Angles of elevation and Trigonometry

Question

Mount Mayon is an active volcano on the island of Luzon in the Philippines. At an unknown distance from the volcano, the angle of elevation to the peak of the volcano is approximately 26.22◦. Exactly 1000 metres closer to the volcano, the angle of elevation to the volcano’s peak increases to approximately 31.62◦. Use the given information to estimate the height of Mount Mayon.

Answer 1

Let X represent the unknown distance to the volcano at which the angle of elevation is 26.22 degrees.

The the point X-1000, the angle of elevation is 31.62.

Let H represent the height of the volcano.

 

Two right triangles are formed, one with base measure X, the other with base measure X-1000

 

H = X*tan26.22

 

tan 31.62 * (x-1000) = H

 

setting them equal and solving for X

 

X*tan26.22 = tan 31.62 * (x-1000)

 

tan 31.62 X - tan 26.22 X = 1000 tan 31.62

 

   X =      1000 tan 31.62

           --------------------------------   =  4980.29

                  tan31.62 - tan 26.22 

 

plugging this into the first equation gives the height of the volcano at 2452.76

 

The answer is verified as follows:

   1) use pythagorean theorem to find the hypotneuse: square-root( height^2 + x^2 ) = 5551.513

 

  2)  2452.76 / 5551.513 = .44181829

 

 3) the arcsin(0.44181829)  , inverse-sine, of .44181829 is 26.22