Surface Integral: Calculating Max Area from Circular Log

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Homework Help Overview

The problem involves calculating the maximum area of a rectangular section that can be inscribed within a circular log with a diameter of 0.5m. The subject area relates to geometry and calculus, particularly in the context of optimization.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the geometric relationship between the sides of the inscribed rectangle and the circle. There are attempts to establish a relation between the dimensions x and y of the rectangle. Some participants suggest using the Pythagorean theorem and consider the properties of squares in relation to maximum area.

Discussion Status

The discussion is progressing with participants exploring geometric representations and relationships. Some guidance has been offered regarding the use of the Pythagorean theorem and the assumption that the maximum area may correspond to a square. There is acknowledgment of the need for calculus to derive a formula for optimization.

Contextual Notes

Participants note the assumption that knowledge of the formula for a circle is required, indicating a potential gap in information for some. The original poster expresses uncertainty about how to begin the problem.

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Homework Statement



A log of wood which is approximately circular in cross section has diameter equal to 0.5m. Calculate the maximum area of the rectangular timber section that can be obtained from the log.

Homework Equations





The Attempt at a Solution


No idea of solution
 
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You have to at least get started. Draw a circle with a rectangle inscribed in it. Label the sides x and y. Can you find a relation between x and y? Use geometry.
 
I have done that before but I can not deduce any relation between x and y.
 
Finding that maximum area should be simple if you can assume that the greatest area amoung rectangles is always taken by a square. When you drew the picture, you should have seen a square with diagonal length equal to the diameter of the log. You can use the Pythagorean theorem to determine the relation between the sides of that square and its diagonal length.

But since this is posted under "Calculus" I presume you are intended to show that by differentiating a formula. And, if you are taking Calculus, I suspect you are assumed to know the formula for a circle of radius R with center at (0,0). If you don't know it, you should try to look it up.
 
Thanks a lot, I can now understand how to solve it.
 

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