Suppose you are standing on the edge of a canyon that is 400 meters high, where two sides of the canyon are at the same height and the walls of the canyon are perfectly vertical. The angle of depression to the bottom of the other side of the canyon is 14.3 degrees. Find the distance to the other side of the canyon. Ignore the height in the computations.

- anonymous

- katieb

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- anonymous

basic tip: the angle of depression is the angle from the horizontal looking down at an object

- mathstudent55

|dw:1437107516201:dw|

- anonymous

why are they right angles

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## More answers

- mathstudent55

|dw:1437107592620:dw|

- mathstudent55

Because the walls of the canyon are perfectly parallel and the same height.

- anonymous

ohhhhhh ok

- mathstudent55

Since the walls are perfectly vertical, they are both perfectly perpendicular to the ground.
The ground is horizontal. Two lines perpendicular to the same line are parallel to each other.

- mathstudent55

Ok.You understand the drawing?

- anonymous

i get it now, i was confused on that part

- mathstudent55

Ok. It means we have a rectangle. We are looking for x.
|dw:1437107821415:dw|

- mathstudent55

We can use trig of right triangles to find x.

- mathstudent55

|dw:1437107892862:dw|

- mathstudent55

The horizontal line at the top of the canyon, and the horizontal line at the floor of the canyon are parallel lines. We already know this because they are opposite sides of a rectangle.
The line of sight is a transversal. This way we know the bottom right angle (marked above) measures 14.3 deg because the two 14.3 deg angles are alternate interior angles of parallel lines cut by a transversal.

- mathstudent55

Now we have a simple right triangle trig problem.

- anonymous

ok

- mathstudent55

We are looking for x.
|dw:1437108150647:dw|

- mathstudent55

Do you remember the sine, cosine, and tangent ratios?

- anonymous

yeah

- mathstudent55

Do you use SOHCAHTOA to remember it?

- anonymous

yes

- mathstudent55

Great.
Here we have the 14.3 angle on the bottom right.
For the 14.3 angle, is the x side the adjacent or the opposite leg?

- anonymous

adjacent

- anonymous

and 400 is the opposite

- anonymous

so we use tangent

- mathstudent55

Great.
That makes the 400 m side the opposite leg since the hypotenuse is unlabeled and we are not interested in it.

- mathstudent55

Yes, we use the tangent.

- anonymous

so basically I got 101.958

- anonymous

so 102f

- anonymous

102m

- mathstudent55

I don't get that.

- anonymous

what did you get

- mathstudent55

Let me show you how just by logic and thinking you can tell that answer cannot be correct.

- mathstudent55

You remember the good old 45-45-90 triangle?

- anonymous

OHHHHHHHH

- anonymous

RIGHT

- anonymous

|dw:1437108626021:dw|

- mathstudent55

If we had a 45-45-90 triangle, the legs would be congruent, so x would also be 400 m, right?
|dw:1437108626987:dw|

- mathstudent55

|dw:1437108699037:dw|

- anonymous

so basically the distance to teh other side is 400 meters

- mathstudent55

No. I'm just explaining why 102 m cannot be correct.
The answer has to be a number much larger than 400 m.

- mathstudent55

In the 45-45-90 case, the two legs would be 400 m.
We don't have a 45-45-90 case.

- anonymous

ok

- mathstudent55

Since our bottom right angle is only 14.3 deg, the upper left angle is its complelement, so it's 75.7 deg.

- anonymous

ok

- anonymous

if possible could you draw your steps

- mathstudent55

|dw:1437108865822:dw|

- mathstudent55

Wit a 45-45-90 triangle, x is the same as 400 m.
Now see how our case more or less looks like above.
As the top left angle gets larger and larger than 45 deg, the distance x also increases.

- mathstudent55

Since 75.7 is much larger than 45, x is much larger than 400 m.

- anonymous

ok

- mathstudent55

Your last equation is incorrect bec you have adj/opp. The tan ratio is opp/adj.
\(\tan \theta = \dfrac{opp}{adj} \)
\(\tan 14.3^o = \dfrac{400~m}{x} \)
\(x = \dfrac{400~m}{\tan 14.3^o}\)

- mathstudent55

All you need to do is calculate x.

- anonymous

1569.3

- mathstudent55

You got it!

- mathstudent55

Don't forget the units.

- anonymous

THNAK YOU SO MUCH

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