document.write( "Question 370575: Liz commutes 30 mi. to her job each day. She Finds that if she drives 10 mi/h faster it takes her 6 minutes less to get to work. Find her new speed.\r
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Algebra.Com's Answer #264133 by jim_thompson5910(35256)\"\" \"About 
You can put this solution on YOUR website!
Let r = original speed.\r
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\n" ); document.write( "\n" ); document.write( "and let t = time to get to work driving that original speed.\r
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\n" ); document.write( "\n" ); document.write( "So if \"Liz commutes 30 mi. to her job each day\", this means that \"30=rt\" because we're using the formula \"d=rt\" where the distance is \"d=30\"\r
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\n" ); document.write( "\n" ); document.write( "So the first equation is \"30=rt\"\r
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\n" ); document.write( "\n" ); document.write( "In addition, because \"she drives 10 mi/h faster it takes her 6 minutes less to get to work\", we can say that \"30=%28r%2B10%29%28t-1%2F10%29\"\r
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\n" ); document.write( "\n" ); document.write( "Notes: since she drives 10 mph faster, her new speed is r+10. Also, because 60 min = 1 hour, this means that 6 min = \"6%2F60=1%2F10\" hours. So if it takes 6 mins or \"1%2F10\" or an hour less, then the new time is \"t-1%2F10\"\r
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\n" ); document.write( "\n" ); document.write( "So the second equation is \"30=%28r%2B10%29%28t-1%2F10%29\"\r
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\n" ); document.write( "\n" ); document.write( "Now let's use both equations to solve for t and r\r
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\n" ); document.write( "\n" ); document.write( "\"30=rt\" Start with the first equation.\r
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\n" ); document.write( "\n" ); document.write( "\"30%2Ft=r\" Divide both sides by t to isolate r.\r
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\n" ); document.write( "\n" ); document.write( "\"r=30%2Ft\" Flip the equation.\r
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\n" ); document.write( "\n" ); document.write( "\"30=%28r%2B10%29%28t-1%2F10%29\" Move onto the second equation.\r
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\n" ); document.write( "\n" ); document.write( "\"30=%2830%2Ft%2B10%29%28t-1%2F10%29\" Plug in \"r=30%2Ft\"\r
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\n" ); document.write( "\n" ); document.write( "\"30=%2830%2Ft%2B10t%2Ft%29%28t-1%2F10%29\" Multiply 10 by \"t%2Ft\"\r
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\n" ); document.write( "\n" ); document.write( "\"30=%28%2830%2B10t%29%2Ft%29%28t-1%2F10%29\" Combine the fractions.\r
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\n" ); document.write( "\n" ); document.write( "\"30=%28%2830%2B10t%29%2Ft%29%28t%29-%28%2830%2B10t%29%2Ft%29%281%2F10%29\" Distribute\r
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\n" ); document.write( "\n" ); document.write( "\"30=30%2B10t-%283%2Bt%29%2Ft\" Multiply.\r
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\n" ); document.write( "\n" ); document.write( "Note: the 't' terms cancel in the first fraction while in the second, we're dividing each term by 10.\r
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\n" ); document.write( "\n" ); document.write( "\"30t=30t%2B10t%5E2-%283%2Bt%29\" Multiply EVERY term by the LCD 't' to clear out the fractions.\r
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\n" ); document.write( "\n" ); document.write( "\"30t=30t%2B10t%5E2-3-t\" Distribute.\r
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\n" ); document.write( "\n" ); document.write( "\"0=30t%2B10t%5E2-3-t-30t\" Subtract 30t from both sides.\r
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\n" ); document.write( "\n" ); document.write( "\"0=10t%5E2-t-3\" Combine like terms.\r
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\n" ); document.write( "\n" ); document.write( "Notice that the quadratic \"10t%5E2-t-3\" is in the form of \"At%5E2%2BBt%2BC\" where \"A=10\", \"B=-1\", and \"C=-3\"\r
