$\"\"$

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(a)

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A rain gutter is to be constructed of aluminum sheets $\"\"$ wide.

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The area $\"\"$ of the opening as a function of $\"\"$ is

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$\"\"$ , $\"\"$ .

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Find the angle $\"\"$ that maximizes $\"\"$ , by solving the equation $\"\"$ for $\"\"$ .

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Double-angle formula: $\"\"$ .

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$\"\"$ .

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Let $\"\"$ .

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$\"\"$

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$\"\"$ and $\"\"$

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$\"\"$ and $\"\"$ .

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$\"\"$ and $\"\"$ .

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$\"\"$ and $\"\"$ .

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Since $\"\"$ , consider $\"\"$ .

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$\"\"$

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(b)

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Maximum area $\"\"$ of opening is at $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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Maximum area $\"\"$ of opening is $\"\"$ .

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$\"\"$

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(c)

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Graph the area function $\"\"$ , $\"\"$ .

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$\"\"$

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Observe the graph:

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Maximum area is at the angle $\"\"$ and maximum area at that point is $\"\"$ .

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Above the result in graph is matched with results in part (a) and (b).

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$\"\"$

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(a) $\"\"$ .

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(b) Maximum area $\"\"$ of opening is $\"\"$ .

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(c) Graph of the area function $\"\"$ , $\"\"$ .

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$\"\"$