Sightings from a helicopter A helicopter hovers at an altitu

Sightings from a helicopter. A helicopter hovers at an altitude of 1000 feet above a short mountain peak which has an altitude of 6580 ft, (refer to figure below). A second, taller peak is viewed from both the shorter mountaintop and from the helicopter . From the helicopter, the angle of depression to the taller peak is 5 degrees and the angle from the taller peak to the shorter peak is 42.5 degrees . If standing on the shorter mountain peak, the angle of elevation to the taller peak is 25 degrees .Remember to set up an equation to solve then in a series of clear steps find a solution.

a) Approximate the distance from peak to peak.

b) Approximate the distance the helicopter is from the taller peak( remember to set up an equation to solve then in a series of clear steps find a solution).

Solution

Here if we draw a perpendicular from smaller peak to upper dash line, it forms a right triangle, so that its height is 1000 ft and its base angle is (42.5 + 5 = 47.5)

So in such triangle, sin 47.5 = Height/Hypotenues

or 0.7372=1000/Hy.

or Hy.= 1000/0.7372 = 1356.43 ft.

so slant side adjcent to 42.5 degree is 1356.43. Further we also have that the left most base angle on down dash line is also equal to 47.5 degree being alternate interior angles. Thus the angle inside triangle at small peak vertex is

180- (47.5 + 25) = 180 - 72.5 = 107.5 degree.

so third angle of this triangle forming at largest peak is 180 - (42.5 + 107.5 ) = 180 - 150 = 30

Now on applying sine rule in such triangle, we have

sin A/a= sin B/b

or sin 42.5/a = sin 30 /1356.43

or 0.6756 /a=(1/2)/1356.43 = 1/2712.86

or on cross product, we get

a= 2712.86( 0.6756) = 1832.78

So surely distance between peak to peak is 1833 ft.

This is the answer of part (a).

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Now on using the formula

sin A =height/hypotenues

in rightmost triangle with base angle 25, we get

sin 25 = h/1833      (where h is the height of the largest peak from below dash line.

or 0.4266= h/1833

or h= 1833 (0.4233) = 775

So clearly the distance of hellicopter from the largest peck = 1000 - 775 = 225 feet

This is the answe of part (b) .

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