When do the Cars Meet?Date: 01/23/97 at 12:55:34 From: Jennifer Subject: Math Write an equation that describes this information: Two cars, an Edsel and a Studebaker, are 635 kilometers apart. They start at the same time and drive toward each other. The Edsel travels at a rate of 70 kilometers per hour and the Studebaker travels 57 kilometers per hour. In how many hours will the two cars meet? I'm stuck on the whole thing. I don't understand this kind of problem. Date: 01/23/97 at 14:52:10 From: Doctor Wilkinson Subject: Re: Math I suppose you know the basic formula for doing this kind of problem (D is distance, R is rate, T is time): D = RT I'll give you two ways to look at this problem. You have two rates given. The rate of the Edsel is 70 and the rate of the Studebaker is 57. So the distance traveled by the Edsel in time T is 70T and the distance traveled by the Studebaker in time T is 57T. Now the question is, when do they meet? The other piece of information is that they started 635 kilometers apart. So they're going to meet when the distance traveled by the Edsel and the distance traveled by the Studebaker adds up to 635. This gives us an equation which you can solve for T: 70T + 57T = 635 Another way of looking at it is to just look at the distance between the two cars. It's decreasing because the cars are driving towards each other. How fast is it decreasing? The rate of decrease is the sum of the speeds of the two cars. So we get the following equation for the time it takes to reduce the distance from 635 to 0: (57 + 70)T = 635 -Doctor Wilkinson, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 01/23/97 at 20:43:39 From: Doctor Wallace Subject: Re: Math Hi Jennifer! I bet you're surprised to get two answers to this one. I originally composed this one, but it got munched by my computer, and before I could send you a new one, my colleague Dr. Wilkinson sent you his thoughts on the problem. I hope my original answer helps you out too. Here goes: The first thing I do when faced with any math problem is to draw a diagram. Here is the diagram I drew for this problem: E -> 70km/h 57km/h <- S ---------------------------------------------------------------------- 635 km The E stands for Edsel, and the S for Studebaker. I listed the speeds of the cars, and their direction of travel. The line represents their path, which is 635 km long We need to understand exactly what is happening in this problem. Pretend we have a stopwatch which reads 0. When we press the button to start the stopwatch, the cars will begin moving toward each other. At some point, they have to meet, and we will then press the button to stop the stopwatch. What we want to find in this problem is what time the stopwatch reads when we stop it. In other words, how long does it take for the two cars to meet? A helpful relation for any problem of this type (where you have something like a car moving at a constant speed) is: D = R x T Distance = Rate x Time For example, if a car is traveling at 60 km per hour for 2 hours, it will have traveled 60 x 2 or 120 km at the end of the 2 hours. (The rate is 60 and the time is 2.) Now how do we solve your problem? Well, there are two ways to go about it. One way uses algebra, and the other doesn't. Since I don't know how old you are, or what grade you're in, or whether you know algebra, I'll show you both ways. If you're not in algebra, you can save the other way until you get there. Way No. 1: Make a table (without algebra) Since your problem seems to hint that the answer will be a nice, round number of hours, a table seems like a good way to solve the problem. It also helps in understanding. We'll make a table of the distance values of each car after various hours of time. We'll start with time zero as our first entry. At time zero, the cars haven't gone anywhere, so their distance is, of course, zero. Notice that, throughout the whole table, the rate of each car stays the same. This is because the cars do not change speed during the trip. Here are the first three entries for our table: Edsel Studebaker Time (hours) Rate Distance Rate Distance 0 70 0 57 0 1 70 70 57 57 2 70 140 57 114 Do you see how we get each entry in the table? We just multiply the rate by the time. Now, the big question: How do we know when the two cars meet? Well, let's take a look at our diagram again. (You can plot the cars' progress on the line if it helps you.) Pick a place on the line where you think the cars will meet. It doesn't matter where. Call it point A. Now, the Edsel will have traveled from the left to point A, and the Studebaker from the right to point A. Notice that wherever you put point A, if we add up the distance that both cars have traveled, we always get 635 km. This is very important. When the cars finally do meet, together they must have traveled the length of the whole path. Individually, each car will only have traveled part of the whole path. But if we add them up, it must equal the whole 635 km. Okay? Now why is this important? Well, it means that all we have to do to find out whether or not the cars have met is to add up their distances in our table. Have they met after 1 hour? Well, after 1 hour, the Edsel has gone 70 km and the Studebaker 57 km. 70 + 57 = 127 km, so they have not met after 1 hour. How about after 2 hours? Well, after 2 hours, the Edsel has gone 140 km and the Studebaker 114 km. 140 + 114 = 254 km, so they have not met after 2 hours either. You can finish the table and check each hour and you'll easily find out when the two cars meet. Way No. 2: Algebra To solve any algebra word problem, you need three things. First, you have to understand the relations between the elements in the problem. Second, you have to be able to translate those relations into equations using symbols. Third, you need to have the skills to solve those equations. Let's look at how these apply to this problem. 1. Relations We've already seen the two important relations for this problem. The first is that Distance = Rate times Time. The second is that the total distance traveled by both cars together once they meet is 635km. 2. Equations This part is harder. We have to choose some variables. First, we'll choose one for the quantity we're looking for. This is the time at which the two cars meet. We'll call this t. What are some other things we don't know? Well, when the two cars meet, we don't know how far each has traveled individually. So we'll call the total distance the Edsel travels x, and the total distance the Studebaker travels y. So now we can use our relation D = R x T and write: x = 70t and y = 57t Why is this true? Well, after t hours, each car will have traveled a distance equal to its rate times the time spent traveling. Now, we have 3 variables here, and we can't solve our equations until we only have 1 variable. So we have to rewrite our equations in terms of only one variable, the one we want to solve for; in this case, t, the time. We do that by using our other relation. Our other relation is that the total distance traveled by both cars is 635 km. Because x and y are the total distances traveled by each car, the total distance traveled by both cars expressed in terms of our variables is: x + y = 635 Now notice that we already have two expressions for x and y. In other words, we already know what x and y are. As above, x is 70t and y is 57t. So we can substitute these into x + y = 635 and we get: 70t + 57t = 635 Now we have an equation in one variable! 3. Solve Now all we have to do is solve this equation and we'll get t, which will be the time that passed before both cars to met. I'll leave that to you. It works out to the same answer as using the table. After you become skilled at algebra, you'll want to use this method instead of constructing a table. This is because your table entries have to get more and more "fine-tuned" if the answer isn't a round number. The cars could meet after, say 2.776 hours. It would take a lot longer to "guess and check" using table entries to get that precise. The algebra method would give you this precision with less work. Well, that was probably a longer answer than you were expecting, but I hope it helps you get an understanding of this and problems like it. If you have any more questions, or need further help, please feel free to write back! -Doctor Wallace, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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