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Mathematical Modeling · Form 5

Mathematical Modeling: Worked Examples (Medium)

This set builds a model and then does the real work: comparing two models, or solving a quadratic by factorisation and checking which root makes sense. It suits students who can form a model and now need to solve and interpret it.

Written by the spmmath.com.my editorial team.

Worked example 1

Company A rents a minibus for a fixed RM120 plus RM0.80 per km. Company B charges a flat RM200 for the same trip.

Form a model for each cost, then find the distance at which the two companies charge the same amount.

  1. Company A: CA = 120 + 0.8d, where d is distance in km.
  2. Company B: CB = 200 (fixed).
  3. Same cost means CA = CB: 120 + 0.8d = 200.
  4. Subtract 120 from both sides: 0.8d = 80.
  5. Divide by 0.8: d = 80 ÷ 0.8 = 100.

Worked example 2

A rectangular vegetable bed has a length that is 2 m longer than its width. Its area is 48 m².

Form a quadratic model and find the width of the bed.

  1. Let the width be w m, so the length is (w + 2) m.
  2. Area model: w(w + 2) = 48.
  3. Expand and rearrange: w² + 2w − 48 = 0.
  4. Factorise: (w + 8)(w − 6) = 0.
  5. So w = −8 or w = 6; reject w = −8 as a width cannot be negative.

Worked example 3

A stall's daily profit RM P from selling a drink at price RM x is modelled by P = −2x² + 40x − 150. Find the two prices at which the profit is zero.

  1. Profit is zero means P = 0: −2x² + 40x − 150 = 0.
  2. Divide every term by −2 to simplify: x² − 20x + 75 = 0.
  3. Factorise: (x − 5)(x − 15) = 0.
  4. So x = 5 or x = 15.
  5. Check at x = 5: P = −2(25) + 40(5) − 150 = −50 + 200 − 150 = 0. ✓

Worked example 4

Gym A charges a one-time registration fee of RM80 plus RM40 per month. Gym B charges RM200 registration plus RM25 per month.

After how many months is the total cost the same for both gyms, and what is that cost?

  1. Let m be the number of months. Cost of A = 80 + 40m; cost of B = 200 + 25m.
  2. Set the costs equal: 80 + 40m = 200 + 25m.
  3. Collect terms: 40m − 25m = 200 − 80 → 15m = 120 → m = 8.
  4. Cost at 8 months: 80 + 40(8) = 80 + 320 = RM400 (check B: 200 + 25(8) = 400).

Worked example 5

A rectangular poster has a length that is 5 cm more than its width. Its area is 84 cm².

Form a quadratic model for the width w (cm) and find the width of the poster.

  1. Length = w + 5, so area = w(w + 5) = 84.
  2. Expand and rearrange into a quadratic: w² + 5w − 84 = 0.
  3. Factorise: (w + 12)(w − 7) = 0, so w = −12 or w = 7.
  4. A width cannot be negative, so reject w = −12; thus w = 7.

Worked example 6

A ball is thrown vertically upwards and its height above the ground is modelled by h = 20t − 5t² metres, where t is the time in seconds after it is thrown. Find the time when the ball returns to the ground and the maximum height it reaches.

  1. The ball is on the ground when h = 0: 20t − 5t² = 0.
  2. Factorise: 5t(4 − t) = 0, so t = 0 (thrown) or t = 4 (returns).
  3. The maximum height is at the vertex, midway between the roots: t = (0 + 4)/2 = 2 s.
  4. Substitute t = 2: h = 20(2) − 5(2²) = 40 − 20 = 20 m.

Worked example 7

A caterer's cost model is C(n) = an + 150, where n is the number of guests and a is the cost per guest in RM. Catering for 40 guests costs RM950.

Find the value of a, then use the model to find the cost for 65 guests.

  1. Substitute n = 40, C = 950 into the model: 950 = 40a + 150.
  2. 40a = 800, so a = 20.
  3. Model becomes C(n) = 20n + 150. For n = 65: C(65) = 20(65)+150 = 1300+150 = 1450.

Worked example 8

A triangular sail has a base that is 4 m longer than its height h metres. Its area is 30 m².

Form a quadratic model and find the height of the sail.

  1. Area of triangle = ½ × base × height, so ½(h+4)(h) = 30.
  2. h(h+4) = 60 → h² + 4h − 60 = 0.
  3. Factorise: (h+10)(h−6) = 0, so h = −10 or h = 6.
  4. Since height cannot be negative, h = 6 m.

Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)

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Frequently asked questions

What's different about medium-level mathematical-modeling questions?

Medium questions usually involve two related quantities or constraints instead of one, so you need to form and solve two equations together, or refine an initial model after checking it against extra given information. You may also be asked whether the model still fits once a condition changes.

What do examiners want to see when a model needs refining?

A clear statement of the original model, the reason it doesn't fully fit (using the extra data given), and a corrected or extended version with working shown at each stage. Simply writing a new equation without explaining why the first one was inadequate earns fewer marks than showing the reasoning.

What mistakes appear most at this medium modeling level?

Solving one equation and forgetting the model actually needs both conditions satisfied together, substituting values into the wrong equation when two are set up, and not re-checking the refined model against all the given data before giving a final answer. Skipping the check step often leaves an inconsistency unnoticed.

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