Mathematics • Year 10 • Unit 2 • Lesson 20
Linear Modelling — Mixed Challenge
Six mixed problems: build models from words, build from two data points, swap the gradient and y-intercept meanings, compare two plans, and reason about extrapolation. Spot a classmate's swapped-gradient-and-intercept error, then design your own model that hits a target prediction.
1. Mixed problems — build and interpret
Show variables, model, and interpretation. 3 marks each
1.1 A pool drains at 25 L/min. It starts with 600 L. Write a linear model V = mt + c for volume remaining after t minutes, then find when it's empty.
1.2 A snow depth measurement shows 30 cm at the start of a storm and 78 cm after 8 hours. Assume linear accumulation. Write a model and predict the depth at hour 12.
1.3 For the model V = −500t + 8000 (depreciation of a piece of equipment, V = value in $, t = years), interpret the gradient and y-intercept in real-world words, and state when the equipment reaches zero book value.
1.4 Plan A: $40 monthly base + $0.05/MB of mobile data. Plan B: $0 monthly base + $0.15/MB. Write a cost model for each and find the data usage at which they cost the same.
1.5 A baby's height was 50 cm at birth and 75 cm at age 1 year. (a) Build a linear model. (b) Predict height at age 3. (c) Predict height at age 30. Comment on which prediction is more reliable.
1.6 A pizza shop earns $4 profit per pizza but has a fixed daily cost of $80. (a) Write a model for daily profit P in terms of pizzas n. (b) How many pizzas must they sell to break even (P = 0)? (c) How many to make $200 profit?
2. Find the mistake
A Year 10 student is given: "A plumber charges $80 call-out plus $60 per hour" and writes the model C = 80h + 60. That model is wrong. Find the mistake. 3 marks
Student's model:
C = 80h + 60
Student says: "$80 per hour and $60 call-out."
(a) What did the student swap?
(b) Explain (in plain words) what the gradient and y-intercept should each represent.
(c) Write the correct model and use it to find the cost of a 3-hour visit.
Stuck? The hourly rate is the gradient (cost per hour); the call-out is the fixed cost at h = 0 (y-intercept).3. Open-ended challenge — design a model
Many valid answers. 4 marks
3.1 Design a real-world linear model y = mx + c that satisfies all four of: (i) the context is a financial/cost/savings situation, (ii) m and c have opposite signs (one positive, one negative), (iii) the model predicts y = 150 when x = 10, (iv) the model has a sensible real-world meaning.
Submit:
(a) The context in one sentence (what's happening).
(b) Your model y = mx + c.
(c) Interpretation of m and c in real-world words.
(d) Verification that y = 150 when x = 10.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Pool drain
V = −25t + 600. Empty: 0 = −25t + 600 → t = 24 min.
1.2 — Snow accumulation
m = (78 − 30)/(8 − 0) = 6 cm/hr. D = 6t + 30. At t = 12: D = 72 + 30 = 102 cm (extrapolation, only modestly beyond data range).
1.3 — Depreciation interpretation
Gradient −500 = the equipment loses $500 in book value each year. y-intercept 8000 = the equipment was bought for $8000. Zero value: 0 = −500t + 8000 → t = 16 years.
1.4 — Mobile data plans
Plan A: C_A = 0.05d + 40. Plan B: C_B = 0.15d. Equal: 0.15d = 0.05d + 40 → 0.10d = 40 → d = 400 MB.
1.5 — Baby height
(a) m = (75 − 50)/(1 − 0) = 25 cm/year. h = 25t + 50. (b) At t = 3: h = 75 + 50 = 125 cm. (c) At t = 30: h = 750 + 50 = 800 cm = 8 m — physically impossible. (b) is far more reliable as it's modest extrapolation; (c) is wild extrapolation, well outside any reasonable range. Real growth slows after age 1 and stops by ~age 18.
1.6 — Pizza profit
(a) P = 4n − 80. (b) 0 = 4n − 80 → n = 20 pizzas. (c) 200 = 4n − 80 → 4n = 280 → n = 70 pizzas.
2 — Find the mistake
(a) The student swapped the gradient and y-intercept.
(b) Gradient = rate of change of cost per hour ($60/hr). y-intercept = cost at h = 0 (just the call-out, no work done) = $80.
(c) Correct model: C = 60h + 80. For a 3-hour visit: C = 60(3) + 80 = 180 + 80 = $260.
3 — Open-ended (sample solution)
(a) Mai begins her week $50 in overdraft (owes the bank $50) and deposits $20 per week of part-time pay into her account.
(b) B = 20x − 50 (B = balance in $, x = weeks).
(c) m = $20/week (the rate she saves); c = −$50 (her starting balance — negative because she's in overdraft).
(d) At x = 10: B = 20(10) − 50 = 200 − 50 = $150 ✓. m and c have opposite signs (m > 0, c < 0) ✓.
Marking: 1 for a sensible financial/cost context; 1 for m and c with opposite signs; 1 for the model giving y = 150 at x = 10; 1 for meaningful interpretation of m and c.