These six questions cover the four lessons in choosing a model from imperfect information. Write a full answer, including assumptions, before you open each block.
Questions
Question 1. A bus charter costs RM300 for the day plus RM2 for each kilometre. A trip is 80 km. The bus is blue, it seats 40 and the driver is named Ravi.
Find the cost.
Answer
Only the fixed charge, the rate and the distance match the pricing rule. The colour, the seating and the driver’s name are extra.
Cost = 300 + 2 × 80 = 300 + 160 = RM460.
Question 2. A phone plan costs RM30 every month plus RM0.10 per minute of calls. Another plan costs RM0.25 per minute with no monthly fee. Find the break-even number of minutes and state which plan is cheaper on each side.
Answer
Let m be the minutes. Plan 1: 30 + 0.10m. Plan 2: 0.25m.
Set equal: 30 = 0.15m, so m = 200.
Test m = 100: Plan 1 costs RM40 and Plan 2 costs RM25, so Plan 2 is cheaper. Test m = 300: Plan 1 costs RM60 and Plan 2 costs RM75, so Plan 1 is cheaper.
Plan 2 is cheaper below 200 minutes, and Plan 1 is cheaper above 200 minutes.
Question 3. The data in the table are for a cooling drink. Find a linear model and predict the temperature at t = 30 minutes.
| t (minutes) | 0 | 5 | 10 | 15 |
|---|---|---|---|---|
| T (°C) | 90 | 80 | 70 | 60 |
Answer
The differences are −10 each 5 minutes, so the gradient is −2 °C per minute and the model is T = 90 − 2t.
At t = 30, T = 90 − 60 = 30 °C. Check the range: a hot drink cools towards room temperature and does not keep falling at a constant rate, so the prediction is an extrapolation and may be too low. The model is reliable only near the measured range of 0 to 15 minutes.
Question 4. A tank has a capacity of 80 litres, starts with 20 litres and fills at 6 litres per minute. State the model and its valid range. What does the model give at t = 15?
Answer
V = 20 + 6t. The tank is full when 20 + 6t = 80, so t = 10. The valid range is 0 ≤ t ≤ 10.
At t = 15 the model gives 20 + 90 = 110 litres, which exceeds the capacity. The real volume is 80 litres.
Question 5. A van seats 12 passengers. A class trip has 30 students.
How many vans are needed? State your assumption.
Answer
30 ÷ 12 = 2.5. Assume each student needs a seat and a van cannot be split, so round up to 3 vans.
If the question allowed smaller vehicles for the last 6 students, a different answer could apply. Under the stated assumption, 3 vans are needed.
Question 6. A parking meter charges RM2 for the first hour and RM1.50 for each further hour. A car parks for 3 hours 20 minutes. Give two defensible answers and recommend one.
Answer
Reading 1: a started hour is charged in full, so the car is charged for 4 hours. Cost = 2 + 1.50 × 3 = RM6.50.
Reading 2: a part-hour is charged in proportion, so the extra time is 2⅓ hours after the first hour. Cost = 2 + 1.50 × 2⅓ = 2 + 3.50 = RM5.50.
Meters usually charge by started hour, so recommend RM6.50, with the assumption stated.
If you got these wrong
- Question 1: read which quantities the fare needs.
- Question 2: read fixed fee versus usage fee.
- Questions 3 and 4: read whether the model works outside the data.
- Questions 5 and 6: read two defensible models.
The timed original practice session builder lets you repeat the set under time. For a teacher to work through your modelling answers, see online one-to-one Mathematics tuition.