Question

A plane is flying at a constant altitude. The angle of depression from the center of the plane to the base of a control tower is 36°. After the plane flies 4 miles toward the control tower, the angle of depression is 43°. At what altitude is the plane flying?

*Round your answer to 2 decimal places*

Answer 1

13.16 miles

Explanation:

The attached diagram is a scale representation of the geometry, but shown upside down from the figure given in the problem statement. The tangent ratio is useful for solving this problem, as it relates the legs of the triangle to the angle of interest.

The basic relation is ...

Tan = Opposite/Adjacent

Applying this to the given angles, we have ...

tan(36°) = EG/AG . . . . . [eq1]

tan(43°) = EG/DG = EG/(AG -4) . . . . . [eq2]

Multiplying both equations by their denominators, we can eliminate the fractions:

AG·tan(36°) = EG . . . . . [eq3]

(AG -4)·tan(43°) = EG . . . . . [eq4]

Equating expressions for EG, we have ...

AG·tan(36°) = (AG -4)·tan(43°)

4·tan(43°) = AG(tan(43°) -tan(36°))

AG = 4·tan(43°)/(tan(43°) -tan(36°)) . . . . . [eq5]

Substituting into [eq3], we find ...

EG = 4·tan(36°)·tan(43°)/(tan(43°) -tan(36°))

EG ≈ 4(0.72644)(0.93252)/(0.93252 -0.72654) ≈ 13.1573 . . . . miles

The plane is flying at an altitude of about 13.16 miles.

_____

Additional comment

That's about 69,500 feet. The operational ceiling of most passenger aircraft is below 51,000 feet. The U2 reconnaissance aircraft reportedly has a service ceiling of about 70,000 feet.