document.write( "Question 1186062: The figure shows an isosceles triangle with equal sides of
\n" ); document.write( "length \"a\" surmounted by a semicircle. What should the measure of angle θ be in order to maximize the total area? \r
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Algebra.Com's Answer #816956 by math_helper(2461) $\"\"$ $\"About$

You can put this solution on YOUR website!


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document.write( "Refer to the figure above, where we've sliced the original picture down the middle.  Here .   If we maximize this figure in terms of , we just need to double that to find .\r
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document.write( " = 
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document.write( " = \r
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document.write( "Observe:
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document.write( "Substituting for r and h:
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document.write( " = 
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document.write( " = \r
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document.write( " = 
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document.write( "=  + \r
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document.write( "...factoring out 
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document.write( "=    (*)\r
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document.write( "Now take the derivative of  with respect to :\r
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document.write( " = 
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document.write( "   
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document.write( "= \r
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document.write( "... substitute  ...
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document.write( "=   \r
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document.write( "This last function is zero at approx.   = 2.575rad, = 5.716rad, ...\r
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document.write( "By graphing, one can see 2.575rad  results in a maximum area, while 5.716 results in a minimal area.   \r
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document.write( "2.575rad is approx 147.54degrees.\r
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document.write( "Ans:  =  maximizes the total area.\r
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document.write( "Just in case...
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document.write( "If you were interested in the value of the total maximal area, note that you would need a modified form of (*) because that equation computes 1/2 of the total area:
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document.write( "  =     \n" );
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