document.write( "Question 587691: A rectangle is bounded by the x-axis and the semicircle y=(4-x2)square root. Write the area A of the rectangle as a function of x and determine the domain of the function. \n" ); document.write( "

Algebra.Com's Answer #374210 by Edwin McCravy(20055) $\"\"$ $\"About$

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document.write( "As you can see by the picture the rectangle has length 2x\r\n" );
document.write( "(the horizontal measurement) and width y (the vertical\r\n" );
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document.write( "              Area = length·width\r\n" );
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document.write( "              Area = (2x)·y\r\n" );
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document.write( "But since y = ,\r\n" );
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document.write( "              Area = (2x)·\r\n" );
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document.write( "              Area =  2x \r\n" );
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document.write( "And now since we have area expressed as a function of x, we can write\r\n" );
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document.write( "              A(x) =  2x\r\n" );
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document.write( "The domain of 2x is the set of all values of x for\r\n" );
document.write( "which 2x is a real number, which will be whenever\r\n" );
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document.write( "              4 - x² > 0\r\n" );
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document.write( "      (2 - x)(2 + x) > 0\r\n" );
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document.write( "Critical numbers are 2 and -2, so we have to get test points in\r\n" );
document.write( "the intervals (, -2), (-2, 2), (2, ). We find\r\n" );
document.write( "that the only interval in which 4 - x² is non-negative is (-2, 2).\r\n" );
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document.write( "We must exclude the endpoints -2, and +2 because the function would be 0\r\n" );
document.write( "there, and no rectangle can have 0 area (unless we allow that a horizontal\r\n" );
document.write( "line segment could be called \"a rectangle with width 0\" or that a\r\n" );
document.write( "vertical line segment could be called \"a rectangle with length 0\").  So the\r\n" );
document.write( "domain is\r\n" );
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document.write( "                 (-2, 2)\r\n" );
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document.write( "Edwin

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