I am really struggling with this question :( Please help :)

Question

You are standing on a mountain that is 6842 feet high. You are looking down at your campsite, creating an angle of depression of 38°. If your eyes are 5.6 feet above the mountain how far is the base of the mountain from the campsite? (Round to the nearest hundredth)

Answer 1

This is a poorly worded question.

The angle of depression is the angle down from looking straight out from your vantage point. Someone at your camp looking up at you would talk about an angle of elevation. So given a rectangle, the line of sight would be its diagonal. (Draw yourself a picture. NOW.)

IF we assume that by "mountain" the writer meant cliff, then it would be more clear because a cliff stands at an angle near 90° (so its base is directly below you) whereas a mountain could really be any angle, which then would need to be stated.

I assumed a cliff, and added your height to the elevation. I then used SOHCAHTOA...

I worked with the lower triangle from the rectangle you should have drawn by now.

You are at the top. The right angle is at the base (of the cliff). The 3rd vertex (and the reference angle) is at your base camp.

BE SURE THAT YOUR CALCULATOR IS SET TO DEGREES NOT RADIANS OR GRADIENTS!!!

We have an angle, the side we have is opposite this angle. The question asks for the adjacent side's length.

The ratio that uses these 2 sides is the tangent ratio. So the equation we have is:

tan 38° = (6847.6 ft)/(A)

(A)/(tan 38°) • tan 38° = (6847.6 ft)/(A) • (A)/(tan 38°)

A = (6847.6 ft) / (tan 38°)

A = (6847.6 ft) / (.7813) ≈ 8764.528

So, if the "mountain" is actually a cliff, then given that height and angle, the base camp is 8764.53 ft from its base.

Answer 2

Hi Jessica. What you want to start by doing is drawing out this triangle if you don’t have it already. Angles of elevation and depression come from right triangles. So draw a right triangle. Since this is an angle of depression problem, draw the triangle below the horizon, basically with the legs drawn like 3 o’clock and 6 o’clock.

An angle of elevation or depression always goes along the horizontal, so write in that angle in the top corner of your triangle, next to the 3 o’clock leg. Next, write in the vertical distance, or height. It’ll be that 6842+the 5.6 for your eye level. Then, write in an x for what you’re solving for, the horizontal distance.

Now, this is just an easy trig set up. Relative to your angle of 38 degrees, you have your opposite and you’re solving for your adjacent side. Opposite and adjacent means tangent.

So we set it up as tan(38)=6847.6/x. Since x is in the denominator, we flip the trig function and that x, leaving us with x=6847.6/tan(38) which you can plug into your calculator to get 8764.5 feet