Answered: A string of length L is cut into…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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A string of length L is cut into arbitrarily many pieces. Some or all pieces are made to circles and other pieces are made to rectangles. What is the maximum possible area enclosed by these shapes?

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Let the string is cut into n equal pieces. Then the length of each piece will be $\frac{L}{n}$ . Now, let x pieces are made to circles and the rest n-x pieces are made into rectangles. Now, it is advisable that each piece is made to a rectangle (better to say, a square) because it will give us the maximum area because the areas of a circle and a rectangle are given by ${πr}^{2} and lb$ and you can see for any fixed length wire it is good to reshape it into a square instead of a circle to maximize the area.

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$Area of a Circle$

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