Practice Maths

Linear Models and Applications

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Key Terms

A linear model uses y = mx + c to represent a real-world situation where one quantity changes at a constant rate relative to another.
The gradient m = rate of change, with units (e.g. dollars per km, litres per minute). Always state the units!
The y-intercept c = initial or starting value (e.g. flag-fall charge, initial volume, starting temperature).
Interpolation
: finding values within the known data range (reliable). Extrapolation: predicting beyond the known range (less reliable).
Domain
: restrict x to sensible values only (e.g. x ≥ 0 for time, x ≤ max capacity for resources).
📚 Building a Linear Model
Step 1: Identify x (independent variable) and y (dependent variable).
Step 2: Find m = rate of change (often stated directly, or calculated from two points).
Step 3: Find c = starting value when x = 0.
Step 4: Write y = mx + c and state the domain.
Step 5: Interpret — always answer “what does the gradient mean?” and “what does the y-intercept mean?” in context.
d (km) C ($) 2 4 6 8 10 0 5 10 15 20 c = $4.50 (flag-fall) run = 2 km rise = $5 ($2.50/km) C = 2.5d + 4.5

Worked Example 1 — Building a model from a description

A plumber charges a $65 call-out fee plus $90 per hour. Write a linear model and find the cost for a 3.5-hour job.

Step 1 — Define variables: Let h = hours worked, C = total cost ($).

Step 2 — Identify gradient and y-intercept:

   Rate of change (gradient) m = $90/hr

   Starting value (y-intercept) c = $65 (call-out fee when h = 0)

Step 3 — Write the model: C = 90h + 65   (for h ≥ 0)

Step 4 — Substitute h = 3.5: C = 90(3.5) + 65 = 315 + 65 = $380

Interpret: The $65 covers the call-out (c = 65); each additional hour adds $90 (m = 90).

Worked Example 2 — Building a model from a table of values

A coffee mug is cooling. Times and temperatures are recorded:

Time (min) 0 5 10 15
Temp (°C) 85 75 65 55

Step 1: Let t = time (min), T = temperature (°C).

Step 2 — Gradient: Temperature decreases by 10°C every 5 minutes: m = −10/5 = −2 °C/min

Step 3 — y-intercept: At t = 0, T = 85, so c = 85.

Model: T = −2t + 85   (for 0 ≤ t ≤ 42.5)

Domain: T ≥ 0 means −2t + 85 ≥ 0, so t ≤ 42.5 min.

Hot Tip — Always include units in your gradient. “m = −2” means nothing on its own; “m = −2 °C/min” tells a full story. Similarly, state what x and y represent and their units when writing a model. Examiners award marks for context, not just algebra.

Introduction

Linear models are used to describe countless real-world situations: taxi fares, water tank levels, temperature changes, phone plans, spring lengths, and more. The power of a linear model is that once you identify the gradient and y-intercept, you can predict any value and answer questions about the situation mathematically.

Identifying Variables and Parameters

The first step is always to define your variables carefully. Ask: “What changes? What does it change with respect to?” The quantity that changes due to another is y (dependent), and the quantity that drives the change is x (independent).

Example: Taxi fare

A taxi charges $3.50 flag-fall plus $2.20 per km. Here:

x = distance (km) — the driver controls how far you go

y = C = cost ($) — depends on distance

Gradient m = 2.20 ($/km) — cost increases $2.20 per km

y-intercept c = 3.50 — cost before moving (flag-fall)

Model: C = 2.20d + 3.50

Finding the Model from Two Points

If you are given a table or two data points, use the gradient formula m = (y&sub2;−y&sub1;)/(x&sub2;−x&sub1;) to find the rate of change, then substitute one point to find c.

Example: Car fuel consumption

A car has 60 L at the start of a trip. After 200 km, it has 40 L.

m = (40 − 60)/(200 − 0) = −20/200 = −0.1 L/km

c = 60 (initial fuel at d = 0)

Model: F = −0.1d + 60

Tank empty when F = 0: 0 = −0.1d + 60 → d = 600 km

Comparing Two Linear Models

To find when two models give the same value, set them equal and solve. This is the break-even point — the value of x where both models produce the same y.

Example: Comparing two phone plans

Plan A: $20/month + $0.10/text. Plan B: $35/month + $0.04/text.

A = 0.10n + 20   B = 0.04n + 35

Set equal: 0.10n + 20 = 0.04n + 35 → 0.06n = 15 → n = 250 texts

For fewer than 250 texts: Plan A is cheaper. For more than 250: Plan B is cheaper.

