document.write( "Question 1128652: Find the area of the largest rectangle that fits inside a circle of radius 10.
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Algebra.Com's Answer #745146 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "Logical reasoning says the rectangle with the greatest area inscribed in a circle is a square. The diameter of the circle is then the diagonal of the square.

\n" ); document.write( "With a diagonal of length 20, the side of the square is 10*sqrt(2); the area is side squared = 200.

\n" ); document.write( "To verify that answer, consider the circle with center at the origin, so the equation is x^2+y^2 = 100. A point on the circle has coordinates $\"x\"$ and $\"sqrt%28100-x%5E2%29%29\"$ .

\n" ); document.write( "The dimensions of the rectangle determined by that point are $\"2x\"$ and $\"2%2Asqrt%28100-x%5E2%29\"$ ; the area is $\"4x%28sqrt%28100-x%5E2%29%29\"$ .

\n" ); document.write( "The maximum area is when the derivative of the area function is 0.

\n" ); document.write( "To simplify the process of finding where the derivative is 0, move all the variables inside the radical:

\n" ); document.write( " $\"A+=+4%2Asqrt%28100x%5E2-x%5E4%29\"$

\n" ); document.write( "Set the derivative equal to 0 and solve:

\n" ); document.write( " $\"%284%28200x-4x%5E3%29%29%2F%282%2Asqrt%28100x%5E2-x%5E4%29%29+=+0\"$
\n" ); document.write( " $\"%2816x%2850-x%5E2%29%29%2F%282%2Asqrt%28100x%5E2-x%5E4%29%29+=+0\"$
\n" ); document.write( " $\"50-x%5E2+=+0\"$
\n" ); document.write( " $\"x%5E2+=+50\"$
\n" ); document.write( " $\"x+=+sqrt%2850%29\"$

\n" ); document.write( "The dimensions of the rectangle with greatest area are

\n" ); document.write( " $\"2x+=+2%2Asqrt%2850%29\"$ and $\"2%28sqrt%28100-x%5E2%29%29+=+2%2Asqrt%2850%29\"$

\n" ); document.write( "The area is

\n" ); document.write( " $\"%282%2Asqrt%2850%29%29%282%2Asqrt%2850%29%29+=+200\"$ \n" ); document.write( "

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