document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1023909>Question 1023909</A>: <I><SPAN STYLE=\"word-break: break-all;\"> THIS IS AN ALGEBRAIC FUNCTIONS VARIATIONS PROBLEM. There is also a constant k that is part of the equation I have to create. Can someone go through the logic for this problem? \r<BR>\n" );
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document.write( "Kepler's Third Law states that the square of time T required for a planet to complete one orbit around the sun (the period, that is, the length of one planetary year) is directly proportional to the cube of average distance d of the planet from the sun. For the planet Earth, assume d = 93 x 10^6 miles an T = 365 days. Find a) the period of Mars, given that Mars is approximately 1.5 times as distant from the sun as Earth; b) the average distance of Venus from the sun, given that the period of Venus is approximately 223 Earth days. \r<BR>\n" );
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document.write( "The units for Part a are in days and the units for part b are in miles.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #639406 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=639406\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Stated in formulas, rather than words,<BR>\n" );
document.write( "Kepler's Third Law states that<BR>\n" );
document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T ALIGN=MIDDLE  ALT=\"T\"   DISABLED_style= \"max-width:90%; height:auto;\" >= time required for a planet to complete one orbit around the sun (the period, that is, the length of one planetary year), and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" >= average distance of the planet from the sun,<BR>\n" );
document.write( "then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5E2=k%2Ad%5E3 ALIGN=MIDDLE  ALT=\"T%5E2=k%2Ad%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the proportionality constant.<BR>\n" );
document.write( "So, for Earth, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BEarth%5D%5E2=k%2Ad%5BEarth%5D%5E3 ALIGN=MIDDLE  ALT=\"T%5BEarth%5D%5E2=k%2Ad%5BEarth%5D%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For the planet Earth, assume <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5BEarth%5D ALIGN=MIDDLE  ALT=\"d%5BEarth%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9%2A10%5E6 ALIGN=MIDDLE  ALT=\"9%2A10%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=miles ALIGN=MIDDLE  ALT=\"miles\"   DISABLED_style= \"max-width:90%; height:auto;\" > an <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BEarth%5D=+365 ALIGN=MIDDLE  ALT=\"T%5BEarth%5D=+365\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=days ALIGN=MIDDLE  ALT=\"days\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since we are going to measure times in Earth days and distances in miles<BR>\n" );
document.write( "(or in whatever units we want, but using the same units all along),<BR>\n" );
document.write( "we do not need to keep writing the units with the calculations.<BR>\n" );
document.write( "NOTE: I did not solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > and I did not use the given average distance between Earth and the sun, because it would only complicate the calculations.<BR>\n" );
document.write( "If your teacher insist that you do it, I am sorry for both of you.<BR>\n" );
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document.write( "a) They tell us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5BMars%5D ALIGN=MIDDLE  ALT=\"d%5BMars%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=approximately ALIGN=MIDDLE  ALT=\"approximately\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.5%2Ad%5BEarth%5D ALIGN=MIDDLE  ALT=\"1.5%2Ad%5BEarth%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and according to Kepler's Third Law<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BMars%5D%5E2=k%2Ad%5BMars%5D%5E3 ALIGN=MIDDLE  ALT=\"T%5BMars%5D%5E2=k%2Ad%5BMars%5D%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BMars%5D%5E2=k%2A%281.5%2Ad%5BEarth%5D%29%5E3 ALIGN=MIDDLE  