In traveling across flat land, you notice a mountain in front of you. Its angle of elevation to the peak is 3.5 degrees. After you drive 13 miles closer to the mountain, the angle of elevation is 9 degrees. Approximate the height of the mountain.
Let the second observation be taken at x miles from the mountain.
Then
xtan(9°) = (13+x)tan(3.5°)
Solve for x after substituting the trig. ratios.
The height of the mountain is therefore
xtan(9°) miles
=5280 x tan(9°) ft.
How do you figure it out from there though? Because I got that far and couldn't figure out how to solve for x.
It's like solving for x in
ax=b(13+x)
x(a-b)=13b
x=13b/(a-b)
where
a=tan(9°)=0.158 approx.
b=tan(3.5°)=0.061 approx.
To approximate the height of the mountain, we can use trigonometry.
Let's assume that the height of the mountain is 'h', and the distance you initially were from the mountain is 'x' miles.
The angle of elevation to the peak of the mountain can be represented as the inverse tangent (arctan) of the ratio of the height of the mountain to the initial distance from the mountain:
tan(3.5 degrees) = h / x
Similarly, after you drive 13 miles closer to the mountain, the angle of elevation becomes 9 degrees:
tan(9 degrees) = h / (x - 13)
Now we have a system of two equations. We can solve these equations simultaneously to find the height of the mountain 'h'.
Divide the second equation by the first equation:
[tan(9 degrees) / tan(3.5 degrees)] = [(h / (x - 13)) / (h / x)]
Simplify the equation:
[tan(9 degrees) / tan(3.5 degrees)] = [(x / (x - 13))]
Now, substitute the given values:
[tan(9 degrees) / tan(3.5 degrees)] = [(x / (x - 13))]
[tan(9 degrees) / tan(3.5 degrees)] = [(x / (x - 13))]
Approximate tan(9 degrees) ≈ 0.1584 and tan(3.5 degrees) ≈ 0.0610:
[0.1584 / 0.0610] = [(x / (x - 13))
Simplify the equation further:
2.6 = [x / (x - 13)]
Cross multiply the equation:
2.6 * (x - 13) = x
2.6x - 33.8 = x
2.6x - x = 33.8
1.6x = 33.8
Divide both sides of the equation by 1.6:
x = 33.8 / 1.6
x ≈ 21.125
Now that we have the value of 'x', we can substitute it back into any of the two original equations to find the height 'h'. Let's use the first equation:
tan(3.5 degrees) = h / x
tan(3.5 degrees) = h / 21.125
Approximate tan(3.5 degrees) ≈ 0.0610:
0.0610 = h / 21.125
Multiply both sides of the equation by 21.125:
0.0610 * 21.125 = h
h ≈ 1.3305
Therefore, the approximate height of the mountain is 1.3305 miles.