Optimisation in context

A square sheet of card, 12 cm by 12 cm. Cut a square of side x from each corner, fold the flaps up, and you have an open box. Make x bigger and two things happen at once: the box gets taller, and its base gets smaller.

x = 1.00

10.0base side, 12 − 2x

1.0height, x

100.0volume, cm³

Somewhere in the middle the volume is as big as it gets. Calculus finds exactly where.

The method

Write the quantity you want as a function of one variable. This is the hard part and it carries the marks.
State the sensible range of that variable.
Differentiate, set it to zero, solve.
Reject any solution that makes no sense in context.
Find the quantity asked for, and answer in a sentence with units.

Worked example: the box

V= x(12 − 2x)² height times base times base with 0 < x < 6, because the base cannot have negative width dVdx= (12 − 2x)(12 − 6x) factorised, which makes solving easy = 0 when x = 6 or x = 2 x = 6→ rejected: the base would be 12 − 12 = 0 and there is no box V(2)= 2 × 8² = 128 cm³ an 8 by 8 base, 2 cm deep

So the largest box has a volume of 128 cm³, made by cutting 2 cm squares from the corners. Check it on the slider: x = 1 gives 100 and x = 3 gives 108, both smaller.

Rejecting a solution is worth a mark, and you have to say why. Writing "x = 6 is not valid" is weaker than "x = 6 is rejected because the base would have zero width". The reason is the mark.

Worked example: fencing against a wall

A farmer has 40 m of fencing for three sides of a rectangular pen; the fourth side is an existing wall. What is the largest area possible?

A= x(40 − 2x) x is each side perpendicular to the wall dAdx= 40 − 4x = 0 when x = 10 A(10)= 10 × 20 = 200 m²

The pen is 10 m out from the wall and 20 m along it. Notice that the side along the wall is twice the other, which is a pattern worth recognising but never worth assuming.

Your turn

1. A 20 cm by 20 cm sheet has squares of side x cut from the corners. Write down the volume when x = 3.

2. In the 12 cm box problem, why is x = 6 rejected?

Because 6 is bigger than 2. Because the base would be 12 − 12 = 0, so there is no box. Because the derivative is negative there. Because x must be a whole number.

3. With 60 m of fencing for three sides against a wall, what is the greatest area in square metres?

Where the marks go

Most of them are before any calculus happens. Setting up the function in one variable, and stating its range, is typically half the question.

Rejecting the impossible solution with a reason is its own mark. So is answering the question that was asked: if it asks for the volume, x = 2 is not the answer, 128 cm³ is.

Units, every time. An area in m² and a volume in cm³. A bare number at the end of an optimisation question is an unfinished answer.

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