Find a rectangle’s largest modelled area

Supplied materials

• Fictional paper rectangle has width w and length 10 − w, measured in units. For this model, 0 ≤ w ≤ 10; zero gives a collapsed boundary shape.
• Area rule: A(w) = w(10 − w) = 10w − w². You need multiplication and squares.

Learn the idea

The area is quadratic because multiplying w by 10 − w produces a squared term. Unlike a linear rule, equal increases in width do not produce a constant area increase.

The expression can also be written A(w) = 25 − (w − 5)². Since a square cannot be negative, the area cannot exceed 25 in this model, and that value occurs when w = 5.

1. Calculate area at widths 2, 5 and 8.
2. Compare widths equally far from 5.
3. Use the completed-square form to explain the maximum.
4. Calculate the changed width in the question.

Check your reasoning

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Take it further

Change the fixed width-plus-length total from 10 to 12. Show that A(w) = 36 − (w − 6)² and the maximum is 36 at w = 6.

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