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How Do You Solve an Opposite-Direction Travel Problem?

Timothy leaves his house at 9 a.m., traveling at 30 mph. Jennifer leaves 30 minutes later, traveling in the opposite direction at 40 mph. At what time will they be 190 miles apart?

Summary

  1. dTIM represents Tim's distance
  2. dJEN represents Jen's distance
  3. dTOT represents total distance
  4. dTIM equals Tim's rate times his time, rTIM•tTIM
  5. dJEN equals Jen's rate times her time, rJEN•tJEN
  6. tJEN=tTIM-0.5
  7. dTOT=rTIM•tTIM +rJEN(tTIM -0.5)
  8. dTOT=190
  9. rTIM=30
  10. rJEN=40

Notes

    1. d will be a variable that represents distance
    2. r will be a variable that represents rate
    3. t will be a variable that represents time
    1. 190 is a distance
    1. Timothy and Jennifer started in the same place and drove in opposite directions
    2. So the distance between them is the total of how far they each traveled
    3. This is 190 miles
    1. We can add Timothy's distance and Jennifer's distance to get the total distance of 190
    1. We can add Timothy's distance and Jennifer's distance to get the total distance of 190
    2. And we can find values for these distances using the formula for uniform motion
    1. The formula for uniform motion gives us distance in terms of rate and time
    2. d is distance
    3. r is rate
    4. t is time
    1. We can use this formula to help us set up our equation
    1. We can put Tim's distance in terms of his rate and time
    2. dTIM will represent Tim's distance
    3. rTIM will represent Tim's rate
    4. tTIM will represent Tim's time
    5. Multiplying Tim's rate times his time will give us his distance
    1. We can put Jen's distance in terms of her rate and time
    2. dJEN will represent Jen's distance
    3. rJEN will represent Jen's rate
    4. tJEN will represent Jen's time
    5. Multiplying Jen's rate times her time will give us her distance
    1. We had that dTOT , the total distance, equals dTIM +dJEN
    2. Now we have values for dTIM and dJEN:
    3. dTIM =rTIM •tTIM
    4. dJEN =rJEN •tJEN
    1. dTOT represents the total distance, which is 190
    2. dTIM =rTIM •tTIM
    3. dJEN =rJEN •tJEN
    1. Since we have so many unknowns, we want to use what we already know to cut down on some of the variables
    2. Since we know Jen left a half our after Tim, we can write Jen's time in terms of Tim's time
    3. tJEN represents Jen's time
    4. tTIM represents Tim's time
    5. Jen left 30 minutes after Tim, but we're measuring time in hours, so 30 minutes is 0.5 hours
    6. Jen has been traveling 0.5 hours less than Tim, so we need to subtract 0.5 from Tim's time
    1. Jen has been traveling a half hour less than Tim, so her time will be a half hour shorter
    1. 30 minutes is a half hour, or 0.5 hours
    1. We now have tJEN =tTIM -.05
    2. dTOT represents the total distance, which is 190
    3. rTIM •tTIM is what we found for Tim's distance
    4. Now we substitute tTIM -0.5 for Jen's time, tJEN
    5. So rJEN (tTIM -0.5) represents Jen's distance
    1. Our equation is dTOT = rTIM •tTIM + rJEN (tTIM -0.5)
    2. Now we can look at our problem and find values for some of these variables
    3. That will make it a little easier to work with!
    1. We represented total distance with dTOT
    2. So dTOT =190
    1. We represented Tim's rate with rTIM
    2. So rTIM =30
    1. We represented Tim's time with tTIM
    1. We represented Jen's rate with rJEN
    2. So rJEN =40
    1. We represented Tim's time with tTIM
    1. We have everything in terms of Tim's time, tTIM!
    1. Now we can solve for our variable, tTIM
    1. Multiply the 40 by each term in the parentheses
    2. 40•tTIM =40tTIM
    3. 40•0.5=20
    1. 30tTIM and 40tTIM are like terms because they have the same variable, tTIM, to the same degree
    2. So we can add them to get 70tTIM
    1. We need to get 70tTIM by itself
    2. Since addition is the opposite of subtraction, we can get rid of the -20 on the right by adding 20 to both sides
    3. We get 190+20=210 on the left
    4. And the 20's cancel to leave us with 70tTIM on the right
    1. We want to get tTIM by itself
    2. Since division is the opposite of multiplication, we can undo the multiplication by dividing both sides by 70
    3. We get 210/70=3 on the left
    4. And we have tTIM by itself on the right!
    1. Remember, tTIM represents Tim's time, which we found to be 3
    1. But our problem wasn't asking us how long Tim was driving
    2. It was asking us when Tim and Jen would be 190 miles apart
    3. We can use Tim's time to figure this out
    1. We know when Tim started driving (9 a.m.), and how long he drove to get 190 miles away (3 hours)
    1. 3 hours after 9 a.m. is 12 p.m., or noon