ for $d$ km. State a sensible domain and one limitation of the model.

$0 = 50 - 0.075d$ gives $d \approx 667$ km.

The tank is empty when F = 0.

Domain: $0 \le d \le 667$

Distance can't be negative and the fuel can't go below zero.

Limitation: fuel use changes with speed, hills, air conditioning and city driving, so the real rate isn't constant.

The gradient assumes a constant 7.5 L per 100 km.

Your turn

Car hire company A charges $\$70$ a day plus $25$ cents per km. Company B charges $\$40$ a day plus $40$ cents per km. For a one-day hire, find the distance at which they cost the same, and the cost.

Show the working

$70 + 0.25k = 40 + 0.4k$, so $30 = 0.15k$ and $k = 200$ km. Cost: $70 + 0.25 \times 200 = \$120$. Under $200$ km B is cheaper, over $200$ km A is cheaper.

Flashcards

Break-even point

Where two lines cross: set the rules equal and solve

Profit

Revenue − cost, including fixed costs

Limitations of a linear model

Domain, whole numbers, rates that change, extrapolation

Practice

Plan A: $A = 4g + 20$. Plan B: $B = 1.5g + 35$. At how many GB do they cost the same?
What is the cost of each plan, in dollars, at that point?
For $10$ GB a month, which plan is cheaper?
Cost $C = 6n + 120$ and revenue $R = 14n$. How many items must be sold to break even?
Using the same rules, what is the profit, in dollars, on $40$ items?
A small business sells $10$ items, but its break-even point is $15$ items. It has:
Casual gym visits cost $\$18$ each. A membership costs $\$45$ a month plus $\$6$ per visit. What is the smallest whole number of visits in a month for which the membership is cheaper?
Which is a limitation of modelling a phone's battery with $B = 100 - 12t$?
Exam style: Hire company A charges $\$70$ a day plus $\$0.25$ per km, and B charges $\$40$ a day plus $\$0.40$ per km. For a one-day hire, at how many km do they cost the same?
Exam style: What is the cost, in dollars, of either hire at that distance?