Creating and Interpreting Linear Models — Solutions

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1. Write linear models

   1. Plumber: C = 50h + 60. Gradient = 50 (cost per hour); y-intercept = 60 (call-out fee).
   2. Gym: C = 15w + 30. Gradient = 15 (cost per week); y-intercept = 30 (sign-up fee).
   3. Car driving home: D = 200 − 80t. Gradient = −80 (approaching home at 80 km/h); y-intercept = 200 (initial distance).
   4. Burning candle: H = 20 − 2t. Gradient = −2 (burns 2 cm/hr); y-intercept = 20 (initial height).
   5. Phone: C = 0.25m. Gradient = 0.25 (cost per minute); y-intercept = 0 (no fixed fee).
   6. Draining tank: V = 500 − 30t. Gradient = −30 (drains 30 L/min); y-intercept = 500 (initial volume).
   7. Baker’s profit: P = 4n − 80. Gradient = 4 (profit per cake); y-intercept = −80 (daily loss before any cakes sold).
   8. Student earnings: S = 12h + 50. Gradient = 12 (earnings per hour); y-intercept = 50 (existing savings).

2. Table to equation to value

   1. C = 15h + 20: C = 15(7) + 20 = $125
   2. D = 300 − 80d: D = 300 − 80(4) = −20 km — the destination is reached before day 4 (reaches 0 after 3.75 days)
   3. P = 5n + 2: P = 5(10) + 2 = $52
   4. V = 8t: 8t = 400 → t = 50 minutes
   5. F = 1.80k + 3.50: F = 1.80(10) + 3.50 = $21.50
   6. H = 25 − 2t: 25 − 2t = 7 → t = 9 hours
   7. S = 25w + 80: 25w + 80 = 330 → w = 10 weeks
   8. C = 25p + 200: C = 25(55) + 200 = $1575

3. Interpret gradient and y-intercept

   1. C = 50h + 80: Gradient = 50 (costs $50 per hour); y-intercept = 80 (call-out fee of $80)
   2. D = 300 − 60t: Gradient = −60 (moving 60 km/h towards destination); y-intercept = 300 (starts 300 km away)
   3. P = 4n − 80: Gradient = 4 ($4 profit per item); y-intercept = −80 (loses $80 if nothing is sold — overhead cost)
   4. V = 800 − 25t: Gradient = −25 (drains 25 L/min); y-intercept = 800 (starts with 800 L)
   5. T = −10 + 3h: Gradient = 3 (warms 3°C per hour); y-intercept = −10 (starts at −10°C at midnight)
   6. S = 12w + 50: Gradient = 12 (saves $12 per week); y-intercept = 50 (started with $50)
   7. C = 1.80k + 3.50: Gradient = 1.80 ($1.80 per km); y-intercept = 3.50 (flag fall fee of $3.50)
   8. H = 30 − 2.5t: Gradient = −2.5 (burns 2.5 cm/hr); y-intercept = 30 (initially 30 cm tall)

4. Two models — find intersection

   1. 30 + 10h = 15h: 30 = 5h → h = 6 hours. Both cost $90 at 6 hours.
   2. 50d + 100 = 40d + 200: 10d = 100 → d = 10 days. Both cost $600 at 10 days.
   3. 8t = 5t + 9: 3t = 9 → t = 3 hours. Both have run 24 km.
   4. 200t = 150t + 500: 50t = 500 → t = 10 minutes. Both pools hold 2000 L.
   5. 60h + 50 = 45h + 95: 15h = 45 → h = 3 hours. Both cost $230.
   6. 20w + 100 = 30w: 100 = 10w → w = 10 weeks. Both have $300 saved.

5. Domain restrictions

   1. H = 25 − 2.5t: Domain: 0 ≤ t ≤ 10 (candle lasts 10 hours before burning out). Range: 0 ≤ H ≤ 25.
   2. V = 400t: Domain: 0 ≤ t ≤ 25 (pool full after 25 min). Range: 0 ≤ V ≤ 10 000.
   3. C = 2h + 3: Domain: 0 ≤ h ≤ 12 (car park open 12 hours). Range: 3 ≤ C ≤ 27.
   4. M = 3q: Domain: 0 ≤ q ≤ 20 (20 questions maximum). Range: 0 ≤ M ≤ 60.
   5. T = −5 + 2h: Domain: 0 ≤ h ≤ 10 (reaches 15°C after 10 hours). Range: −5 ≤ T ≤ 15.
   6. C = 4.5p: Domain: 1 ≤ p ≤ 8 (between 1 and 8 people). Range: 4.50 ≤ C ≤ 36.

