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Joan’s house has a window in a shape of rectangle surmounted with a semi-circle. For a given perimeter,what are the dimensions o fthe window if it allows the smallest amount of light? the answer is A= P/4+pi (2P/4+pi),,,please need the whole solution.By the way ,this topic is maxima and minima

Joan’s house has a window in a shape of rectangle surmounted with a semi-circle. For a given perimeter,what are the dimensions o fthe window if it allows the smallest amount of light? the answer is A= P/4+pi (2P/4+pi),,,please need the whole solution.By the way ,this topic is maxima and minima 

Grade:12th pass

1 Answers

Riddhish Bhalodia
askIITians Faculty 434 Points
5 years ago
Ok now lets proceed step by step

Variables: = R = radius of the semicircle, length = l and height = h

Assume semicircle on the length => 2R = l
and the perimeter is known
hence
P = constant =2\pi R + 2h + 2R
h = P/2 - \pi R - R

Now we want to minimize the area (the light that is)
so
A = \pi R^2/2 + 2Rh = PR -3\pi R^2/2 - 2R^2
to minimize
\frac{dA}{dR} = P -3\pi R - 4R = 0
hence we get the R as
R = P/(4+3\pi)
put in the A and get answer

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