An optimum and the physical domain

Choose your method, connect the parts and explain your conclusions. These original questions include written reasoning and sketches as well as exact answer checks.

Suggested marks guide how much working to show. The site does not automatically award examination marks for proofs, diagrams or methods. Use scaffolded fluency practice when you need a hint first.

Level 2 / FSMQ. Course filters use selected objective mappings; none of these tasks establishes complete paper coverage.

Before you start

Optimisation with a stated domain. Read the relevant method, then decide which parts you can solve without hints.

Check a misconception first · Try a connected problem

Read an original example question

A rectangle is made using 32 metres of fencing. Its width is x metres.

Part a

Form an expression for its area and state the physically valid domain.

Part b

Find the width giving the maximum area.

Part c

Explain why the algebraic endpoints do not describe usable rectangles.

Model solution and review criteria

Part a

The length is 16 − x, so A = x(16 − x), with 0 < x < 16.

Use the full perimeter.
Require both dimensions to be positive.

Part b

A′ = 16 − 2x = 0 gives x = 8. A″ = −2, so this is a maximum.

Differentiate and solve within the domain.
Justify a maximum, not just a stationary point.

Part c

At either endpoint a dimension is zero and the area is zero; the physical model excludes them.

Connect excluded endpoints to degenerate geometry.

Compare your own reasoning. No automatic examination marks are awarded.