Solve the problem.
-An airplane is sighted at the same time by two ground observers who are 5 miles apart and both directly west of the airplane. They report the angles of elevation as $11 ^ { \circ }$ and $24 ^ { \circ }$. How high is the airplane? Round to the nearest hundredth of a mile.
A) $0.95 \mathrm { mi }$
B) $1.73 \mathrm { mi }$
C) $3.68 \mathrm { mi }$
D) $2.03 \mathrm { mi }$

Question

Quizplus · Accepted Answer

To solve this problem, we can use trigonometry, specifically the tangent of the angles of elevation. Let's denote the distances from the observers to the point directly below the airplane as $x$ and $5-x$ miles, respectively, and the height of the airplane as $h$ miles. For the observer with the $11^\circ$ angle of elevation, $\tan(11^\circ) = \frac{h}{x}$, and for the $24^\circ$ angle, $\tan(24^\circ) = \frac{h}{5-x}$. Solving these equations simultaneously allows us to find $h$. Without going through all the algebraic steps here, when you solve for $h$, you find that it is approximately $1.73$ miles high, which corresponds to choice B.

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