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\n" ); document.write( "\n" ); document.write( "Let's use the quadratic formula to solve for \"t\":\r
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\n" ); document.write( "\n" ); document.write( "\"t+=+%28-B+%2B-+sqrt%28+B%5E2-4AC+%29%29%2F%282A%29\" Start with the quadratic formula\r
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\n" ); document.write( "\n" ); document.write( "\"t+=+%28-%28-1%29+%2B-+sqrt%28+%28-1%29%5E2-4%2810%29%28-3%29+%29%29%2F%282%2810%29%29\" Plug in \"A=10\", \"B=-1\", and \"C=-3\"\r
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\n" ); document.write( "\n" ); document.write( "\"t+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%2810%29%28-3%29+%29%29%2F%282%2810%29%29\" Negate \"-1\" to get \"1\". \r
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\n" ); document.write( "\n" ); document.write( "\"t+=+%281+%2B-+sqrt%28+1-4%2810%29%28-3%29+%29%29%2F%282%2810%29%29\" Square \"-1\" to get \"1\". \r
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\n" ); document.write( "\n" ); document.write( "\"t+=+%281+%2B-+sqrt%28+1--120+%29%29%2F%282%2810%29%29\" Multiply \"4%2810%29%28-3%29\" to get \"-120\"\r
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\n" ); document.write( "\n" ); document.write( "\"t+=+%281+%2B-+sqrt%28+1%2B120+%29%29%2F%282%2810%29%29\" Rewrite \"sqrt%281--120%29\" as \"sqrt%281%2B120%29\"\r
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\n" ); document.write( "\n" ); document.write( "\"t+=+%281+%2B-+sqrt%28+121+%29%29%2F%282%2810%29%29\" Add \"1\" to \"120\" to get \"121\"\r
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\n" ); document.write( "\n" ); document.write( "\"t+=+%281+%2B-+sqrt%28+121+%29%29%2F%2820%29\" Multiply \"2\" and \"10\" to get \"20\". \r
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\n" ); document.write( "\n" ); document.write( "\"t+=+%281+%2B-+11%29%2F%2820%29\" Take the square root of \"121\" to get \"11\". \r
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\n" ); document.write( "\n" ); document.write( "\"t+=+%281+%2B+11%29%2F%2820%29\" or \"t+=+%281+-+11%29%2F%2820%29\" Break up the expression. \r
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\n" ); document.write( "\n" ); document.write( "\"t+=+%2812%29%2F%2820%29\" or \"t+=++%28-10%29%2F%2820%29\" Combine like terms. \r
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\n" ); document.write( "\n" ); document.write( "\"t+=+3%2F5\" or \"t+=+-1%2F2\" Simplify. \r
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\n" ); document.write( "\n" ); document.write( "So the possible solutions are \"t+=+3%2F5\" or \"t+=+-1%2F2\" \r
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\n" ); document.write( "However, since a negative time isn't possible, this means that the only solution for 't' is \"t=3%2F5\"\r
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\n" ); document.write( "\n" ); document.write( "Remember that 't' is the time in hours. So convert to minutes to get \"%283%2F5%29%2A%2860%29+=+180%2F5+=+36\" minutes\r
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\n" ); document.write( "\n" ); document.write( "So the time it takes to travel 30 miles at the original speed is 36 minutes.\r
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\n" ); document.write( "\n" ); document.write( "Recall that we made \"r=30%2Ft\". So plug \"t=3%2F5\" into the equation to get \"r=30%2F%283%2F5%29=%2830%2F1%29%285%2F3%29=150%2F3=50\"\r
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\n" ); document.write( "\n" ); document.write( "So her original speed is 50 mph. Add 10 mph to this speed to get 50+10 = 60 mph\r
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\n" ); document.write( "\n" ); document.write( "So her new speed is 60 mph.\r
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\n" ); document.write( "\n" ); document.write( "I'll leave the check to you. Remember to use the formula \"d=rt\"\r
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\n" ); document.write( "\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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\n" ); document.write( "\n" ); document.write( "Also, feel free to check out my tutoring website\r
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\n" ); document.write( "\n" ); document.write( "Jim
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