A helicopter hovers at an altitude that is 2000 feet above a mountain peak of altitude 5280 feet. A second taller peak, is viewed from both the mountain top and the helicopter. From the helicopter, the angle of depression is 43° 55'. From the mountain top, the angle of elevation is 18° 40'. Find:
a) The distance from peak to peak.
b) Altitude of the peak of the taller mountain.
If the distance is x and the other peak has height h, note that
(7280-h)/x = tan43°55'
(h-5280)/x = tan18°40'
Now you can easily find x and h.
To find the distance from peak to peak, we can use the concept of trigonometry. Let's denote the altitude of the taller mountain as 'x' feet.
a) The distance from peak to peak:
From the helicopter, the angle of depression is given as 43° 55'. This means that the line of sight from the helicopter to the peak of the taller mountain is 43° 55' below the horizontal line.
From trigonometry, we know that the tangent of an angle is given by the ratio of the length of the opposite side to the length of the adjacent side. In this case, if we consider the horizontal line as the adjacent side and the distance from the helicopter to the taller peak as the opposite side, the tangent of the angle of depression can be used to find the value of the distance.
We can use the formula:
tan(angle of depression) = (altitude of the helicopter - altitude of the mountain top)/distance between the helicopter and the mountain top
Let's plug in the values:
tan(43° 55') = (2000 feet - 5280 feet)/distance
Using a calculator or trigonometry tables, we can find that tan(43° 55') is approximately 1.02.
1.02 = -3280/distance
Cross-multiplying, we get:
distance ≈ -3280/1.02
distance ≈ -3215.69 feet
However, distance cannot be negative. So we take the absolute value:
distance ≈ 3215.69 feet
Therefore, the distance from the peak to peak is approximately 3215.69 feet.
b) The altitude of the peak of the taller mountain:
From the mountain top, the angle of elevation is given as 18° 40'. This means that the line of sight from the mountain top to the peak of the taller mountain is 18° 40' above the horizontal line.
Using the same trigonometric concept, we can use the tangent of the angle of elevation to find the altitude of the peak of the taller mountain.
tan(angle of elevation) = altitude of the taller mountain/distance between the mountain top and the taller peak
Let's plug in the values:
tan(18° 40') = x/distance
Again, using a calculator or trigonometry tables, we find that tan(18° 40') is approximately 0.3306.
0.3306 = x/3215.69
x ≈ 1063.23 feet
Therefore, the altitude of the peak of the taller mountain is approximately 1063.23 feet.