document.write( "Question 880779: please help solve with solutions: two cars travels at a rate of 30 kph and 40 kph respectively at the same point in a right angle position. how long will they be 100 kms apart
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Algebra.Com's Answer #531697 by KMST(5347) $\"\"$ $\"About$

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The distances (in km) traveled by one car an the other form the legs of a right triangle.
\n" ); document.write( "The distance from one car to the other would be the hypotenuse.
\n" ); document.write( "The Pythagorean theorem says that the lengths of the legs and hypotenuse of a right triangle ( $\"leg%5B1%5D\"$ , $\"leg%5B2%5D\"$ , and $\"hypotenuse\"$ )
\n" ); document.write( "are related by $\"leg%5B1%5D%5E2%2Bleg%5B2%5D%5E2=hypotenuse%5E2\"$ .
\n" ); document.write( "You are expected to write some equation using that and solve it.
\n" ); document.write( "It is good to know how to do that
\n" ); document.write( "However, if you are faced with this problem on a timed test, a mental math approach would be very efficient, and with that approach I knew that the answer is that it will take $\"highlight+%282hours%29\"$ for the cars to be 100 kms apart.
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\n" ); document.write( "THE EXPECTED WAY TO SOLVE IT:
\n" ); document.write( " $\"t\"$ = time in hours from the time the cars start traveling away from their common starting point.
\n" ); document.write( " $\"30t\"$ = distance (in km) traveled by the car moving at a rate of 30 kph.
\n" ); document.write( " $\"40t\"$ = distance (in km) traveled by the car moving at a rate of 40 kph.
\n" ); document.write( "Applying the Pythagorean theorem we get our equation:
\n" ); document.write( " $\"%2830t%29%5E2%2B%2840t%29%5E2=100%5E2\"$ .
\n" ); document.write( "Now, we simplify and solve:
\n" ); document.write( " $\"%2830t%29%5E2%2B%2840t%29%5E2=100%5E2\"$
\n" ); document.write( " $\"30%5E2t%5E2%2B40%5E2t%5E2=100%5E2\"$
\n" ); document.write( " $\"900t%5E2%2B1600t%5E2=10000\"$
\n" ); document.write( " $\"%28900%2B1600%29t%5E2=10000\"$
\n" ); document.write( " $\"2500t%5E2=10000\"$
\n" ); document.write( " $\"t%5E2=10000%2F2500\"$
\n" ); document.write( " $\"t%5E2=4\"$
\n" ); document.write( "That equation has the solutions $\"highlight%28t=2%29\"$ and $\"t=-2\"$ ,
\n" ); document.write( "but the way the problem asks the question, $\"t=-2\"$ makes no sense, because you want a positive number for the answer,
\n" ); document.write( "so it will take $\"highlight+%282hours%29\"$ for the cars to be 100 kms apart.
\n" ); document.write( "In reality, what the $\"t=2\"$ and $\"t=-2\"$ answers tell you is that if two cars are moving at those speeds on perpendicular roads, and cross paths narrowly missing a collision, they will be 100 km apart 2 hours before and 2 hours after they cross paths.
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\n" ); document.write( "THE MENTAL MATH TRICK:
\n" ); document.write( "Math problems are usually carefully designed to give results that are look nice and are easy to calculate. If possible, we like integer numbers.
\n" ); document.write( "There are some trios of natural numbers (positive integers) that fit the Pythagorean relation.
\n" ); document.write( "They are called Pythagorean triples, and you can search online and find lists of the first few. (I believe nobody listed the expected infinity of them).
\n" ); document.write( "The only one that all math teachers remember is (3,4,5).
\n" ); document.write( "(It is the only one I remember.
\n" ); document.write( "Because $\"3%5E2%2B4%5E2=5%5E2\"$ , a right triangle with legs measuring 3 and 4 of whatever unit will have a hypotenuse measuring 5 of the same unit.
\n" ); document.write( "Of course, you can scale up that 3-4-5 triangle to get one with sides measuring 6, 8, and 10, or 60, 80, and 100. (I can do that calculation in my head, can you?)
\n" ); document.write( "The problem asked for a hypotenuse measuring 100 km, and that meant legs measuring 60 km and 80 km.
\n" ); document.write( "For a car moving at 30 kph to travel 60 kn it would take $\"60km%2F30kph=2hours\"$ . (I can do that calculation in my head, can you?)
\n" ); document.write( "Right triangles with side length in a 3:4:5 ratio show up in many problems, so it is a useful ratio to keep in mind. \n" ); document.write( "

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