Mathematics • Year 8 • Unit 2 • Lesson 15
Linear Modelling — Mixed Challenge
Pull together everything from Lesson 15: cost models, distance-speed-time, conversion graphs, interpreting m and c, comparing two plans (break-even). Six mixed problems, one "find the mistake", one open-ended challenge.
1. Mixed problems
For each, identify m and c, write the equation, and answer with units. 3 marks each
1.1 A pool drains at 50 L per minute and starts with 1200 L. Write V = mt + c, then find when V = 0 (the pool empties).
1.2 An online tutor charges a $25 booking fee plus $40 per hour. Write the equation and find the cost of a 2.5-hour session.
1.3 A car going 90 km/h has just passed kilometre marker 60 on a highway. Write d in terms of t (hours) and find d after 2 hours.
1.4 A gym's revenue model is R = 50m, where m is members and R is dollars. State c, explain why it's 0, and find R when m = 80.
1.5 — Break-even. Plan A: C = 10x + 20. Plan B: C = 15x. Find x where both plans cost the same. Which plan is cheaper for large x?
1.6 A water bill is W = 1.50k + 15. Interpret m and c in the context, then explain (in one sentence each) what changes to your bill if (i) the per-kL rate doubles, and (ii) the service fee is cut to $0.
2. Find the mistake
A student is asked: a candle burns 0.3 cm per minute and starts at 12 cm tall. Find the candle's height after 20 minutes. Their working is shown. Exactly one line contains a mistake. Spot it, explain why, then re-do correctly. 3 marks
Student's working — candle height:
Line 1: m = 0.3 (cm per minute), c = 12 (starting height).
Line 2: H = 0.3t + 12.
Line 3: H = 0.3 × 20 + 12 = 6 + 12 = 18.
Line 4: The candle is 18 cm tall after 20 minutes.
(a) Which line contains the mistake?
(b) Explain in one or two sentences what is wrong with that line.
(c) Write out the corrected working, including the corrected equation and the corrected height after 20 minutes.
Stuck? A candle gets SHORTER as it burns — its height goes DOWN with time. What does that mean for the sign of m?3. Open-ended challenge — invent three pricing plans
This question has more than one valid answer. 4 marks
3.1 A streaming service is launching THREE different pricing plans. Each must be linear in the form C = mx + c, where x is hours watched per month. Design three plans that all cost $25 for 10 hours of watching.
For each plan you invent:
(i) Choose a per-hour rate m and a sign-up fee c that satisfy C = 10m + c = 25.
(ii) Write the equation.
(iii) Compute the cost for 0 hours and for 20 hours.
Bonus: Of your three plans, which is cheapest for a heavy user (x large) and which is cheapest for a light user (x small)? Justify in one sentence.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Pool draining
m = −50 (draining → negative), c = 1200. V = −50t + 1200. Empties when 0 = −50t + 1200 → t = 24 minutes.
1.2 — Online tutor
C = 40t + 25. At t = 2.5: C = 40(2.5) + 25 = 100 + 25 = $125.
1.3 — Highway car
m = 90 (km/h), c = 60 (starting marker). d = 90t + 60. At t = 2: d = 180 + 60 = 240 km.
1.4 — Gym revenue
c = 0 — with zero members the gym earns zero revenue (no fixed fee). At m = 80: R = 50(80) = $4000.
1.5 — Break-even
10x + 20 = 15x → 20 = 5x → x = 4. The two plans cost the same at x = 4 (both cost $60). For large x (e.g. x = 100): A = 1020, B = 1500. Plan A is cheaper for large x (smaller per-x rate beats the higher rate as x grows).
1.6 — Water bill interpretation
m = 1.50 = cost per kilolitre of water used. c = 15 = fixed service fee (paid even with 0 kL used).
(i) Doubling the per-kL rate to $3 → bill becomes W = 3k + 15; usage costs grow twice as fast.
(ii) Zero service fee → bill becomes W = 1.50k; the line passes through the origin and there's no charge if you use no water.
2 — Find the mistake
(a) The mistake is on Line 1 (and is then carried into Lines 2–4).
(b) The candle is burning down, so its height DECREASES with time. The gradient must be NEGATIVE: m = −0.3 cm/min, not +0.3. Using a positive m makes the line go uphill — the candle would grow taller, which is impossible.
(c) Corrected: m = −0.3, c = 12 → H = −0.3t + 12. At t = 20: H = −0.3(20) + 12 = −6 + 12 = 6 cm. The candle is 6 cm tall after 20 minutes.
3 — Open-ended challenge (sample solution)
Each plan must satisfy 10m + c = 25, i.e. c = 25 − 10m.
Plan 1 (low rate, high fee): m = 1, c = 15 → C = x + 15. At x = 0: $15. At x = 20: $35.
Plan 2 (medium rate, medium fee): m = 2, c = 5 → C = 2x + 5. At x = 0: $5. At x = 20: $45.
Plan 3 (flat fee, no per-hour): m = 0, c = 25 → C = 25. At x = 0: $25. At x = 20: $25.
Bonus: For a heavy user (large x) the flat-fee Plan 3 is cheapest (every hour after 10 is free). For a light user (small x) Plan 2 is cheapest (only $5 at 0 hours, since the sign-up fee is small).
Marking: 1 mark per valid plan with all three parts (3 marks). 1 bonus mark for correctly identifying cheapest for heavy vs light user.