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Optimisation problem: the flower bed by the fence

This page was translated from German with AI. It has not been checked by a native speaker. Original page in German

You have \(24\,\text{m}\) of fence for a bed against the house wall. How wide must the bed be so that the area is as large as possible?
📄 Open worksheet (PDF, German)

The video

In the video we set up the area function and read the maximum at the vertex.

Subtitles are available in ten languages. Choose them at the bottom right of the video.

Contents of the video

  1. 0:00 Introduction
  2. 0:17 Setting up the function
  3. 0:49 Calculating the maximum
  4. 1:28 The result on the graph
  5. 1:52 Careful, mistake!
  6. 2:14 Try it yourself

The most important points

Setting up the function

Width \(x\); the two short sides need \(2x\) of fence; the long side is \(24 - 2x\). Area: \(A(x) = x\cdot(24 - 2x) = -2x^2 + 24x\). Because \(a < 0\), there is a maximum.

Sketch: width x, width x, long side 24 - 2x.
Sketch: width \(x\), width \(x\), long side \(24 - 2x\).

Calculating the maximum

\(A(x) = -2(x-6)^2 + 72\), \(S(6\,|\,72)\). Width \(6\,\text{m}\), length \(24 - 12 = 12\,\text{m}\), area \(72\,\text{m}^2\).

The highest point of the graph is S(6\,|\,72). At x = 0 and x = 12 there is no area.
The highest point of the graph is \(S(6\,|\,72)\). At \(x = 0\) and \(x = 12\) there is no area.

Careful: The long side is not \(24 - x\), because the width \(x\) occurs twice. Check: \(x + x + (24 - 2x) = 24\).

The worksheet

Here you can see the worksheet for the video (in German). First it explains everything again, then there are practice tasks at three levels. You can also open it as a PDF and print it.

📄 Open worksheet (PDF, German)

The worksheet – Page 1
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The worksheet – Page 2
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The worksheet – Page 3
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Show the worksheet solutions
The worksheet – Page 4
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Try it yourself

Task

You have \(40\,\text{m}\) of fence for a bed against the house wall. How wide must it be so that the area is maximal? How big is it?

Show solution

\(A(x) = x(40-2x) = -2(x-10)^2 + 200\): width \(10\,\text{m}\), area \(200\,\text{m}^2\).

Keep learning with a language model

Du bist meine Mathe-Lehrkraft. Ich lerne Deutsch als Zweitsprache. Unser Thema: Extremwertaufgabe (Fläche maximal, Zaun an der Hauswand). Stelle mir nacheinander fünf kurze Aufgaben. Warte immer auf meine Antwort. Sag mir dann, ob sie richtig ist, und erkläre kurz, warum. Benutze einfache deutsche Sätze.

How this was made: The idea, structure and examples of the series are mine. I made them with the help of AI: a language model (LLM) built the slides of the video following my instructions. The voice was generated with Qwen3-TTS, modelled on my own voice.

The subtitles, the translations, the worksheet and this page were also made with the help of AI. The translations have not been checked by native speakers.

The videos, worksheets, texts and prompts in this series are licensed under CC BY 4.0: you may use, change and share them if you name the source.