MHB Maximum light through the window

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The discussion focuses on optimizing the dimensions of a window, which consists of a rectangle topped by a semi-circle, to maximize light admission while adhering to a total perimeter of 10 meters. The area of the window, which is the sum of the rectangle's area and the semi-circle's area, is the key factor in determining light admission. The variables defined include the width of the rectangle (x) and its height (y), with the radius of the semi-circle derived from these dimensions. The conversation emphasizes the relationship between the perimeter constraint and the area calculation to achieve maximum light. Ultimately, the goal is to find the optimal dimensions that maximize the window's area.
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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