6. Full modelling tasks

   1. Phone plans:
      1. Plan A: C = 0.15m + 25; Plan B: C = 40
      2. 0.15m + 25 = 40 → 0.15m = 15 → m = 100 minutes
      3. Plan A: C = 0.15(90) + 25 = $38.50 < $40. Plan A is better for Erica.
      4. Plan A: m ≥ 0 (can’t have negative call time). Plan B: any m ≥ 0 (flat fee regardless).
   2. Car hire:
      1. Cars R Us: C = 60d; Hire4Less: C = 40d + 100
      2. 60d = 40d + 100 → 20d = 100 → d = 5 days. After 5 days, Hire4Less is cheaper.
      3. Cars R Us: 7×60=$420; Hire4Less: 7×40+100=$380. Save $40 with Hire4Less.
   3. Swimming pool:
      1. After 20 minutes: 300 × 20 = 6000 L
      2. Both hoses: 300+200=500 L/min. V = 6000 + 500(t−20) = 500t − 4000, for t > 20.
      3. 500t − 4000 = 24 000 → 500t = 28 000 → t = 56 minutes total.
   4. Gardening:
      1. C = 45h + 30
      2. Domain: 2 ≤ h ≤ 6; Range: $120 ≤ C ≤ $300
      3. C = 45(4) + 30 = $210. This exceeds the $200 budget, so the job cannot be completed within budget at the standard rate.

7. Break-even analysis — food truck

   1. Daily cost model: C = 120 (fixed daily costs — constant, independent of meals sold)
   2. Revenue model: R = 8m (gradient = 8, revenue per meal; y-intercept = 0, no revenue if no meals sold)
   3. Break-even point: 8m = 120 → m = 15 meals. The food truck breaks even at 15 meals per day.
   4. At least $200 profit: Profit = R − C = 8m − 120 ≥ 200 → 8m ≥ 320 → m ≥ 40 meals. Domain: m must be a whole number ≥ 0 (can’t sell a fraction of a meal).

8. Critique a model — sunflower growth

   1. Gradient: The sunflower grows at 3.5 cm per week.
   2. y-intercept −10: The model predicts a height of −10 cm at planting (week 0), which is physically impossible. It represents the fact that the seed is below ground level at the start — the model is only valid once the plant has emerged above ground.
   3. First appears above ground: 3.5w − 10 > 0 → 3.5w > 10 → w > 2.86. The plant emerges during week 3 (at w ≈ 2.86 weeks).
   4. Domain restriction: The model is valid from approximately w = 2.86 (emergence) to w = (200 + 10)/3.5 ≈ 60 weeks (max height). Beyond this, the plant stops growing and the linear model overestimates height.

9. Build a model from cyclist data

   1. Gradient and model: Gradient = (45 − 60)/(1 − 0) = −15. y-intercept = 60. Model: D = −15t + 60.
   2. Interpret: Gradient −15: the cyclist is travelling towards home at 15 km/h. y-intercept 60: the cyclist starts 60 km from home.
   3. Arrival time: −15t + 60 = 0 → t = 4 hours. The cyclist arrives home after 4 hours.
   4. Valid for t > 4: No. Once home (D = 0), the model would give negative distances, which are meaningless. The domain is 0 ≤ t ≤ 4.

10. Energy bills — comparing two providers

    1. Cost models: Provider A: C = 0.28k + 25. Provider B: C = 0.35k.
    2. Equal cost: 0.28k + 25 = 0.35k → 25 = 0.07k → k = 25/0.07 ≈ 357 kWh.
    3. At 380 kWh: A: C = 0.28(380) + 25 = $131.40. B: C = 0.35(380) = $133. Provider A is cheaper by $1.60.
    4. Graph description: Both lines start at different y-intercepts (A at $25, B at $0). They intersect at approximately (357, 125). Below 357 kWh, Provider B is cheaper; above 357 kWh, Provider A is cheaper. The intersection point is the break-even usage level.