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A window is in the form of a rectangle surmounted by a semicir

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A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

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A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

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Text solutionVerified

Solution:

2 a + b + π \frac{b}{2} = 10

a = \frac{10 - b - π \frac{b}{2}}{2}

area =

a b + \frac{π}{2} \cdot \frac{b ^{2}}{4}

= \frac{( 10 - b - π \frac{b}{2} ) b}{2} + \frac{π b ^{2}}{8}

= 5 b^{-} \frac{b ^{2}}{2} - \frac{π b ^{2}}{4} + \frac{π b ^{2}}{8}

= 5 b - \frac{b ^{2}}{2} - \frac{π b ^{2}}{8}

\frac{d A}{d b} = 0

\Rightarrow 5 - b - \frac{π}{4} \cdot b = 0

b = \frac{5}{1 + \frac{π}{4}}

\frac{d ^{2} A}{d b ^{2}} = 0 - 1 - \frac{π}{4} < 0

a = \frac{10 - b - π \frac{b}{2}}{2} = \frac{10}{π + 4}

area =

(\frac{20}{π + 4}) (\frac{10}{π + 4}) + \frac{π}{2} \cdot \frac{400}{4 ( π + 4 ) ^{2}}

= \frac{200 + 50 π}{( π + 4 ) ^{2}} = \frac{50}{π + 4} u n i t^{2}

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Question Text	A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.
Answer Type	Text solution:1
Upvotes	150

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

Text solutionVerified

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