document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=208577>Question 208577</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I was hoping to get some help with this problem......\r<BR>\n" );
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document.write( "The gravitational attraction between two masses varies inversely as the square of the distance between them. The force of attraction is 4 lbs when the masses are 3ft apart, What is the attraction when the masses are 6ft apart?\r<BR>\n" );
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document.write( "Thank you so much!!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #158430 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=158430\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> variation equations can be found here:<BR>\n" );
document.write( "http://www.purplemath.com/modules/variatn.htm<BR>\n" );
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document.write( "stick with the basic definitions of directly proportional or inversely proportional and this website should help.  i'll explain the difference below.<BR>\n" );
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document.write( "if y is directly proportional to x, then y = k*x where k is a constant of proportionality.<BR>\n" );
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document.write( "if y is inversely proportional to x, then y = k/x where k is a constant of proportionality.<BR>\n" );
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document.write( "let's see how these work in practice.<BR>\n" );
document.write( "if you want to know how it applies to your problem and skip the tutorial, then just jump to the bottom, otherwise continue.<BR>\n" );
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document.write( "y is directly proportional to x with a constant factor of 1<BR>\n" );
document.write( "y = x<BR>\n" );
document.write( "x = 1,2,3 means y = 1,2,3<BR>\n" );
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document.write( "y is directly proportional to x with a constant factor of 10<BR>\n" );
document.write( "y = 10x<BR>\n" );
document.write( "x = 1,2,3 means y = 10,20,30<BR>\n" );
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document.write( "y is directly proportional to x with a constant factor of .5<BR>\n" );
document.write( "y = .5x<BR>\n" );
document.write( "x = 1,2,3 means y = .5,1,1.5<BR>\n" );
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document.write( "y is directly proportional to x with a constant factor of -10<BR>\n" );
document.write( "y = -10x<BR>\n" );
document.write( "x = 1,2,3 means y = -10,-20,-30<BR>\n" );
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document.write( "in all of the above, an increase in the value of x resulted in an increase in the value of y if you disregard the direction of the increase (just look at absolute values).<BR>\n" );
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document.write( "now let's look at the same examples using inverse proportion.<BR>\n" );
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document.write( "y is inversely proportional to x with a constant factor of 1<BR>\n" );
document.write( "y = 1/x<BR>\n" );
document.write( "x = 1,2,3 means y = (1),(1/2),(1/3)<BR>\n" );
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document.write( "y is inversely proportional to x with a constant factor of 10<BR>\n" );
document.write( "y = 10/x<BR>\n" );
document.write( "x = 1,2,3 means y = (10/1),(/10/2),(10/3)<BR>\n" );
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document.write( "y is inversely proportional to x with a constant factor of .5<BR>\n" );
document.write( "y = .5/x<BR>\n" );
document.write( "x = 1,2,3 means y = (.5/1),(.5/2),(.5/3)<BR>\n" );
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document.write( "y is inversely proportional to x with a constant factor of -10<BR>\n" );
document.write( "y = -10/x<BR>\n" );
document.write( "x = 1,2,3 means y = (-10/1),(-10/2),(-10/3)<BR>\n" );
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document.write( "in all of the above, an increase in the value of x resulted in a decrease in the value of y if you disregard the direction of the increase (just look at absolute values).<BR>\n" );
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document.write( "with direct proportion, the absolute value of y increases as the absolute value of x increases, and decreases as the absolute value of x decreases.<BR>\n" );
document.write( "if abs(x) goes up, abs(y) goes up, if abs(x) goes down, abs(y) goes down.<BR>\n" );
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document.write( "with inverse proportion, the absolute value of y decreases as the absolute value of x increases, and increases as the absolute value of x decreases.<BR>\n" );
document.write( "if abs(x) goes up, abs(y) goes down, if abs(x) goes down, abs(y) goes up.<BR>\n" );
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document.write( "NOW TO YOUR PROBLEM<BR>\n" );
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document.write( "The gravitational attraction between two masses varies inversely as the square of the distance between them. <BR>\n" );
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document.write( "The force of attraction is 4 lbs when the masses are 3ft apart. <BR>\n" );
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document.write( "What is the attraction when the masses are 6ft apart?<BR>\n" );
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document.write( "let d = distance between them.<BR>\n" );
document.write( "let g = the gravitational attraction between them.<BR>\n" );
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document.write( "since this is an inverse relationship, the general form of our equation should be:<BR>\n" );
document.write( "g = k/d^2<BR>\n" );
document.write( "where g is the gravitational attraction between them and d is the distance between them and k is the constant of proportionality.<BR>\n" );
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document.write( "the problem states that the force of attraction is 4 pounds when the masses are 3 feet apart.<BR>\n" );
document.write( "putting this in our equation, we get:<BR>\n" );
document.write( "4 = k/3^2<BR>\n" );
document.write( "to solve for k, we multiply both sides of this equation by 3^2 to get:<BR>\n" );
document.write( "4 * 3^2 = k<BR>\n" );
document.write( "this means that k = 4 * 9 = 36<BR>\n" );
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document.write( "we now know what k is and we can solve further.<BR>\n" );
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document.write( "the question is what is the attraction when the masses are 6 feet apart.<BR>\n" );
document.write( "k = 36<BR>\n" );
document.write( "d = 6<BR>\n" );
document.write( "formula is:<BR>\n" );
document.write( "g = 36/d^2<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "g = 36/36<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "g = 1<BR>\n" );
document.write( "this says that the gravitational attraction between them is 1 pound.<BR>\n" );
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document.write( "at 3 feet it's 4 pounds and at 6 feet it's 1 pound.  as they get further apart the gravitational attraction gets weaker.<BR>\n" );
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document.write( "let's go back to the original equation to see if we got it right.<BR>\n" );
document.write( "what is the gravitational pull at 3 feet?<BR>\n" );
document.write( "k = 36<BR>\n" );
document.write( "d = 3<BR>\n" );
document.write( "g = 36/9 = 4<BR>\n" );
document.write( "this means the value of k is confirmed as good.<BR>\n" );
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document.write( "a word about units.<BR>\n" );
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document.write( "g is in pounds and d is in feet.<BR>\n" );
document.write( "k = g*d^2 means that k is in square feet pounds.<BR>\n" );
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document.write( "our equation of 4 = 36 / 9 is really:<BR>\n" );
document.write( "4 pounds = 36 square feet pounds / 9 square feet<BR>\n" );
document.write( "when you divide 36 square feet pounds by 9 square feet, the square feet part of the equation cancels out and you are left with 4 pounds.<BR>\n" );
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document.write( "you don't normally track the types of units as you go along, but sometime it helps to understand what is going on.<BR>\n" );
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document.write( " </SPAN>\n" );
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