Practice Maths

Topic Review — Problem Solving and Modelling

Mixed Practice — L49 – L51

This review consolidates mathematical modelling, mixed problem solving, and Year 9 exam-style questions. Show all working and check answers in context.

Review Questions

  1. Mathematical modelling — set up and solve. Fluency

    1. An electrician charges a $90 call-out fee and $65 per hour. Write an equation for total cost C in terms of hours h and find the cost of a 2.5-hour job.
    2. A tank contains 6 000 L of water and is draining at 150 L/min. Write an equation for volume V after t minutes and find when the tank is empty.
    3. For the situation in (b), state one assumption and one limitation of the model.
    4. Two internet plans: Plan A costs $45/month + 2c/MB used. Plan B costs $60/month flat (unlimited). At how many MB of usage are the plans equal in cost?

  2. Multi-topic problem solving. Understanding

    1. Simplify: √48 + 2√12 − √75
    2. Solve: 5(x − 2) = 3x + 4
    3. A right triangle has legs of 5 cm and 12 cm. Find the hypotenuse and the angle opposite the 12 cm leg (to 1 d.p.).
    4. Find the volume of a cylinder with diameter 10 cm and height 8 cm (in terms of π and as a decimal to 2 d.p.).
    5. A $2 200 laptop is discounted by 15%. Find the sale price.

  3. Algebra and geometry combined. Problem Solving

    A rectangular room has a length that is (x + 4) m and a width of (x − 1) m. The area of the room is 36 m².

    1. Write and expand an equation for the area: (x + 4)(x − 1) = 36.
    2. Rearrange to form a quadratic: x² + 3x − 40 = 0.
    3. Factorise and solve to find x (taking the positive solution).
    4. Find the dimensions and perimeter of the room.
    5. Carpet costs $45 per m². Find the cost to carpet the room.

  4. Evaluating a model and interpreting results. Understanding

    A student models the profit P (in dollars) from selling x handmade candles as P = 8x − 120.

    1. What does the value −120 represent in context?
    2. How many candles must be sold to break even?
    3. If the student sells 30 candles, what profit do they make?
    4. Is a linear model appropriate here? State one assumption and one potential limitation.

  5. Comprehensive Year 9 problem. Problem Solving

    A wheelchair ramp must reach a doorway that is 0.75 m above ground level. Safety guidelines require the angle of inclination to be no more than 5°.

    1. Find the minimum horizontal length of the ramp (to 2 d.p.) using trigonometry.
    2. Find the length of the ramp surface (the hypotenuse) to 2 d.p.
    3. If the ramp is 1.2 m wide, find the area of the ramp surface.
    4. The ramp surface costs $85 per m² to build. Find the total cost.
    5. State two assumptions made in this model and explain why they matter in a real construction context.
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