Mathematics • Year 10 • Unit 2 • Lesson 20
Linear Models in the Real World
Build linear models for electricity bills, hire-car rental, savings goals, water-tank drainage and population growth. Interpret every gradient and intercept in plain words, and reason about interpolation versus extrapolation.
1. Word problems
For each: define variables, write the linear equation, interpret m and c in plain words, then answer the asked question.
1.1 — Electricity bill. A bill has a fixed daily supply charge of $1.20 plus $0.28 per kWh of electricity used. Let u = kWh, C = daily cost in $.
(a) Write the linear model C = ____.
(b) Interpret each of m and c in real-world words.
(c) Find the cost if 25 kWh is used in a day. 3 marks
1.2 — Hire car. One company charges $80 for the day plus $0.40/km. A second company charges no daily fee but $0.65/km. Let d = km driven.
(a) Write a linear model for each company's cost.
(b) For what distance are the two companies' costs equal? (Find the break-even d.)
(c) For a 250 km trip, which is cheaper, and by how much? 4 marks
1.3 — Savings goal. Marco has $150 saved and adds $35 per week. He wants to reach $850 to buy a guitar. Let B = balance after t weeks.
(a) Write the linear model B = ____.
(b) After how many weeks will he have $850? 3 marks
1.4 — Water tank drainage. A 500 L water tank drains at a constant rate. After 20 minutes it holds 380 L. Let W = litres remaining, t = minutes after the drain starts.
(a) Find the gradient (L/min — note the sign).
(b) Write the linear model W = mt + c (use t = 0, W = 500 as a data point).
(c) When is the tank empty? 3 marks
1.5 — Population growth. A town has 8,000 people in 2010 and 12,500 in 2020. Assume linear growth. Let t = years since 2010, P = population.
(a) Find the gradient (people per year).
(b) Write the model P = mt + c.
(c) Predict the population in 2025. (d) Predict in 2080. Which prediction is more reliable, and why? Use the word "extrapolation". 4 marks
2. Explain your thinking
Communication, not just numbers. 4 marks
2.1 Mai's plant grew from 5 cm to 15 cm over its first 10 days. A linear model gives h = 1d + 5. Using the words interpolation, extrapolation and linear model, explain (i) what the model predicts the plant's height to be on day 6, and is this prediction reliable? (ii) What the model predicts on day 100, and is that prediction reliable? (iii) Give one real-world reason why a linear plant-growth model breaks down for very long times.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Electricity bill
(a) C = 0.28u + 1.20. (b) m = $0.28/kWh (cost per kilowatt-hour used); c = $1.20 (daily supply charge — paid even if no electricity is used). (c) C = 0.28(25) + 1.20 = 7 + 1.20 = $8.20.
1.2 — Hire car
(a) Co A: C_A = 0.40d + 80. Co B: C_B = 0.65d. (b) 80 + 0.40d = 0.65d → 80 = 0.25d → d = 320 km. (c) At d = 250: C_A = 100 + 80 = $180; C_B = 162.50. Co B is cheaper by $17.50.
1.3 — Savings goal
(a) B = 35t + 150. (b) 850 = 35t + 150 → 35t = 700 → t = 20 weeks.
1.4 — Water tank
(a) m = (380 − 500)/(20 − 0) = −120/20 = −6 L/min. (b) W = −6t + 500. (c) 0 = −6t + 500 → t = ≈ 83.3 minutes.
1.5 — Population
(a) m = (12500 − 8000)/(10 − 0) = 450 people/year. (b) P = 450t + 8000. (c) At 2025 (t = 15): P = 6750 + 8000 = 14,750. (d) At 2080 (t = 70): P = 31500 + 8000 = 39,500. The 2025 prediction is much more reliable — it sits just outside the data range (0–10) and is closer to interpolation, while the 2080 prediction is far-out extrapolation. Real towns don't grow at a constant rate for 70 years (immigration patterns, infrastructure limits, etc).
2.1 — Explain (sample response)
(i) Day 6 is within the data range (0–10 days), so using h = 1(6) + 5 = 11 cm is interpolation — reliable, because we have data on both sides. (ii) Day 100 gives h = 1(100) + 5 = 105 cm. This is far beyond the data, so it is extrapolation — unreliable. (iii) Real plants stop growing once they reach a mature height; the linear model ignores this. A 105 cm-tall version of a small herb is physically impossible — the model is a useful approximation only over a small early range, not for the lifetime of the plant.
Marking: 1 for interpolation interpretation on day 6; 1 for extrapolation interpretation on day 100; 1 for using all three required words; 1 for a realistic reason why linear growth fails over long times.