Question

an airplane is flying at an altitude of 6000 m. the angle of depression from the airplane to the airport is 3.7°. determine the direct distance from the airplane to the airport to the nearest kilometre.
91 km
92 km
93 km
94 km

Explanation:

Step1: Identify the trigonometric relationship

We know that the altitude of the airplane is the opposite - side of the right - triangle formed and we want to find the hypotenuse. The angle of depression is equal to the angle of elevation from the airport to the airplane. Using the sine function $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 3.7^{\circ}$ and the opposite side $y = 6000$ m. Let the direct distance (hypotenuse) be $d$. So, $\sin(3.7^{\circ})=\frac{6000}{d}$.

Step2: Solve for $d$

We can re - arrange the formula $\sin(3.7^{\circ})=\frac{6000}{d}$ to get $d=\frac{6000}{\sin(3.7^{\circ})}$. We know that $\sin(3.7^{\circ})\approx0.0645$. Then $d=\frac{6000}{0.0645}\approx93023$ m.

Step3: Convert to kilometers

Since $1$ km = $1000$ m, then $d=\frac{93023}{1000}=93.023$ km. Rounding to the nearest kilometer, $d\approx93$ km.

Answer:

93 km