document.write( "Question 1087752: A farmer is fencing off space for his chickens using two rolls of wire fence that are equal in length. One entire roll of wire fence is used to enclose space A in the shape of a regular polygon with 8 sides, and the entire second roll is used to enclose space B in the shape of a regular polygon with 5 sides. If each side of B is 6 feet longer than each side of A, what is the length, in feet, of one roll of wire fence?\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #702046 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
Start by defining a variable:
\n" ); document.write( " $\"x\"$ = length (in feet) of each side of the regular polygon with 8 sides (space A).
\n" ); document.write( "Now, you know that the length (in feet) of the roll used to fence space A is
\n" ); document.write( " $\"8x\"$ ,
\n" ); document.write( "which is what you want to find.
\n" ); document.write( "You also know that
\n" ); document.write( " $\"x%2B6\"$ = length (in feet) of each side of the regular polygon with 5 sides (space B),
\n" ); document.write( "and as a consequence, you can calculate the length (in feet) of the roll used to fence space B as
\n" ); document.write( " $\"5%28x%2B6%29\"$ .
\n" ); document.write( "Because the two rolls were \"two rolls of wire fence that are equal in length\",
\n" ); document.write( "the equation to write and solve is
\n" ); document.write( " $\"5%28x%2B6%29=8x\"$ .
\n" ); document.write( "You solve that to find $\"x\"$ , and then use it to find $\"8x\"$ .
\n" ); document.write( " $\"8x=5%28x%2B6%29\"$
\n" ); document.write( " $\"8x=5x%2B30\"$
\n" ); document.write( " $\"8x-5x=30\"$
\n" ); document.write( " $\"3x=30\"$
\n" ); document.write( " $\"x=30%2F3\"$
\n" ); document.write( " $\"x=10\"$
\n" ); document.write( " $\"8x=8%2A10\"$
\n" ); document.write( " $\"8x=highlight%2880%29\"$
\n" ); document.write( "The length, in feet, of one roll of wire fence is $\"highlight%2880%29\"$ .
\n" ); document.write( "
\n" ); document.write( "ANOTHER WAY:
\n" ); document.write( "Rolls of wire fence probably come in lengths that are integer numbers of feet,
\n" ); document.write( "and farmers like integers, so the lengths of the sides (in feet) of fenced spaces are probably whole numbers, like 1, 3, 5, 10.
\n" ); document.write( "Then the length (in feet) of a roll of wire fence must be a multiple of 8, and a multiple of 5.
\n" ); document.write( "The smallest one is $\"8%2A5=40\"$ , and the others are multiples such as
\n" ); document.write( " $\"2%2A40=80\"$ , $\"3%2A40=120\"$ , $\"4%2A40=160\"$ , and so on.
\n" ); document.write( "If the length of a toll were $\"5%2A8=40\"$ feet,
\n" ); document.write( "the lengths (in feet) of the polygons' sides would be
\n" ); document.write( " $\"5%2A8%2F8=5\"$ for space A, and
\n" ); document.write( " $\"5%2A8%2F5=8\"$ for space B,
\n" ); document.write( "but 8 is not 6 more that 5,
\n" ); document.write( "so a 40 ft roll length does not make the side of space B 6 feet longer than the side of space A.
\n" ); document.write( "What about a length of $\"80=2%2A40=2%2A5%2A8\"$ feet for the roll?
\n" ); document.write( "That would make the lengths (in feet) of the polygons' sides
\n" ); document.write( " $\"25%2A8%2F8=2%2A5=10\"$ for space A, and
\n" ); document.write( " $\"2%2A5%2A8%2F5=2%2A8=16\"$ for space B,
\n" ); document.write( "and $\"16\"$ is $\"16-10=6\"$ more that $\"10\"$ ,
\n" ); document.write( "so an $\"80\"$ -foot roll length is the solution.
\n" ); document.write( "It does make the side of space B exactly 6 feet longer than the side of space A. \n" ); document.write( "

\n" );