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Extended Only Algebra and Graphs

Practical Optimisation for IGCSE Maths

Using differentiation to solve real-world maximum/minimum problems. This subtopic is part of Differentiation in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding practi

What You Need to Know

Using differentiation to solve real-world maximum/minimum problems. This subtopic is part of Differentiation in the Cambridge IGCSE Mathematics 0580 syllabus (Extended tier only). Understanding practical optimisation is essential for achieving a strong grade in your IGCSE Maths exam.

Understanding Practical Optimisation

Optimisation uses calculus to find the maximum or minimum value of a quantity in real-world problems — such as maximum volume of a box, minimum surface area of a container, or maximum profit. The strategy is to express the quantity to be optimised as a function of one variable, differentiate, set dy/dx = 0, solve for the optimum value, and verify it is the required maximum or minimum. This is one of the most applied topics in IGCSE Extended Paper 4.

Step-by-Step Method

  1. 1

    Define a variable and write the objective function

    Let x be the unknown. Write the quantity to be maximised/minimised as a function of x only.

  2. 2

    Use any constraints to eliminate other variables

    If given a fixed total (e.g. fixed area, fixed perimeter), express one dimension in terms of x and substitute.

  3. 3

    Differentiate the objective function

    Find dy/dx (or dA/dx, dV/dx etc.).

  4. 4

    Set the derivative to zero and solve

    dy/dx = 0 gives the critical x-value. Solve for x.

  5. 5

    Verify the nature and find the optimum value

    Use d²y/dx² (or test points on either side). Calculate the optimum value by substituting x back.

Worked Example

Question

A farmer has 60 m of fencing to enclose a rectangular field. One side of the field is a wall and needs no fencing. Find the dimensions that maximise the area.

Solution

Step 1: Let width = x m. Then the length = 60 − 2x m (two widths + one length = 60). Step 2: Area A = x(60 − 2x) = 60x − 2x² Step 3: Differentiate. dA/dx = 60 − 4x Step 4: Set to zero. 60 − 4x = 0 → x = 15 Step 5: d²A/dx² = −4 < 0 → Maximum. Length = 60 − 30 = 30 m. Maximum area = 15 × 30 = 450 m² Answer: Width = 15 m, Length = 30 m, Maximum area = 450 m²

Exam Tips for Practical Optimisation

  • Always write down the objective function (the thing you are maximising or minimising) before differentiating.
  • Use the constraint to eliminate variables so you have exactly one variable in the objective function.
  • Verify it is a maximum or minimum using d²y/dx² or by checking the sign change of dy/dx on either side.
  • Give the final answer as asked — usually both the critical x-value AND the maximum/minimum quantity.

Practice Questions

Q1: A box with no lid has a square base of side x cm and height h cm. The total surface area is 300 cm². Show that V = 75x − x³/4 and find the value of x that maximises the volume.

Show hint

Surface area: x²+4xh=300 → h=(300−x²)/(4x). V=x²h=x²(300−x²)/(4x)=x(300−x²)/4=75x−x³/4. dV/dx=75−3x²/4=0 → x²=100 → x=10.

Q2: y = 30x − x² for 0 < x < 30. Find the value of x that gives the maximum y and state this maximum.

Show hint

dy/dx = 30−2x = 0 → x=15. Max y = 450−225 = 225.

Q3: A cylinder has volume 500π cm³. Find the radius that minimises the total surface area.

Show hint

V=πr²h=500π → h=500/r². SA=2πr²+2πrh=2πr²+1000π/r. dSA/dr=4πr−1000π/r²=0 → r³=250.

Frequently Asked Questions

What is practical optimisation in IGCSE Maths?

Using differentiation to solve real-world maximum/minimum problems.

Is practical optimisation in the Core or Extended syllabus?

Practical Optimisation is part of the Extended only syllabus for IGCSE Mathematics 0580.

How do I revise practical optimisation effectively?

Start with the revision notes to understand key concepts, then work through the worked examples step by step. Finally, practise past paper questions under timed conditions. Teacher Rig recommends spending focused revision sessions on practical optimisation rather than trying to cover everything at once.

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