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Here's another interesting word problem that was

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sent by [? Cortagio ?].

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A woman cycles to work alongside a railroad track

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at 6 kilometers per hour.

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That's an interesting piece of data.

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6 kilometers per hour.

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Every day she arrives at a crossing the same

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time that a train does.

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OK.

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One day she was 50 minutes late, and was overtaken by

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the train 6 kilometers from the crossing.

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In how many minutes will the train reach the crossing.

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Let's see if we can draw a diagram here that can somewhat

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describe what's going on in this problem.

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So what's on a normal day?

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On a normal day, she leaves from home, she leaves from

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home, and she goes and she travels alongside

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a railroad track.

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Not too different than my own commute on my own bicycle.

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Actually, I actually do travel on a bike path that's right

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along a railroad track.

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Anyway, let's say she leaves here at time is equal to 0.

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Time is equal to 0.

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And right at this point right here, maybe that's where the

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railroad crossing is, right there, the train passes her up.

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And she's traveling at 6 kilometers per hour.

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6 kilometers per hour.

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6 kilometers per hour.

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And what else does it tell us?

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It doesn't tell us how long it takes her, or

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how far she travels.

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So let's just say she arrives here at t is equal to a hours.

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t is equal to a.

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We'll do time in hours, since we're in kilometers per hour.

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So this path takes her a hours to travel.

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So it takes her a hours.

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If we care about the time, a hours.

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Now on this next day, she's 50 minutes late.

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So on this day, she has to travel the same distance in

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time, well in both distance and time.

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But this is this day.

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She leaves not a t equals 0.

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She leaves 50 minutes late.

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Since we're dealing with hours, let's write

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50 minutes in hours.

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So she leaves at-- if this was time equal 0, that's when she

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normally leaves --she's now leaving at that time is equal

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to 50 minutes, or time is equal to 5/6 hours.

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Right?

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50 minutes is just 50/60 hours or 5/6 hours of time is

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equal to 5 over 6 hours.

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This time is equal to 0 hours.

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And let's see, it says that she was overtaken by the train 6

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kilometers from the crossing.

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So she was overtaken by the train some place over

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here, and this distance.

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Right this is where the crossing is, that's

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that crossing.

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This is 6 kilometers.

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6 kilometers.

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In how many minutes will the train reach the crossing?

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So if we can figure out some way of expressing this time, if

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we can figure out with this time is, we know that the train

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reaches the crossing a t is equal to a.

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So if we know this time, we can just find the difference

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between these two times and we would have solved the problem.

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So let's see if we can figure out this time.

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So we know that it normally takes her a hours to travel

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this whole distance.

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It doesn't matter when she leaves it takes her a hours.

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So in this case it should also take her a hours.

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And how long does she have to go when the train passes up on

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this day that she was late?

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How long will it take her to go this distance right there?

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Well she's traveling at 6 kilometers per hour, she

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has to go 6 kilometers.

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So in time, this is going to take her 1 hour.

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To get-- you know, the train will have long passed her --but

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to get back to the crossing, she's going to have to

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travel for another hour.

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So if her whole route takes her a hours, this whole length

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in time is a hours.

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Then this distance right here is going to have

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to take her, what?

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She has an hour left, her whole trip is going to be a hours, so

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this is going to be a minus 1 hours.

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Let me scroll down a little bit.

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This is going to take her a minus 1 hours.

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So if on the day she was late it took her a minus 1 hours to

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get passed by the train, what time are we talking

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about right here?

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What time is this?

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This is t is equal to-- well we started at 5/6

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--t is equal to 5/6.

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That's where we started.

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Plus the amount of hours she traveled plus a minus 1.

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So there we, I think, we now have all the information we

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need to solve this problem.

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On the day she is late, the train passes or at time is

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equal to 5/6 plus a minus 1 hours, where a is the amount of

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time it normally takes her to travel her entire route

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to the crossing.

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We know that the train passes this point at t is equal to a.

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So if we want to know how many minutes does it take the train

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to go from this point to this point, we just have to

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subtract the two times.

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So the minutes it takes, well the hours because

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we're doing it hours.

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And remember I said 50 minutes was 5/6 of an hour, so the

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minutes it takes is a, that a, minus this.

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Minus 5/6 plus a minus 1.

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Right?

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That's just the difference in time for the train, in hours.

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So this is a minus 5/6 minus a plus 1.

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Luckily the a's cancel out.

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a minus a, and you're left with 1 minus 5/6 hours, which is

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just equal to 1/6 hours.

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There are 60 minutes in an hour, so 1/6 of

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that is 10 minutes.

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So it takes the train 10 minutes to get to the

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crossing from the point that it passed her.

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Another neat problem, and I thank [? Cortagio ?].

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for that.

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And then you can actually use this information if you want

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figure out how fast the train is going, or you could even

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think of other types of derivative problems

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off of this one.

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Anyway, another good problem.

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