document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=70334>Question 70334</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two circular regions are tangent to each other, one being larger than the other.  The distance between the centers is 10 feet.<BR>\n" );
document.write( "Find the radius of each circle if the combined area is 52pi square feet. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #50167 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=50167\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Without going into the geometric detail, if you draw a diagram of the two tangent circles<BR>\n" );
document.write( "mark their centers, and draw a line connecting the two centers, the line will pass through<BR>\n" );
document.write( "the point of tangency.  Therefore, the line joining the two centers will be comprised<BR>\n" );
document.write( "of the two radii.  Call one of the radii x feet.  Since the two radii will add together to be 10 feet,<BR>\n" );
document.write( "the second radius will be 10-x feet.<BR>\n" );
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document.write( "It might not be a bad idea to make a sketch of the two circles with the line joining<BR>\n" );
document.write( "their centers so you can understand what is happening.<BR>\n" );
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document.write( "You know that the Area of a circle is given by the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+pi%2AR%5E2 ALIGN=MIDDLE  ALT=\"A+=+pi%2AR%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "where A represents the Area and R is the radius of the circle.  So for this problem we<BR>\n" );
document.write( "can compute the combined Area of the two circles by substituting x for one radius<BR>\n" );
document.write( "and 10-x for the other and add the resulting two Areas.  In equation form in which we<BR>\n" );
document.write( "use \"At\" for total Area is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=At+=+%28pi%2Ax%5E2%29+%2B+%28pi%2A%2810-x%29%5E2%29 ALIGN=MIDDLE  ALT=\"At+=+%28pi%2Ax%5E2%29+%2B+%28pi%2A%2810-x%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Factor a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > out of each of the two terms on the right side:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=At+=+pi%2A%28x%5E2+%2B+%2810-x%29%5E2%29 ALIGN=MIDDLE  ALT=\"At+=+pi%2A%28x%5E2+%2B+%2810-x%29%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Square the (10-x) term to make the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=At+=+pi%2A%28x%5E2+%2B+100+-+20x+%2B+x%5E2%29 ALIGN=MIDDLE  ALT=\"At+=+pi%2A%28x%5E2+%2B+100+-+20x+%2B+x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Simplify the expression in the parentheses and rearrange it in descending powers of x:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=At+=+pi%2A%282x%5E2+-+20x+%2B+100%29 ALIGN=MIDDLE  ALT=\"At+=+pi%2A%282x%5E2+-+20x+%2B+100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "At this point we use the fact that the combined Area of the two circles is:<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=52%2Api ALIGN=MIDDLE  ALT=\"52%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" > square feet.<BR>\n" );
document.write( "by substituting this for At in the equation.  Our equation then becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=52%2Api+=++pi%2A%282x%5E2+-+20x+%2B+100%29 ALIGN=MIDDLE  ALT=\"52%2Api+=++pi%2A%282x%5E2+-+20x+%2B+100%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Cancel the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > multipliers on both sides by dividing both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=52+=+2x%5E2+-+20x+%2B+100 ALIGN=MIDDLE  ALT=\"52+=+2x%5E2+-+20x+%2B+100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Subtract 52 from both sides and then transpose the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+-+20x+%2B+48+=+0 ALIGN=MIDDLE  ALT=\"2x%5E2+-+20x+%2B+48+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Divide both sides by 2 to simplify the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+10x+%2B+24+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+10x+%2B+24+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This equation factors to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-6%29%2A%28x-4%29+=+0 ALIGN=MIDDLE  ALT=\"%28x-6%29%2A%28x-4%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "And this equation will be true if either of the factors equals zero.  Therefore, to solve<BR>\n" );
document.write( "for x, set each factor equal to zero.<BR>\n" );
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document.write( "If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-6=0 ALIGN=MIDDLE  ALT=\"x-6=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x-4=0 ALIGN=MIDDLE  ALT=\"x-4=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=6 ALIGN=MIDDLE  ALT=\"x=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > then the<BR>\n" );
document.write( "other radius is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10-6+=+4 ALIGN=MIDDLE  ALT=\"10-6+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4 ALIGN=MIDDLE  ALT=\"x=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > then the other radius is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10+-+4=6 ALIGN=MIDDLE  ALT=\"10+-+4=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >}.<BR>\n" );
document.write( "From this, it is obvious that one radius must be 6 feet long and the other radius <BR>\n" );
document.write( "must be 4 feet long.<BR>\n" );
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document.write( "Hope this helps to clarify the problem for you. </SPAN>\n" );
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