document.write( "Question 1128389: A piece of string l is bent to form the sector of a circle radius r. Show that the area of the sector is maximised when r=1/4 l \n" ); document.write( "

Algebra.Com's Answer #744852 by math_helper(2461) $\"\"$ $\"About$

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The proportion of the sector to a full circle determines the area:
\n" ); document.write( " $\"+A+=+%28theta%2F360%29%28pi%29r%5E2+\"$
\r
\n" ); document.write( "\n" ); document.write( "The value of $\"+theta+\"$ is the ratio of the arc length to the circumference of a full circle. Noting that the arc length is $\"+l-2r+\"$ :
\n" ); document.write( " $\"+theta+%2F+360+=++%28l-2r%29+%2F+%282%28pi%29r%29+\"$
\r
\n" ); document.write( "\n" ); document.write( "Substituting the latter into the former:
\n" ); document.write( " $\"+A+=+%28%28l-2r%29%2F2%28pi%29r%29+%28%28+pi%29+r%5E2+%29+\"$
\n" ); document.write( " $\"+A+=+%28lr%2F2%29+-+r%5E2+\"$
\r
\n" ); document.write( "\n" ); document.write( "Taking the derivative of A with respect to r:
\n" ); document.write( " $\"+dA%2Fdr+=+l%2F2+-+2r+\"$
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\n" ); document.write( "\n" ); document.write( "Setting this to zero:
\n" ); document.write( " $\"+l%2F2+-+2r+=+0+\"$ Ч> $\"+highlight%28+r+=+l%2F4+%29+\"$ (<<<ЧЧ that's an \"ell\" on top)
\r
\n" ); document.write( "\n" ); document.write( "ЧЧ
\n" ); document.write( "Check:
\n" ); document.write( " $\"+d%5E2A%2Fdr%5E2+=+-2+\"$ Ч> the function is concave down, thus the point dA/dr=0 we found is a maximum.\r
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