I have a problem here from my cousin.
Let's see what it says.
It says Milwaukee, Wisconsin is directly west of Grand
Haven, Michigan, on opposite sides of Lake Michigan.
So let me draw that.
So if we say that this right here is Milwaukee.
It's due west of Grand Haven, Michigan.
So if I were to draw-- let me draw a horizontal line.
We're going over water so let me do it in blue.
It's due west of Grand Haven, Michigan.
You know, you go straight east.
If I were to go straight east, I would get to
Grand Haven, Michigan.
Which is right here.
Let me just label that, g.
All right.
On a foggy night, a law enforcement boat leaves
from Milwaukee on a course of 105 degrees.
The hardest part of these problems, in my opinion, is
really just trying to figure out the convention
they're using.
When they say a course of 105 degrees, what does that mean?
What direction is it?
And I checked with my cousin, and her book says that in, I
guess in the boating world, the course is how many
degrees clockwise you're going, of due north.
So due north is 0 degrees.
So 105 degrees.
So due north would be 0 degrees.
Due north would be-- that's 0 degrees.
So he's going 105 degrees clockwise of that.
So if we 105 degrees-- so he's going 105 degrees.
So that's like 105 degrees.
Something like that.
He's going 105 degrees.
And I'll do him in magenta.
His course is 105 degrees.
So that's 105 degrees clockwise of due north.
And what is that-- oh, it's like my screen just backed up.
So what is that angle, in kind of what we're familiar with?
Well, this would be 90 degrees right here.
Going here.
And then he goes 15 more degrees.
So in kind of unit circle terms, this would be
negative 15 degrees.
Or if we wanted to figure out the angle of this vertex right
here, that would be 90, and then we'd go another 15.
So this angle right here is going to be what?
It's going to be 15 degrees.
Because he said his course is 105.
So 90 plus 15 is 105.
So how much south he's going of, kind of, straight
west-east, is 15 degrees.
This whole thing is 105.
OK, let me keep reading the problem.
He leaves from Milwaukee at a course of 105 degrees at the
same time that a small smuggling craft steers a
course of 195 degrees from Grand Haven.
195 degrees.
So once again, due north is 0 degrees.
And this guy's going 195.
So we figure-- this is just a convention.
You figure out-- well, he's going 195 degrees
clockwise of due north.
So 195 degrees is going to be-- let's see, it's going to be
a 180 degrees and then some.
It's going to be like that.
So his course is going to look something like this.
His course is going to look something like that.
And let's see if we can figure out what this
angle right here is.
Because we, as you can kind of see where this is going, we're
trying to figure out probably the size of this triangle,
if I had to guess.
I haven't even read the whole problem yet.
Let's see.
So he's going 195 degrees.
So if we were to drop, well, like here.
This right here is a 180 degrees, to go clockwise from
straight up to straight down.
This is 180.
And so he went 195.
So this is going to be-- that's going to be 15 degrees.
And if this angle is 15 degrees, what is this
angle going to be?
Well, this entire angle is 90 degrees, right?
It's kind of the third quadrant when we're thinking
in unit circle terms.
So this angle right here is going to be 90
minus this 15 degrees.
So what's 90 minus 15?
It's 75, right?
75 degrees.
And if we wanted to convert his, kind of, course angles
into unit circles-- you know, with unit circles, you start
here and you go all the way around this way.
So I think you would get something like 255 degrees.
But anyway.
So we figured out that this angle is 15 degrees, this
angle is 75 degrees.
What's this angle going to be?
This angle is going to be-- these all have to add
up to 180, right?
So this is going to be 180 minus 15 minus 75.
And what's that?
That's 180 minus 90.
So 180 minus 90 is 90 degrees!
So this angle here is 90 degrees.
It's a right angle.
Interesting.
OK, so what do they tell us?
They tell us the law enforcement boat
averages 23 knots.
So he's traveling in this direction at 23 knots.
All right, that's a little bit faster than 23 miles per hour.
And collides with the smuggling craft.
What was the smuggling boat's average speed?
So they both leave their respective sites at the
same time, and they both collide, right?
So the time they traveled is the same.
Let's call that time, t.
Right?
I don't know.
They both left at the same time and it took some
time for them to collide.
So let's say that the time between when they left and
the time they collided is t.
So how far did the patrol boat travel?
Well, he traveled at a speed of 23 knots, and it took him time,
t, to get to the collision.
So the distance he traveled is 23t.
Speed times time is equal to distance.
So the length of this side is 23t.
Similarly, this guy, we don't know his speed.
Let's call it, I don't know.
Let's call it x.
His speed is x.
But the distance he travels is x times t.
x times t, right?
So that's the length of this side.
So let's see if we can figure out what x is.
So what do we know?
We know a lot about this.
We know this is a right triangle, et cetera.
We know this angle.
So if we wanted to solve for xt and use this 23t
information, let's see.
We know-- look at this angle.
If we use the 75 degrees, we know the opposite angle.
The opposite side, sorry.
Which is 23t.
And we know the adjacent side, which is xt.
So let me write SOHCAHTOA here.
So what deals with opposite and adjacent?
Well, that's tangent, right?
TOA.
So if we say the tan of 75 degrees is going to be equal to
the opposite side-- 23t-- over the adjacent side-- that's this
side, opposite over adjacent-- xt.
Well, the t's cancel out, right?
The t's cancel out.
And let's see if we can solve for x.
Multiply x times both sides.
You get x tangent of 75 is equal to 23.
And then divide both sides by the tan of 75 and you get x is
equal to 23 divided by the tangent of 75 degrees.
And so that's our answer.
And if I had to-- well, actually, I have-- let's see.
Tangent of 75 degrees.
I don't have a calculator in front of me.
You could calculate it.
It's actually going to be a pretty high number.
So you could try to fit, you know.
If you have a calculator, just type in 75 degrees.
Take the tangent of it, and perform this calculation.
But we've essentially solved this problem.
I'll see you in the next video.