Question

score: 1/8 answered: 1/6 question 2 note: the picture is not drawn to scale. ancient alien theorists manning two radar stations located b = 61 mi apart both detect a ufo located between them. the angle of elevation measured by the first station (a) is α = 87.3° and the angle of elevation measured by the second station (c) is γ = 83.5°. how far is the ufo from station a? distance from a = mi how high is the ufo above the ground? h = mi enter your answer as a number; your answer should be accurate to 2 decimal places. question help: video message instructor submit question

Explanation:

Step1: Find angle \( \beta \)

The sum of angles in a triangle is \( 180^{\circ} \). So, \( \beta=180^{\circ}-\alpha - \gamma\).
Substituting \( \alpha = 87.3^{\circ}\) and \( \gamma = 83.5^{\circ}\), we get \( \beta=180^{\circ}-87.3^{\circ}-83.5^{\circ}=9.2^{\circ}\).

Step2: Use the Law of Sines to find \( c \) (distance from \( A \))

The Law of Sines states that \( \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}\).
We want to find \( c \), so \( c=\frac{b\sin\gamma}{\sin\beta}\).
Substituting \( b = 61\) mi, \( \gamma = 83.5^{\circ}\), and \( \beta=9.2^{\circ}\), we have \( c=\frac{61\times\sin(83.5^{\circ})}{\sin(9.2^{\circ})}\).
Since \( \sin(83.5^{\circ})\approx0.9925\) and \( \sin(9.2^{\circ})\approx0.1606\), then \( c=\frac{61\times0.9925}{0.1606}\approx375.24\) mi.

Step3: Find \( h \) (height)

We know that \( \sin\alpha=\frac{h}{c}\).
So, \( h = c\sin\alpha\).
Substituting \( c\approx375.24\) mi and \( \alpha = 87.3^{\circ}\) (\( \sin(87.3^{\circ})\approx0.9989\)), we get \( h=375.24\times0.9989\approx374.73\) mi.

Answer:

distance from \( A = 375.24\) mi
\( h = 374.73\) mi