Maximum light through the window

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    Light Maximum Window
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Discussion Overview

The discussion revolves around optimizing the dimensions of a window, which consists of a rectangle topped by a semi-circle, to maximize the area for light admission given a fixed perimeter of 10 meters. The focus is on the relationship between the geometry of the window and the area it provides for light entry.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant suggests that the area of the window is the key factor in determining the amount of light that comes through.
  • Another participant confirms that the area is indeed the relevant property to consider.
  • A further contribution proposes a method to express the area as the sum of the rectangle and the semi-circle, introducing variables $x$ for the width of the rectangle and $y$ for its height, and questions how to relate the semi-circle's radius to these dimensions.

Areas of Agreement / Disagreement

Participants generally agree that the area is the critical factor for maximizing light admission, but the specific approach to relate the dimensions and perimeter remains under discussion without a consensus on the method.

Contextual Notes

The discussion does not clarify the assumptions regarding the relationship between the rectangle's dimensions and the semi-circle's radius, nor does it resolve how to incorporate the perimeter constraint into the area calculation.

marutkpadhy
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A window is in the form of rectangle surmounted by a semi circle. The total perimeter of the window is 10m. Find the dimensions of the window to admit maximum light through the whole opening.
 
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What property of the window is it that determines the amount of light that comes through?
 
Area
 
marutkpadhy said:
Area

Correct! :D

So, we need to find the area of the window, which is the sum of a rectangle and a semi-circle, and somehow get the given perimeter involved. Suppose we let $x$ be the width of the rectangle and $y$ be the height. If the semi-circle sits on top of the rectangle, what then is its radius?
 

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