document.write( "Question 468784: Hello I have a uniform motion problem involving rational expressions, I normally get these but, I can't find the set up for this one. I know the whole d=rt, r=d/t and t=r/d but I can't seam to figure out whats what,anyways here it goes: \r
\n" ); document.write( "\n" ); document.write( "\"Suppose that Jeremy Shockey of the New Orlean Saints can run 100yards in 12 seconds. Further suppose Brian Urlacher of the Chicago Bears can run 100yards in 9 seconds. Suppose that Shockey catches a pass at his own 20-yard line in stride and starts running away from Urlacher, who is at the 15-yard line directly behind Shockey. At what yard line will Urlacher catch up to Shockey and tackle him?\"\r
\n" ); document.write( "\n" ); document.write( "Thanks for the help! \n" ); document.write( "
Algebra.Com's Answer #321671 by Theo(13342)\"\" \"About 
You can put this solution on YOUR website!
THERE IS A LONG WAY TO CALCULATE THIS AND THERE IS A SHORT WAY TO CALCULATE THIS.
\n" ); document.write( "FIRST THE LONG WAY.
\n" ); document.write( "shockey rate of travel is 100 yards in 12 seconds = 100/12.
\n" ); document.write( "urlacher rate of travel is 100 yards in 9 seconds = 100/9.
\n" ); document.write( "urlacher will catch shockey in 1.8 seconds.
\n" ); document.write( "in 1.8 seconds, shockey will have traveled 1.8 * (100/12) = 15 yards.
\n" ); document.write( "in 1.8 seconds, urlacher will have traveled 1.8 * (100/9) = 20 yards.
\n" ); document.write( "the formulas that we used to solve this problem are:
\n" ); document.write( "R1*T = D for shockey
\n" ); document.write( "R2*T = D+5 for urlacher.
\n" ); document.write( "We know R1 and R2, but we don't know D or T, although we do know that urlacher had to travel 5 more yards than shockey in order to catch him.
\n" ); document.write( "Since T will be the same for both, we solve each of these equations in terms of T to get:
\n" ); document.write( "T = D/R1
\n" ); document.write( "T = (D+5)/R2
\n" ); document.write( "since the expressions on the right hand side of each equation are both equal to T, they we can set these expressions equal to each other to get:
\n" ); document.write( "D/R1 = (D+5)/R2
\n" ); document.write( "Cross multiply to get:
\n" ); document.write( "D*R2 = (D+5)*R1
\n" ); document.write( "Expand the expression on the right side of the equation to get:
\n" ); document.write( "D*R2 = D*R1 + 5*R1
\n" ); document.write( "subtract D*R1 from both sides of the equation to get:
\n" ); document.write( "D*R2 - D*R1 = 5*R1
\n" ); document.write( "factor out the D from the left side of the equation to get:
\n" ); document.write( "D*(R2-R1) = 5*R1
\n" ); document.write( "divide both sides of the equation by (R2-R1) to get:
\n" ); document.write( "D = (5*R1)/(R2-R1)
\n" ); document.write( "This formula will get us D from which we can then derive T.
\n" ); document.write( "Since R1 = (100/12) and R2 = (100/9), this formula becomes:
\n" ); document.write( "D = (5*(100/12)) / ((100/9) - (100/12)) which gets us:
\n" ); document.write( "D = 15
\n" ); document.write( "This makes D + 5 = 20
\n" ); document.write( "shockey runs 15 yards in the same time that urlacher runs 20.
\n" ); document.write( "since we know D, we can solve for T.
\n" ); document.write( "Either equation should do it because the time is the same in both.
\n" ); document.write( "We'll use Shockey's equation.
\n" ); document.write( "That is R1 * T = D which becomes:
\n" ); document.write( "(100/12) * T = 15
\n" ); document.write( "Divide both sides of this equation by (100/12) to get:
\n" ); document.write( "T = 15/(100/12) which becomes:
\n" ); document.write( "T = 1.8 seconds.
\n" ); document.write( "shockey runs 15 yards in 1.8 seconds.
\n" ); document.write( "urlacher runs 20 yards in 1.8 seconds.
\n" ); document.write( "urlacher catches shockey in 1.8 seconds.
\n" ); document.write( "***********************************************
\n" ); document.write( "NOW THE SHORT WAY
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\n" ); document.write( "We already derived our formulas above using rate * time = distance.
\n" ); document.write( "They are:
\n" ); document.write( "R1 * T = D for shockey.
\n" ); document.write( "R2 * T = D + 5 for urlacher.
\n" ); document.write( "we want to solve these equations simultaneously since the same solution will apply to both equations.
\n" ); document.write( "we subtract the first equation from the second equation to get:
\n" ); document.write( "R2*T - R1*T = 5
\n" ); document.write( "We factor out the T to get:
\n" ); document.write( "T * (R2-R1) = 5
\n" ); document.write( "We divide both sides of this equation by (R2-R1) to get:
\n" ); document.write( "T = 5/(R2-R1)
\n" ); document.write( "Since R2 = 100/12 and R1 = 100/9, we substitute in this equation to get:
\n" ); document.write( "T = 5/((100/12)-(100.9)) to get:
\n" ); document.write( "T = 1.8
\n" ); document.write( "Notice that we solved for T but did not solve for D.
\n" ); document.write( "If we want D, we still have to go back in to either equation to solve for D.
\n" ); document.write( "If we used the second equation (the first would have been a little easier), we would get:
\n" ); document.write( "R2 * T = D + 5
\n" ); document.write( "We know R2 and we know T so we substitute to get:
\n" ); document.write( "(100/9) * 1.8 = D + 5
\n" ); document.write( "we subtract 5 from both sides of this equation to get:
\n" ); document.write( "((100/9)*1.8) - 5 = D
\n" ); document.write( "We solve for D to get:
\n" ); document.write( "D = 15
\n" ); document.write( "THIS PROBLEM CAN BE SOLVED EITHER WAY AND YOU'LL GET THE SAME ANSWER.
\n" ); document.write( "THE SHORT WAY SEEMS EASIER IF YOU RECOGNIZE THAT YOU CAN DO IT THAT WAY.\r
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