document.write( "Question 798185: A 5 foot long pole leans against a wall. The bottom is 3 feet from the wall. How much farther should the bottom be pulled away from the wall so that the top moves the same amount down the wall? \n" ); document.write( "

Algebra.Com's Answer #482092 by KMST(5328) $\"\"$ $\"About$

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THE MENTAL MATH SOLUTION:
\n" ); document.write( " $\"highlight%281foot%29\"$ because (3,4,5) (with $\"3%5E2%2B4%5E2=5%5E2\"$ ) is the most famous Pythagorean triple.
\n" ); document.write( "(It is the only one I can remember and a favorite of math teachers).
\n" ); document.write( "A Pythagorean triple is a set of 3 natural numbers (positive integers) that could be the measures of the sides of a right triangle, because the squares of the first two numbers add up to the square of the third number.
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\n" ); document.write( "A 5 meter pole leaning against a wall, with the bottom at 3 meters from the wall has its top 4 meters high on the wall.
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$\"3%5E2%2B4%5E2=9%2B16=25=5%5E2\"$
\n" ); document.write( "A 5 meter pole leaning against a wall, with the bottom at 4 meters from the wall has its top 3 meters high on the wall.
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It's a flipped version of the same right triangle.
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\n" ); document.write( "THE INTENDED SOLUTION:
\n" ); document.write( "Of course it is harder, as you should expect from a math teacher.
\n" ); document.write( "We find that the pole reaches a height of $\"h=+4\"$ meters up the wall when its bottom is 3 meters from the wall by using the Pythagorean theorem.
\n" ); document.write( " $\"3%5E2%2Bh%5E2=5%5E2\"$ --> $\"9%2Bh%5E2=25\"$ --> $\"h%5E2=25-9\"$ --> $\"h%5E2=16\"$ --> $\"h=4\"$
\n" ); document.write( "As the base of the right triangle is increased by a distance $\"x\"$ (in meters), from $\"3\"$ to $\"3%2Bx\"$ , the height is decreased by a distance $\"x\"$ (in meters), from $\"4\"$ to $\"4-x\"$ .
\n" ); document.write( "According to the Pythagorean theorem:
\n" ); document.write( " $\"%283%2Bx%29%5E2%2B%284-x%29%5E2=5%5E2\"$
\n" ); document.write( " $\"9%2B6x%2Bx%5E2%2B16-8x%2Bx%5E2=25\"$
\n" ); document.write( " $\"25-2x%2B2x%5E2=25\"$
\n" ); document.write( " $\"2x%5E2-2x=0\"$
\n" ); document.write( " $\"2x%28x-1%29=0\"$
\n" ); document.write( "The solution $\"2x=0\"$ --> $\"x=0\"$ means not moving the pole at all, and is not what the problem requires.
\n" ); document.write( "The other solution, $\"x-1=0\"$ --> $\"highlight%28x=1%29\"$ means moving the bottom of the pole 1 meter away from the wall, so it will be 4 meters from the wall, which will get the top to reach 3 meters up the wall. \n" ); document.write( "

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