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About Modeling with Mathematics — Class 8 IB

Use mathematical models to represent and solve real-world problems across disciplines. This topic is part of the IB Class 8 mathematics syllabus (chapter: Unit 10). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Modeling with Mathematics — solved examples for Class 8 IB

Example 1easy

A taxi charges a flat fee of 3plus3 plus 2 per kilometre. Write an expression for the cost \(C\) of travelling \(d\) kilometres.
  1. A)\(C = 2d\)
  2. B)\(C = 3d + 2\)
  3. C)\(C = 3 + 2d\)
  4. D)\(C = 5d\)

Step-by-step solution

  1. Flat fee = $3 (constant term)
  2. Per-km charge = $2 × d (variable term)
  3. Total cost:
    C=3+2dC = 3 + 2d

Answer: \(C = 3 + 2d\)

Example 2medium

A rectangular field has a perimeter of 100 m. If the length is \(l\) metres, express the area in terms of \(l\).
  1. A)\(l(50 - l)\)
  2. B)\(l(100 - l)\)
  3. C)\(50l - l^2\)
  4. D)Both A and C

Step-by-step solution

  1. Perimeter: 2(l + w) = 100, so l + w = 50
  2. Width: w = 50 − l
  3. Area = l(50 − l) = 50l − l²
    A=l(50−l)=50l−l2A = l(50 - l) = 50l - l^2
  4. Options A and C are equivalent expressions.

Answer: Both A and C

Example 3hard

A projectile's height is \(h = -5t^2 + 30t + 10\). What is the maximum height reached?

Step-by-step solution

  1. Vertex at t = −30/(2(−5)) = 3 seconds
  2. h(3) = −5(9) + 30(3) + 10 = −45 + 90 + 10 = 55 m

Answer: 55

Practice questions on Modeling with Mathematics

  1. Q1.easy

    A pool is being filled at a rate of 50 litres per minute. It already contains 200 litres. How many litres will it contain after \(t\) minutes?
    1. A)\(50t\)
    2. B)\(200t + 50\)
    3. C)\(200 + 50t\)
    4. D)\(250t\)
    Show answer

    Answer: \(200 + 50t\)

    Hint: Start with what's already in the pool and add the rate multiplied by time.

  2. Q2.easy

    A rectangle's length is 3 more than twice its width \(w\). What is the perimeter in terms of \(w\)?
    1. A)\(6w + 6\)
    2. B)\(4w + 3\)
    3. C)\(6w + 3\)
    4. D)\(2w + 6\)
    Show answer

    Answer: \(6w + 6\)

    Hint: Perimeter = 2(length + width). Express length as \(2w + 3\).

  3. Q3.easy

    Tom earns 12perhour.Healsogetsa12 per hour. He also gets a 20 bonus each day. If he works \(h\) hours in a day, what is his daily earning?
    1. A)\(12h\)
    2. B)\(20h + 12\)
    3. C)\(12h + 20\)
    4. D)\(32h\)
    Show answer

    Answer: \(12h + 20\)

    Hint: Daily earning = hourly wage × hours + bonus.

  4. Q4.medium

    A car depreciates in value by 15% each year. If the car is worth $20,000 now, what is its value after 2 years? Round to the nearest dollar.
    Show answer

    Answer: 14450

    Hint: Each year, multiply by 0.85. After 2 years: 20000 × 0.85².

  5. Q5.medium

    Two cyclists start from the same point. One rides north at 12 km/h and the other rides east at 16 km/h. How far apart are they after 2 hours?
    Show answer

    Answer: 40

    Hint: They form a right triangle. Use the Pythagorean theorem on their distances.

  6. Q6.medium

    A ball is thrown upward and its height in metres after \(t\) seconds is \(h = 20t - 5t^2\). At what time does the ball reach its maximum height?
    Show answer

    Answer: 2

    Hint: The maximum of a quadratic \(at^2 + bt + c\) occurs at \(t = -b/(2a)\).

  7. Q7.hard

    A company finds that demand \(d\) for a product at price \(p\) is \(d = 500 - 4p\). Revenue is \(R = pd\). What price maximises revenue?
    Show answer

    Answer: 62.5

    Hint: R = p(500 − 4p) = −4p² + 500p. Find vertex.

  8. Q8.hard

    Radioactive material decays by 10% each year. How many years until only half remains? Give the answer as a whole number (round up).
    Show answer

    Answer: 7

    Hint: 0.9ⁿ = 0.5. Take logarithms: n = log(0.5)/log(0.9).

  9. Q9.hard

    A cone-shaped tank has radius 4 m and height 8 m. The volume of water when filled to height \(h\) is \(V = \frac{\pi h^3}{24}\). What height gives a volume of \(\frac{27\pi}{3}\) m³?
    Show answer

    Answer: 6

    Hint: Set πh³/24 = 9π. Solve for h.

These are 9 of the 60 questions available for Modeling with Mathematics. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.