document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=931541>Question 931541</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two spheres M and N have a volume of 36 PI cubic cm and 972 PI respectively. What is the ratio of their radii? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #565618 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=565618\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the volume of the smaller sphere is 36 * pi.<BR>\n" );
document.write( "the volume of the larger sphere is 972 * pi.\r<BR>\n" );
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document.write( "the formula for the volume of a sphere is equal to 4/3 * pi * r^3\r<BR>\n" );
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document.write( "solve this formula for r and you get r = cube root of (volume / (4/3 * pi)) which can be simplified to (3*v) / (4*pi).\r<BR>\n" );
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document.write( "the radius of the smaller sphere is equal to 2.04835219<BR>\n" );
document.write( "the radius of the larger sphere is equal to 6.145056569\r<BR>\n" );
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document.write( "the ratio of the radius of the smaller sphere to the radius of the larger sphere is therefore equal to 1/3.\r<BR>\n" );
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document.write( "there's an easier way to do this.\r<BR>\n" );
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document.write( "the ratio of any corresponding side of two similar figures is equal to the cube root of the ratio of their volumes.\r<BR>\n" );
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document.write( "if the figures are similar, then the ratio of any of their corresponding sides is the same.\r<BR>\n" );
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document.write( "with a circle, the corresponding sides would be the radius and the circumference.\r<BR>\n" );
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document.write( "the ratio of the volume of the smaller sphere to the volume of the larger sphere is equal to 36 / 972 which is equal to 1/27.\r<BR>\n" );
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document.write( "the cube root of 1/27 is equal to 1/3.\r<BR>\n" );
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document.write( "the ratio of any of the corresponding sides must be equal to 1/3.\r<BR>\n" );
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document.write( "i have already determined that the radii are in the ratio of 1/3.\r<BR>\n" );
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document.write( "what remains is to examine the circumference.\r<BR>\n" );
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document.write( "what also remains is to examine the the surface area.\r<BR>\n" );
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document.write( "the circumference of the smaller sphere is equal to 2*pi*r1<BR>\n" );
document.write( "the circumference of the larger sphere is equal to 2 * pi * r2<BR>\n" );
document.write( "the ratio of the circumference of the smaller sphere to the larger sphere is therefore equal to (2*pi*r1) / (2*pi*r2).<BR>\n" );
document.write( "the 2*pi in the numerator and the denominator cancel out and you are left with r1/r2 which we already know is the ratio of 1/3.\r<BR>\n" );
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document.write( "the ratio of the surface area of the sphere should be equal to the square of the ratio of the sides.\r<BR>\n" );
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document.write( "that would make the ratio of the surface area equal to 1/9.\r<BR>\n" );
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document.write( "ratio of the sides is 1/3.<BR>\n" );
document.write( "radio of the surface areas is 1/9<BR>\n" );
document.write( "ratio of the volumes is 1/27.\r<BR>\n" );
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document.write( "the formula for the surface area of the sphere is 4*pi*r^2\r<BR>\n" );
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document.write( "once again, the ratio of the surface area of sphere1 and sphere2 becomes (4*pi*r1^2) / (4*pi*r2^2) which simplifies to r1^2/r2^2 which becomes 1/9.\r<BR>\n" );
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document.write( "the general rules are:\r<BR>\n" );
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document.write( "if the two figures are similar, then the ratio of their corresponding sides is the same.<BR>\n" );
document.write( "the ratio of their corresponding areas is equal to the square of the ratio of their corresponding sides.<BR>\n" );
document.write( "the ratio of their corresponding volumes is equal to the cube of the ratio of their corresponding sides.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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