Question

Find height using right triangles?

From a point A, the angle of elevation of a mountain is 13.7 degrees, and from a point B directly behind point A, the angle of elevation is 10.4 degrees. If the distance between A and B is 5 miles, what is the height of the mountain to the nearest hundredth of a mile?

Answer 1

`color(maroon)("Height of the mountain " = bar(CD) = 3.7138 " miles"`

Explanation:

CD is the mountain.

`hat (DAC) = 13.7^@, hat (DBC) = 10.4^@, AB = 5 " miles"`

To find the height of mountain CD.

`tan (13.7) = (CD) / (AC) " or " AC = (CD) / tan (13.7)`

`tan (10.4) = (CD) / (BC) = (CD) / (AB + AC) = (CD) / (5 + AC)`

Substituting value of AC in terms of CD,

`tan (10.4) = (CD) / (5 + (CD) / tan 13.7)`

`(CD) / (CD + 5 tan 13.7) = tan 10.4 / tan 13.7`

`(CD + 5 tan 13.7) / (CD) = tan 13.7 / tan 10.4`

`1 + (5 tan 13.7) / (CD) = tan 13.7 / tan 10.4 = 1.3282`

`(5 tan 13.7) / (CD) = 1.3282 - 1 = 0.3282`

`CD = (5 tan 13.7) / 0.3282 = 3.7138 " miles"`