Unit 2 Topic 1 Review — Applications of Linear Equations

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Review Questions

1. Q1 — Taxi fare model

   Fluency

   A taxi company charges according to the model C = 3.20 + 2.10d, where C is the cost in dollars and d is the distance in kilometres.

   (a) Find the cost for a 12 km trip.

   (b) Find the distance travelled if the fare is $25.40.

2. Q2 — Piecewise electricity bill

   Fluency

   An electricity plan charges $0.22 per kWh for the first 500 kWh used each month, and $0.31 per kWh for any usage above 500 kWh.

   (a) Find the monthly bill for 350 kWh.

   (b) Find the monthly bill for 720 kWh.

3. Q3 — Step graph parking costs

   Fluency

   A car park uses the following stepped pricing:

   • Up to 1 hour: $4
   • More than 1 hour up to 2 hours: $7
   • More than 2 hours up to 3 hours: $10
   • More than 3 hours up to 4 hours: $13

   Find the parking cost for: (a) 45 minutes; (b) exactly 2 hours; (c) 2 hours 10 minutes; (d) 4 hours.

4. Q4 — Break-even analysis

   Fluency

   A business has total cost TC = 7x + 1400 and revenue R = 21x, where x is the number of units produced and sold.

   (a) Find the break-even quantity.

   (b) Find the profit when x = 180 units.

5. Q5 — Building a linear model from a table

   Understanding

   A plumber charges based on time. The table shows costs for different job lengths:

   Time (hr)	0	1	2	3
   Cost ($)	80	140	200	260

   (a) Confirm the relationship is linear.

   (b) Write the linear model for cost C in terms of time t.

   (c) Find the cost for a job taking 4.5 hours.

   (d) Interpret the gradient and y-intercept in context.

6. Q6 — Piecewise gym membership

   Understanding

   A gym charges $15 per visit for the first 8 visits per month, then $9 per visit for any additional visits beyond 8.

   (a) Write a piecewise model for monthly cost C in terms of number of visits v.

   (b) Find the cost for 5 visits in a month.

   (c) Find the cost for 14 visits in a month.

   (d) How many visits in a month results in a total cost of exactly $111?

7. Q7 — Step graph mobile data plan

   Understanding

   A mobile phone plan uses stepped data pricing:

   • 0–10 GB: $35/month
   • 10–30 GB: $55/month
   • 30–60 GB: $75/month
   • More than 60 GB: $95/month

   (a) Find the monthly cost for: 8 GB, 25 GB, 55 GB, 72 GB.

   (b) Which plan tier is best for someone who uses an average of 40 GB per month?

8. Q8 — Canteen break-even analysis

   Understanding

   A school canteen has fixed daily costs of $180 and variable costs of $2.40 per item. Items are sold for $7.50 each.

   (a) Write equations for Total Cost (TC) and Revenue (R) in terms of items sold x.

   (b) Find the break-even number of items.

   (c) Find the profit when 50 items are sold.

   (d) How many items must be sold to achieve a profit of $200?

9. Q9 — Water tank drainage model

   Understanding

   At the start of draining, a water tank holds 8400 litres. Two hours later it holds 5600 litres.

   (a) Write a linear model for the volume V (litres) in terms of time t (hours).

   (b) What is the drain rate per hour?

   (c) When does the tank reach 1000 litres?

   (d) State the practical domain of your model.

10. Q10 — Comparing two delivery options

    Understanding

    Two courier companies quote for delivery:

    • Option A: $60 base charge + $0.35 per km
    • Option B: $30 base charge + $0.50 per km

    (a) Write cost equations for each option in terms of distance d (km).

    (b) Find the distance at which both options cost the same.

    (c) Which option is cheaper for a 200 km trip?

11. Q11 — Piecewise overtime pay

    Understanding

    An employee is paid:

    • $26/hr for the first 38 hours (normal time)
    • $39/hr (time-and-a-half) for the next 4 hours (hours 39–42)
    • $52/hr (double time) for any hours beyond 42

    Find the weekly pay for: (a) 35 hours; (b) 40 hours; (c) 46 hours.

12. Q12 — Income tax as a step function

    Problem Solving

    Income tax is calculated using these brackets:

    • $0 – $18 200: Nil
    • $18 201 – $45 000: 19c per $1 over $18 200
    • $45 001 – $120 000: $5092 + 32.5c per $1 over $45 000

    (a) Calculate the tax payable on an income of $27 500.

    (b) Calculate the tax payable on an income of $65 000.

13. Q13 — Production decision and machine upgrade

    Problem Solving

    A business has fixed costs of $3600/month and variable costs of $14/unit. Each unit sells for $38.

    (a) Find the break-even quantity.

    (b) The business currently produces 200 units/month. Find the current monthly profit.

    (c) A machine upgrade costs an extra $500/month in fixed costs but reduces variable cost to $10/unit. Is the upgrade worth it at 200 units/month?

    (d) At what monthly production volume does the upgraded setup become more profitable than the current setup?

14. Q14 — Piecewise car journey model

    Problem Solving

    A car makes a journey: it travels at 80 km/h for 2 hours, stops for 30 minutes (0.5 hr), then travels at 100 km/h for 1.5 hours.

    (a) Write a piecewise model for distance D (km) from the start in terms of time t (hours).

    (b) Find the total distance travelled.

    (c) At what time(s) is the car exactly 160 km from the start?

15. Q15 — Business profitability and pricing decision

    Problem Solving

    A business currently sells 450 units/month at $32 each. Variable costs are $12/unit and fixed costs are $4500/month.

    (a) Find the current monthly profit.

    (b) A price cut to $28 is expected to increase sales to 600 units/month. Would the price cut increase profit?

    (c) Find the break-even quantity at the new price of $28.

    (d) With current production fixed at 450 units, what selling price would produce a monthly profit of exactly $5000?