anonymous
  • anonymous
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.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
basic tip: the angle of depression is the angle from the horizontal looking down at an object
mathstudent55
  • mathstudent55
|dw:1437107516201:dw|
anonymous
  • anonymous
why are they right angles

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More answers

mathstudent55
  • mathstudent55
|dw:1437107592620:dw|
mathstudent55
  • mathstudent55
Because the walls of the canyon are perfectly parallel and the same height.
anonymous
  • anonymous
ohhhhhh ok
mathstudent55
  • 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
  • mathstudent55
Ok.You understand the drawing?
anonymous
  • anonymous
i get it now, i was confused on that part
mathstudent55
  • mathstudent55
Ok. It means we have a rectangle. We are looking for x. |dw:1437107821415:dw|
mathstudent55
  • mathstudent55
We can use trig of right triangles to find x.
mathstudent55
  • mathstudent55
|dw:1437107892862:dw|
mathstudent55
  • 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
  • mathstudent55
Now we have a simple right triangle trig problem.
anonymous
  • anonymous
ok
mathstudent55
  • mathstudent55
We are looking for x. |dw:1437108150647:dw|
mathstudent55
  • mathstudent55
Do you remember the sine, cosine, and tangent ratios?
anonymous
  • anonymous
yeah
mathstudent55
  • mathstudent55
Do you use SOHCAHTOA to remember it?
anonymous
  • anonymous
yes
mathstudent55
  • 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
  • anonymous
adjacent
anonymous
  • anonymous
and 400 is the opposite
anonymous
  • anonymous
so we use tangent
mathstudent55
  • mathstudent55
Great. That makes the 400 m side the opposite leg since the hypotenuse is unlabeled and we are not interested in it.
mathstudent55
  • mathstudent55
Yes, we use the tangent.
anonymous
  • anonymous
so basically I got 101.958
anonymous
  • anonymous
so 102f
anonymous
  • anonymous
102m
mathstudent55
  • mathstudent55
I don't get that.
anonymous
  • anonymous
what did you get
mathstudent55
  • mathstudent55
Let me show you how just by logic and thinking you can tell that answer cannot be correct.
mathstudent55
  • mathstudent55
You remember the good old 45-45-90 triangle?
anonymous
  • anonymous
OHHHHHHHH
anonymous
  • anonymous
RIGHT
anonymous
  • anonymous
|dw:1437108626021:dw|
mathstudent55
  • 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
  • mathstudent55
|dw:1437108699037:dw|
anonymous
  • anonymous
so basically the distance to teh other side is 400 meters
mathstudent55
  • 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
  • mathstudent55
In the 45-45-90 case, the two legs would be 400 m. We don't have a 45-45-90 case.
anonymous
  • anonymous
ok
mathstudent55
  • 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
  • anonymous
ok
anonymous
  • anonymous
if possible could you draw your steps
mathstudent55
  • mathstudent55
|dw:1437108865822:dw|
mathstudent55
  • 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
  • mathstudent55
Since 75.7 is much larger than 45, x is much larger than 400 m.
anonymous
  • anonymous
ok
mathstudent55
  • 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
  • mathstudent55
All you need to do is calculate x.
anonymous
  • anonymous
1569.3
mathstudent55
  • mathstudent55
You got it!
mathstudent55
  • mathstudent55
Don't forget the units.
anonymous
  • anonymous
THNAK YOU SO MUCH

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