Domain and Reasonableness

Always restrict the domain to values that make physical sense. A water tank cannot have negative volume; time cannot be negative; the number of items bought must be a non-negative integer. State the domain explicitly.

💡 Extrapolation warning: A model that fits data well over a certain range may not hold outside it. For example, a cooling coffee model predicts T = 0°C at some point — in reality the coffee reaches room temperature and stops cooling. Always check whether extrapolation makes physical sense.

Summary

To build a linear model: define x and y with units; find m (rate of change) and c (initial value); write y = mx + c; state the domain. To compare models: set equal and solve. Always interpret m and c in context. Be cautious when extrapolating beyond the data.

Mastery Practice

See Answers ➔
  1. Fluency

    A taxi charges a flag-fall of $3.50 plus $2.20 per kilometre.

    1. (a) Write the cost function C as a linear equation in terms of distance d (km).
    2. (b) Find the cost of a 15 km trip.
    3. (c) If a trip costs $25.50, how far was the journey?
  2. Fluency

    A phone plan charges $30 per month plus $0.15 per text message.

    1. (a) Write the monthly cost equation C in terms of n (number of texts).
    2. (b) Find the cost if 80 texts are sent.
    3. (c) How many texts were sent if the monthly bill was $51?
  3. Fluency

    A car travels at constant speed. The table below records distance over time.

    Time (hr)0123
    Distance (km)12223242
    1. (a) Write the linear model for distance D as a function of time t.
    2. (b) Interpret the gradient and y-intercept in context.
    3. (c) How far will the car have travelled after 5 hours?
  4. Understanding

    A car starts a journey with 65 L of fuel. After 120 km, the fuel gauge reads 50 L.

    1. (a) Write a linear model F = md + c for the fuel remaining after d km.
    2. (b) Interpret the gradient in context (include units).
    3. (c) What distance can be travelled before the tank is empty? What assumption are you making?
  5. Understanding

    A water tank starts with 4800 L and loses 12 L per minute due to a leak.

    1. (a) Write a linear model for the volume V (litres) remaining after t minutes.
    2. (b) How many minutes until the tank holds only 1200 L?
    3. (c) When will the tank be completely empty?
    4. (d) State the domain of the model.
  6. Understanding

    Two internet plans are available:

    Plan A: $29/month + $0.05 per GB used    Plan B: $45/month + $0.02 per GB used
    1. (a) Write cost equations for each plan in terms of g (GB used per month).
    2. (b) Find the monthly usage where both plans cost the same.
    3. (c) Which plan is better for a heavy user who uses 300 GB/month? How much cheaper?
  7. Understanding

    The temperature on a winter evening drops linearly from 22°C at 6 pm to 10°C at midnight.

    1. (a) Find the rate of cooling per hour.
    2. (b) Write a linear model T = mh + c, where h = hours after 6 pm.
    3. (c) Predict the temperature at 2 am.
    4. (d) At what time will the temperature reach 5°C?
  8. Understanding

    A spring has a natural length of 8 cm. When a force of 5 N is applied, it stretches to 13 cm (Hooke’s Law).

    1. (a) Write a linear model for the spring length L (cm) in terms of force F (N).
    2. (b) Find the length when a force of 8 N is applied.
    3. (c) What force would cause the spring to reach 20 cm?
    4. (d) Is it reasonable to use this model at 100 N? Explain.
  9. Problem Solving

    Two gyms offer different membership structures:

    Gym A: $200 joining fee + $15/month    Gym B: $50 joining fee + $30/month
    1. (a) Write total cost equations for each gym in terms of n (months).
    2. (b) Find the number of months at which the total costs are equal.
    3. (c) Describe (in words) what a sketch of both lines would look like — where they start, which is steeper, and where they cross.
    4. (d) Which gym is cheaper for a 2-year membership? By how much?
  10. Problem Solving

    A car rental company charges $35 per day plus $0.22 per kilometre. The daily budget is $200.

    1. (a) Write the daily cost equation C in terms of km travelled k.
    2. (b) Find the maximum number of kilometres that can be driven in one day within budget.
    3. (c) Over 5 days at an average of 180 km/day, find the total cost.
    4. (d) An upgraded car costs $55/day + $0.12/km. Compare the total 5-day costs for both cars at 180 km/day. Which is cheaper?