document.write( "Question 798186: A concrete walk of uniform width is to be built around a circular pool, as shown in the figure. The radius of the pool is 16 meters, and enough concrete is available to cover $ {\color{black}144} \pi $ square meters. If all the concrete is to be used, how wide should the walk be? \n" ); document.write( "
Algebra.Com's Answer #482147 by KMST(5328)\"\" \"About 
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PROBABLE EXPECTED SOLUTION:
\n" ); document.write( "The area of the pool is the area of a circle of radius \"16m\", so it is
\n" ); document.write( "\"pi%2A16%5E2\"\"m%5E2=256pi\"\"m%5E2\"
\n" ); document.write( "The area (in square meters) of the pool plus a surrounding walkway of width \"x\" meters would be the area of a circle of radius \"16%2Bx\" meters:
\n" ); document.write( "
\n" ); document.write( "The difference is the area of the walkway. In square meters, it is
\n" ); document.write( "\"256pi%2Bpi%2832x%2Bx%5E2%29-256pi=%2832x%2Bx%5E2%29pi\"
\n" ); document.write( "If we can use enough concrete to cover \"144pi\" square meters,
\n" ); document.write( "\"%2832x%2Bx%5E2%29pi=144pi\"
\n" ); document.write( "and \"x%5E2%2B32x=144\" is our equation.
\n" ); document.write( "We can solve it by completing the square, by factoring, or by using the quadratic formula.
\n" ); document.write( "
\n" ); document.write( "Completing the square:
\n" ); document.write( "\"x%5E2%2B32x=144\"
\n" ); document.write( "\"x%5E2%2B32x%2B16%5E2=144%2B16%5E2\"
\n" ); document.write( "\"%28x%2B16%29%5E2=144%2B254\"
\n" ); document.write( "\"%28x%2B16%29%5E2=400\"
\n" ); document.write( "\"%28x%2B16%29%5E2=20%5E2\"
\n" ); document.write( "Since we are looking for a positive number \"x\",
\n" ); document.write( "\"x=16=20\" --> \"x=20-16\" --> \"highlight%28x=4%29\"
\n" ); document.write( "
\n" ); document.write( "Factoring:
\n" ); document.write( "\"x%5E2%2B32x=144\"
\n" ); document.write( "\"x%5E2%2B32x-144=0\"
\n" ); document.write( "\"%28x-4%29%28x%2B36%29=0\" --> \"x-4=0\" --> \"highlight%28x=4%29\"
\n" ); document.write( "because \"x%2B36=0\" <--> \"x=-36\" is not an acceptable solution.
\n" ); document.write( "
\n" ); document.write( "Applying the quadratic formula:
\n" ); document.write( "The solutions to \"+ax%5E2%2Bbx%2Bc=0\" are given by \"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"
\n" ); document.write( "In the case of \"x%5E2%2B32x-144=0\", \"a=1\", \"b=32\", and \"c=-144\", so
\n" ); document.write( "
\n" ); document.write( "The two solutions are
\n" ); document.write( "\"x=%28-32-40%29=2=-72%2F2=-36\" which does not make sense as width of a walkway,
\n" ); document.write( "and \"x=%28-32%2B40%29%2F2=8%2F2=highlight%284%29\" which is the width, in meters, of the walkway we want.
\n" ); document.write( "
\n" ); document.write( "MENTAL MATH QUICK SOLUTION:
\n" ); document.write( "The area of a circle or radius \"R\" is \"pi%2AR%5E2\"
\n" ); document.write( "The area of the pool, \"pi%2A16%5E2\" square meters,
\n" ); document.write( "plus the area of the walkway, \"pi%2A144=pi%2A12%5E2\" square meters,
\n" ); document.write( "adds up to the area, \"pi%2AR%5E2\", of a larger circle of radius \"R\" meters. So \"pi%2A16%5E2%2Bpi%2A12%5E2=pi%2AR%5E2\" --> \"16%5E2%2B12%5E2=R%5E2\"
\n" ); document.write( "That reminds me of the Pythagorean theorem.
\n" ); document.write( "Since \"16=4%2A4\" and \"12=4%2A3\", I can think of \"16%5E2%2B12%5E2=R%5E2\" as
\n" ); document.write( "\"%284%2A4%29%5E2%2B%284%2A3%29%5E2=R%5E2\" --> \"4%5E2%2A4%5E2=4%5E2%2A3%5E2=R%5E2\" --> \"4%5E2%284%5E2%2B3%5E2%29=R%5E2\"
\n" ); document.write( "Since \"4%5E2%2B3%5E2=16%2B9=25=5%5E2\" has been used in so many problems, I remember it better than I remember the times tables, so
\n" ); document.write( "\"4%5E2%284%5E2%2B3%5E2%29=R%5E2\" --> \"4%5E2%2A5%5E2=R%5E2\" --> \"%284%2A5%29%5E2=R%5E2\" --> \"R=20\"
\n" ); document.write( "and that means that the width of the sidewalk is \"20-16=highlight%284%29\" meters \n" ); document.write( "
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