What is the solution to Dido's problem?

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In summary, the conversation discusses the optimal geometric shape for maximizing the enclosed area given a fixed perimeter. MathWorld and other sources claim that a semi-circle is the answer, but simple calculations with a perimeter of 100m show that a circle would have a larger area. The discrepancy may be due to the requirement in Dido's problem for one side of the enclosure to be a straight line. However, if all closed curves are allowed, then a circle would be the optimal shape.
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Given a fixed perimeter, what is the geometric shape (non-self-intersecting closed curve) that would maximize the enclosed area? MathWorld (I'm not allowed to link) and a few other sites insist that the answer is a semi-circle, but simple math with perimeter=100 m would show that a circle would have area 2500/[itex]\pi[/itex][itex]\approx[/itex]795.8 m[itex]^{2}[/itex] whereas a semicircle wold have area [itex]\approx[/itex]594.2 m[itex]^{2}[/itex].

What am I missing?
 
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Dido's problem demands that one side of the enclosure is a straight line of fixed length. A circle would not qualifiy, since it has no straight sides.

If you allow all closed curves, then the circle would be optimal.
 

Related to What is the solution to Dido's problem?

1. What is Dido's problem?

Dido's problem refers to a mathematical conundrum posed by the ancient Greek mathematician, Dido of Carthage. The problem involves finding the optimal shape of a fence that encloses the largest possible area, given a fixed length of fencing.

2. Why is Dido's problem important?

Dido's problem is important because it serves as a fundamental problem in the field of optimization, which has applications in various fields such as engineering, economics, and computer science. It also highlights the creativity and ingenuity of ancient mathematicians in tackling complex problems.

3. Has Dido's problem been solved?

Yes, Dido's problem has been solved using a mathematical technique known as calculus. The solution involves creating a semicircle with a diameter equal to the length of the fencing, and then connecting the endpoints of the semicircle to the midpoint of the diameter. This creates a shape known as a 'stadium' with the largest possible area.

4. Are there any variations of Dido's problem?

Yes, there are several variations of Dido's problem, such as finding the optimal shape of a fence with a fixed perimeter, but different constraints, or finding the optimal shape for a given area. These variations add complexity to the problem and require different mathematical approaches for their solutions.

5. What is the significance of Dido's problem in modern mathematics?

Dido's problem is significant in modern mathematics as it serves as a precursor to the development of calculus and optimization techniques. It also sheds light on the ancient origins of mathematical concepts that are still relevant and applied in modern-day mathematics and other fields.

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