document.write( "Question 255263: If an arc of 45 degrees on circle A has the same length as an arc of 30 degrees on circle B, find the ratio of the area of circle A to the area of circle B. \n" ); document.write( "

Algebra.Com's Answer #187544 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
The length of an arc is, in general:
\n" ); document.write( "

\r\n" );
document.write( "degrees in the arc\r\n" );
document.write( "------------------ * 2 * pi * r\r\n" );
document.write( "       360\r\n" );
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\n" ); document.write( "If $\"r%5BA%5D\"$ represents the radius of circle A then the arc length is:
\n" ); document.write( " $\"%2845%2F360%29%2A2%2Api%2Ar%5BA%5D\"$
\n" ); document.write( "which simplifies to
\n" ); document.write( " $\"%281%2F4%29pi%2Ar%5BA%5D\"$

\n" ); document.write( "If $\"r%5BB%5D\"$ represents the radius of circle B then the arc length is:
\n" ); document.write( " $\"%2830%2F360%29%2A2%2Api%2Ar%5BB%5D\"$
\n" ); document.write( "which simplifies to
\n" ); document.write( " $\"%281%2F6%29pi%2Ar%5BB%5D\"$

\n" ); document.write( "We are told that these arc lengths are equal so
\n" ); document.write( " $\"%281%2F4%29pi%2Ar%5BA%5D+=+%281%2F6%29pi%2Ar%5BB%5D\"$

\n" ); document.write( "We are asked to find the ratio of the areas of the two circles. The area for circle A would be $\"pi%2Ar%5BA%5D%5E2\"$ and for circle B it would be $\"pi%2Ar%5BB%5D%5E2\"$ . The ratio then would be:
\n" ); document.write( " $\"%28pi%2Ar%5BA%5D%5E2%29%2F%28pi%2Ar%5BB%5D%5E2%29\"$
\n" ); document.write( "The pi's cancel leaving:
\n" ); document.write( " $\"r%5BA%5D%5E2%2Fr%5BB%5D%5E2\"$
\n" ); document.write( "which can be rewritten as
\n" ); document.write( " $\"%28r%5BA%5D%2Fr%5BB%5D%29%5E2\"$
\n" ); document.write( "So if we can find the ratio of the radii then we can find the ratio of the areas. Our arc length equation tells us that
\n" ); document.write( " $\"%281%2F4%29pi%2Ar%5BA%5D+=+%281%2F6%29pi%2Ar%5BB%5D\"$
\n" ); document.write( "We can use this to find the ratio of the radii. Multiplying both sides by 4 we get:
\n" ); document.write( " $\"pi%2Ar%5BA%5D+=+%284%2F6%29pi%2Ar%5BB%5D\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"pi%2Ar%5BA%5D+=+%282%2F3%29pi%2Ar%5BB%5D\"$
\n" ); document.write( "Dividing both sides by pi we get:
\n" ); document.write( " $\"r%5BA%5D+=+%282%2F3%29r%5BB%5D\"$
\n" ); document.write( "Dividing both sides by $\"r%5BB%5D\"$ we get:
\n" ); document.write( " $\"r%5BA%5D%2Fr%5BB%5D+=+2%2F3\"$
\n" ); document.write( "So the ratio of the radii is 2/3. We can put this into our area ratio expression:
\n" ); document.write( " $\"%28r%5BA%5D%2Fr%5BB%5D%29%5E2\"$
\n" ); document.write( "and we get:
\n" ); document.write( " $\"%282%2F3%29%5E2\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"4%2F9\"$

\n" ); document.write( "Note that we were able to find the ratios of the radii and the areas without being able to find any radius or any area!?\r
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