ALT=\"T%5BMars%5D%5E2=k%2A%281.5%2Ad%5BEarth%5D%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BMars%5D%5E2=k%2A1.5%5E3%2Ad%5BEarth%5D%5E3 ALIGN=MIDDLE  ALT=\"T%5BMars%5D%5E2=k%2A1.5%5E3%2Ad%5BEarth%5D%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BMars%5D%5E2=T%5BEarth%5D%5E2%2A1.5%5E3 ALIGN=MIDDLE  ALT=\"T%5BMars%5D%5E2=T%5BEarth%5D%5E2%2A1.5%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BMars%5D%5E2=%28k%2Ad%5BEarth%5D%5E3%29%2A1.5%5E3 ALIGN=MIDDLE  ALT=\"T%5BMars%5D%5E2=%28k%2Ad%5BEarth%5D%5E3%29%2A1.5%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BMars%5D%5E2=365%5E2%2A1.5%5E3 ALIGN=MIDDLE  ALT=\"T%5BMars%5D%5E2=365%5E2%2A1.5%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BMars%5D%5E2=133225%2A3.375 ALIGN=MIDDLE  ALT=\"T%5BMars%5D%5E2=133225%2A3.375\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BMars%5D%5E2=449634.375 ALIGN=MIDDLE  ALT=\"T%5BMars%5D%5E2=449634.375\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BMars%5D=sqrt%28449634.375%29 ALIGN=MIDDLE  ALT=\"T%5BMars%5D=sqrt%28449634.375%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BMars%5D=about671 ALIGN=MIDDLE  ALT=\"T%5BMars%5D=about671\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded).<BR>\n" );
document.write( "So, the period of Mars is approximately 671 days.<BR>\n" );
document.write( " <BR>\n" );
document.write( "b) They tell us that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BVenus%5D=223 ALIGN=MIDDLE  ALT=\"T%5BVenus%5D=223\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and according to Kepler's Third Law<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BVenus%5D%5E2=k%2Ad%5BVenus%5D%5E3 ALIGN=MIDDLE  ALT=\"T%5BVenus%5D%5E2=k%2Ad%5BVenus%5D%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5BVenus%5D%5E2=k%2Ad%5BVenus%5D%5E3 ALIGN=MIDDLE  ALT=\"T%5BVenus%5D%5E2=k%2Ad%5BVenus%5D%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so with distances in miles this time<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=223%5E2=k%2Ad%5BVenus%5D%5E3 ALIGN=MIDDLE  ALT=\"223%5E2=k%2Ad%5BVenus%5D%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=365%5E2=k%2A%2893%2A10%5E6%29%5E3 ALIGN=MIDDLE  ALT=\"365%5E2=k%2A%2893%2A10%5E6%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Dividing one equation by the other,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=223%5E2%2F365%5E2=k%2Ad%5BVenus%5D%5E3%2F%28k%2A%2893%2A10%5E6%29%5E3%29 ALIGN=MIDDLE  ALT=\"223%5E2%2F365%5E2=k%2Ad%5BVenus%5D%5E3%2F%28k%2A%2893%2A10%5E6%29%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49729%2F133225=d%5BVenus%5D%5E3%2F%2893%5E3%2A10%5E18%29 ALIGN=MIDDLE  ALT=\"49729%2F133225=d%5BVenus%5D%5E3%2F%2893%5E3%2A10%5E18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49729%2F133225=d%5BVenus%5D%5E3%2F%28804357%2A10%5E18%29 ALIGN=MIDDLE  ALT=\"49729%2F133225=d%5BVenus%5D%5E3%2F%28804357%2A10%5E18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49729%2A804357%2A10%5E18%2F133225=d%5BVenus%5D%5E3 ALIGN=MIDDLE  ALT=\"49729%2A804357%2A10%5E18%2F133225=d%5BVenus%5D%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49729%2A804357%2A10%5E18%2F133225=d%5BVenus%5D%5E3 ALIGN=MIDDLE  ALT=\"49729%2A804357%2A10%5E18%2F133225=d%5BVenus%5D%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=300243%2A10%5E18=d%5BVenus%5D%5E3 ALIGN=MIDDLE  ALT=\"300243%2A10%5E18=d%5BVenus%5D%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=root%283%2C300243%2A10%5E18%29=d%5BVenus%5D ALIGN=MIDDLE  ALT=\"root%283%2C300243%2A10%5E18%29=d%5BVenus%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5BVenus%5D=root%283%2C300243%29%2Aroot%283%2C10%5E18%29 ALIGN=MIDDLE  ALT=\"d%5BVenus%5D=root%283%2C300243%29%2Aroot%283%2C10%5E18%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5BVenus%5D=67%2A10%5E6 ALIGN=MIDDLE  ALT=\"d%5BVenus%5D=67%2A10%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded).<BR>\n" );
document.write( "So, the the average distance of Venus from the sun, is approximately 67 x 10^6 miles.<BR>\n" );
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document.write( "</TD></TR></TABLE>\n" );