T4T3 Review — Problem Solving and Modelling

Mixed review covering L46 (Mathematical Modelling), L47 (Applying Functions to Real Problems) and L48 (Optimisation and Decision Making).

1. Formulating a model. Fluency

   A plumber charges a $75 call-out fee plus $55 per hour.

   • (a) Write the cost function C(h).
   • (b) Find the cost for a 2-hour job.
   • (c) How many hours does a $350 job take?
   • (d) State one assumption in the model.

2. Profit function. Fluency

   A bakery sells loaves for $7 each. Fixed costs are $140/day. Variable cost is $3/loaf.

   • (a) Write R(n) and C(n).
   • (b) Find the profit function P(n).
   • (c) Find the break-even number of loaves.
   • (d) Find the profit when 60 loaves are sold.

3. Quadratic model. Fluency

   A stone is thrown upward from a 5 m ledge. Its height h(t) = −5t² + 15t + 5.

   • (a) Find the height at t = 1 and t = 2.
   • (b) Find the maximum height and the time it occurs.
   • (c) When does it hit the ground?
   • (d) State the domain and range in context.

4. Optimisation — maximum area. Fluency

   A rectangle with perimeter 100 m. Let the width be x m.

   • (a) Write A(x).
   • (b) Find x that maximises A.
   • (c) State the maximum area.
   • (d) What shape gives the maximum area?

5. Composite and inverse. Understanding

   A taxi converts km to minutes using m(k) = 2k + 5 (average journey). The cost is c(m) = 0.5m + 4 dollars.

   • (a) Find c(m(k)) — total cost for a k km journey.
   • (b) Find the cost of a 12 km journey.
   • (c) If the total cost is $24.50, how far was the journey?
   • (d) Find the inverse of c(m(k)) and interpret it.

6. Model validation. Understanding

   A population model predicts P(t) = 5000 + 200t for a town (t = years from 2020).

   • (a) What does the model predict for 2025 and 2030?
   • (b) The actual 2025 population was 6 400. Calculate the percentage error.
   • (c) Suggest a reason the model underestimates the population.
   • (d) An improved model is P(t) = 5000 × 1.04^t. What type of model is this, and what does the 1.04 represent?

7. Piecewise pricing. Understanding

   A theme park charges: $30 for adults, $15 for children. Group discount: groups of 10+ get 20% off.

   • (a) A family of 2 adults and 3 children: find the total cost.
   • (b) A group of 4 adults and 8 children: total cost (use group discount).
   • (c) Write a cost formula C(a, c) for a adults and c children if a+c ≥ 10.
   • (d) A group of 6 adults and 4 children. Does the discount apply? What do they pay?

8. Ticket pricing for maximum revenue. Understanding

   A theatre sells 500 tickets at $20 each. For each $2 price rise, 25 fewer tickets are sold.

   • (a) Write R(n) for n price increases of $2.
   • (b) Find the price increase that maximises revenue.
   • (c) Find the maximum revenue.
   • (d) How many tickets are sold at the optimal price?

9. Multi-step modelling. Problem Solving

   A company produces x units with cost C(x) = x² − 10x + 50 dollars and sells each for $(30 − x).

   • (a) Find the revenue function R(x).
   • (b) Find the profit function P(x) = R − C.
   • (c) Find the production level that maximises profit and the maximum profit.
   • (d) Find the break-even production levels.

10. Extended: choosing the best model. Problem Solving

    A scientist records temperature T (°C) of a cooling liquid every 5 minutes:

    Time (min)	0	5	10	15	20
    Temperature (°C)	80	64	51	41	33

    • (a) Calculate the temperature drop in each 5-minute interval. Is the rate of cooling constant?
    • (b) Calculate the ratio T(n+1)/T(n) for each step. What do you notice?
    • (c) Using T(0)=80 and ratio 0.80 per 5-minute period, write an exponential model T(n) where n is the number of 5-minute intervals.
    • (d) Predict the temperature at 30 minutes (n=6). What temperature does the model approach as n → ∞, and is this physically reasonable?