At a point on a level ground, the angle of elevationof a mountain top B is 42deg36min. At a pointD350 meters nearer the mountain, the angle of elevation is 51deg 43 min. Find the height of the mountain?
just set it up. If D is x from the base of the mountain of height h, then
h/x = tan 51°43'
h/(x+350) = tan 42°36'
solving for x, we get
h/tan51°43' = h/tan42°36' - 350
Now just solve for h.
I get h=1173
To find the height of the mountain, we can use the concept of trigonometry and set up two right triangles. Let's label the points as follows:
A: The observer standing on the level ground
B: The mountain top
D: The position 350 meters closer to the mountain
First, let's consider triangle ABD, where BD represents the height of the mountain.
In triangle ABD, we have two given angles:
Angle ABD = 42°36' (angle of elevation from point A)
Angle BAD = 90° (as it is a right angle)
We can now find the value of angle ADB using the fact that the sum of all angles in a triangle is equal to 180 degrees.
Angle ADB = 180° - Angle ABD - Angle BAD
Angle ADB = 180° - 42°36' - 90°
To compute this value, we need to convert 42°36' to decimal degrees:
42° + (36' / 60') = 42 + 0.6 = 42.6 degrees
Therefore,
Angle ADB = 180° - 42.6° - 90°
Angle ADB = 47.4°
Now, let's consider triangle ACD, where CD represents the horizontal distance between points A and D.
In triangle ACD, we have the given angle:
Angle ACD = 51°43' (angle of elevation from point D)
We know that the angle ACD is vertical to triangle ABD, which means it is equal to angle ADB.
Angle ADB = Angle ACD = 51°43'
With this information, we can now find the horizontal distance, CD.
Using the trigonometric tangent function, we have the following relationship:
tan(ADB) = BD / CD
We can plug in the known values:
tan(47.4°) = BD / CD
To isolate BD, we rearrange the equation:
BD = tan(47.4°) * CD
Next, let's find the length of CD. We are given that point D is 350 meters closer to the mountain top B than point A.
So, CD = AC - AD = 350 meters
Now, we can substitute this value into the equation for BD:
BD = tan(47.4°) * 350
Using a calculator, find the value of tan(47.4°), then multiply it by 350 to get the value of BD.
Finally, the height of the mountain, BD, is the length we calculated in meters.