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Production uses 2 units per x item and 1 per y item, with 2t units available. Profit is 3x+y. Find maximum continuous profit.

Read the idea, work independently, then explain what changed.

T04高二理組數學思維本(2026).pdf · III. Simple linear programming · PDF 11 / printed page 10

TOPIC 01

Linear programming and discrete feasibility

Draw a feasible region and optimise a linear objective, then check integer restrictions when required.

Draw a feasible region and optimise a linear objective, then check integer restrictions when required.
Justify the method and check the conditions in a new situation.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Transfer#Worked example

x,y\ge0,\ 2x+y\le8

A vertex search needs a nonempty bounded polygon or a justified finite optimum; integer solutions may differ from continuous optima.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Vertices (0,0),(t,0),(0,2t).

Worked solution

Write the defining equation, retain exclusions, and then solve or compare.
Apply the stated relation and retain its conditions.
$z=0,12,8\Rightarrow z_{\max}=12$
Compare profit per resource; distinguish the stated continuous model from an integer production model.

The requested value is 12.

Checks and common pitfalls: Compare profit per resource; distinguish the stated continuous model from an integer production model.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Draw a feasible region and optimise a linear objective, then check integer restrictions when required.
Which condition is essential in linear programming and discrete feasibility?
Must rounding a continuous optimum preserve feasibility?

Model or definition: Draw a feasible region and optimise a linear objective, then check integer restrictions when required.
$z=ax+by$
Conditions: A vertex search needs a nonempty bounded polygon or a justified finite optimum; integer solutions may differ from continuous optima.

An objective has no optimum if it is unbounded in an improving feasible direction.