Word Problem?

Question

A surveyor has been hired by a resort developer to determine if a mountain is high enough to develop with a ski lift. The surveyor approaches the problem as follows:

In the valley below he sets up a transit at point K and sights an angle of elevation to a prominent rock R at the top to be 48degrees. He moves the transit to a second point M and measures the angle of elevation to R to be 45.3 degrees.

A. If distance KM is 100m across level ground, and the line MK when extended meets a perpendicular from R, determine the height of the mountain.
B. Find the length of the ski run from R to K.

I have no idea how to get this..could you please help me understand the problem?

Answer 1

I can't attach the diagram here, but I can show you the formulae:

Height of mountain = h

x = horizontal component of distance between R & K

Therefore,

h / x = tan (48)

x / h = cot (48) ...........(eq.1)

Also,

h / (x + 100) = tan (45.3)

(x + 100) / h = cot (45.3)..........(eq.2)

Subtracting eq.1 from eq.2, we get:

100 / h = cot (45.3) - cot (48)

100 / h = 0.98958 - 0.90040 = 0.08918

h / 100 = 1 / 0.08918

h = 100 / 0.08918 = 1121.33 metres

Height of mountain is roughly 1121 metres

Answer 2

The real trick here is to understand how to draw out what they're saying in words. I'll try to "paint" a picture with words:

Draw a line, from left to right, beginning with M, then move through K, then stop at a point (which will be directly below R). Next, draw a line starting at the end of the line you just drew, going straight up to a point labeled R. Then, draw a line connecting R and M.

You should end up with a right triangle, where the "height" can be labeled 'y' and the base is labeled 100 +x (because the problem says that K to M is 100 m, but then extended to the point just below R, which I'm calling a distance of 'x').

Now, the angle between M and R is 45.3 degrees. The angle between K and R is 48 degrees.

Remember from trig that tangent = (opposite)/(adjacent)

Therefore, you can construct two equations: tan(48) = y/(100=x) and tan(45.3) = y/x

You now have enough information to solve for the height of the mountain as well as the distance between M and the point below R (the 100 + x part).

For part B, just use Pythagorean Theorem RK^2 = y^2 +(100+x)^2

Answer 3

I won't give you the answer, but I will tell you how to set it up.

First you need to visualize the problem by drawing a diagram. Draw a right triangle.
........R
......../|
......./.|
....../..|
...../...|
..../....|
K/__-|

The angle at K is 48 degrees. By the SOH-CAH-TOA, tan(k) = opposite / adjacent.

Now you move the transit 100m farther to a point M. Now you have tan(m) = opposite / (adjacent + 100). Again you have a known value of m = 45.3 degrees.

You have two equations, and two unknowns so you should be able to solve this. The value of 'opposite' is the height of the mountain.

For part B, you need to switch to SOH. sin(k) = opposite / hypotenuse. Opposite is the height you just calculated. Hypotenuse is the length of the ski run from R to K (assuming a straight line). Alternatively, just use the Pythagorean theorem with the opposite and adjacent + 100 lengths to get the hypotenuse.