From a point on the ground a person notices that an antenna that is d=109
d
=
109
feet tall on the top of a mountain subtends an angle of D=0.7∘
D
=
0.7
∘
. If the angle of elevation to the top of the mountain is A=35∘
A
=
35
∘
, find the height of the mountain h
h
.
Round to the nearest tenth of a foot.
I see you didn't learn anything about text formatting from your first post.
Again, a diagram should show you that
(h+109) cot35.7° = h cot35°
To find the height of the mountain, we need to use trigonometry and apply the concept of similar triangles.
First, draw a diagram to visualize the problem. Label the point on the ground where the person is standing as point O, the top of the antenna as point T, and the top of the mountain as point M.
Now, we have two right triangles: triangle OTM and triangle OMT, where angle OTM is equal to angle A and angle OMT is equal to angle D.
We know that the height of the antenna (d) is 109 feet. We need to find the height of the mountain (h).
Using trigonometry, we can write two equations:
1. tan(A) = OT / OM (equation 1) --> This relates angle A to the sides of triangle OTM.
2. tan(D) = MT / OM (equation 2) --> This relates angle D to the sides of triangle OMT.
Since tan(A) = OT / OM, we can rearrange equation 1 to solve for OT:
OT = tan(A) * OM
Similarly, using equation 2, we can rearrange to solve for MT:
MT = tan(D) * OM
Since OT + MT = d (the height of the antenna), we can substitute the values we have into this equation:
tan(A) * OM + tan(D) * OM = d
Simplifying further:
OM * (tan(A) + tan(D)) = d
Now, we can solve for OM (the height of the mountain):
OM = d / (tan(A) + tan(D))
Substituting the given values:
OM = 109 / (tan(35) + tan(0.7))
Using a calculator, evaluate the value of tan(35) and tan(0.7):
OM ≈ 109 / (0.7002 + 0.0123)
OM ≈ 109 / 0.7125
OM ≈ 152.955
So, the height of the mountain (h) is approximately 152.955 feet, rounded to the nearest tenth.
Therefore, the height of the mountain is h ≈ 152.9 feet.