Pre-Calculus: Word Problems with Sine or Cosine

Very helpful!

06/01/2009

~ ht1979

This lesson really helped me understand how to practically apply the math to this particular real-life application. Very clear and very helpful - great!

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The Trigonometric Functions
Graphing Sine and Cosine Functions
Solving Word Problems Involving Sine or Cosine Functions Page [1 of 1]
Let’s take a look at an application where we can actually sort of see these ideas of periodicity and so forth, sort of in
action. Now, you know, you know what a tidal wave is, by the way? A tidal wave is what happens after an
earthquake. You know, you have an earthquake and of course, everything sort of gets shaken up a little bit, including
the water. And, of course you have this huge mass, you know, of water, so all of a sudden, it either drops or it goes
up and starts these huge waves, also known as tsunami waves. And it turns out that these waves are enormous. In
fact, here are some samples. Look at those things. Those are very threatening looking. And they start and they fall
and zoom, zoom, zoom. And now, okay, let’s suppose that we have one of these tsunami things going on here. And
suppose that the waves, each individual wave, is traveling at a speed of, let’s say, 540 feet per second., which is
about 370 mile per hour. So really fast. These things are just ripping through. Suppose that the height of the peak of
the wave is 50 feet. So, you can see -- if this is 50, you can see that that’s a huge fish. That’s like a whale, if that’s 50
feet, because that fish is you know, basically 50 feet high. Anyway, now the question is, if the period of the wave --
let’s say it’s 20 minutes. So, what that means is that it takes 20 minutes basically to go from sort of crest to crest. So,
for this wave to get to here will take 20 minutes.
The questions is, what is the length between each wave? So, what is this length right here? Well, I gave you a lot of
information, right? We know for example that the wave is traveling at a rate of 540 feet per second. I would tell you
that the height is 50, and I’d tell you that it takes 20 minutes to get from here to here. So, the question is, how long is
this? Well, there’s a lot of information there. We have to synthesize what’s going on. These waves are very much
like sound waves in that they’re sinusoidal. So they have this, sort of, sine function type aspect to them. But all I care
about here is finding the length of the period. So let’s think about how to find the length of the period. First of all, is
the amplitude, which is 50 here, at all relevant? The answer is no. The amplitude doesn’t play a role in this at all.
That’s just sort of a dead rat, which is just there to throw us off the scent of the real mission. The real mission is to try
to find this length. What do I know? I know that the waves are traveling at a rate of 540 feet per second and I know
that it takes each of these waves sort of 20 minutes to get to its next period. So, the period of the wave is 20 minutes.
Now, I’m going to have to probably get all this in the same units. Here we’re talking about feet per second. So, to get
this into seconds, I better take this and multiply it by 60 and that equals 1200 seconds. So that how long it takes from
here to here. And, I know how fast we’re traveling so the question is, what’s that distance? Well, it’s just distance =
rate x time. The rate I know is 540 and the time is 1200 seconds and so what’s this? Well, you just multiply these two
numbers together and we get 648,000 feet. And what is that in miles? Well, you’d have to divide this by 5,280 and
you’d get 122.7 miles. So that’s how far these are at part. These are really huge distances. Notice the key to doing
this so was just to realize I want a period length from this sort of sine like function type thing and to get the period
length, notice the amplitude played no rule in this. That was the complete red herring. And all I do is I have the speed
and I know how long it takes and so I can actually figure out the period length. This is sort of like the 2? over
absolute value of b, to find the period length. But, now it’s sort of encoded in this rate type thing, so I need to use the
fact that distance equals rate times time.
Anyway, you can see in the real world, these sort of sinusoidal type things do occur and figuring out these periods
turns out to be a valuable thing. But not very hard. I just wanted to show you a little thing about periods. Up next,
we’ll take a look at some more exotic trig functions. Like what does tangent look like anyway? I don’t know, we’ll see.