Question

note: the picture is not drawn to scale. ancient alien theorists manning two radar stations located b = 43 km apart both detect a ufo located between them. the angle of elevation measured by the first station (a) is α = 87.2° and the angle of elevation measured by the second station (c) is γ = 84.1°. how far is the ufo from station a? distance from a = km how high is the ufo above the ground? h = km enter your answer as a number; your answer should be accurate to 2 decimal places. question help: video message instructor

Explanation:

Step1: Find angle $\beta$

In a triangle, the sum of angles is $180^{\circ}$. So, $\beta=180^{\circ}-\alpha - \gamma$.
Substituting $\alpha = 87.2^{\circ}$ and $\gamma=84.1^{\circ}$, we get $\beta=180^{\circ}-87.2^{\circ}-84.1^{\circ}=8.7^{\circ}$.

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

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 $b = 43$ km, $\gamma = 84.1^{\circ}$, and $\beta=8.7^{\circ}$, we have $c=\frac{43\times\sin(84.1^{\circ})}{\sin(8.7^{\circ})}$.
Since $\sin(84.1^{\circ})\approx0.9945$ and $\sin(8.7^{\circ})\approx0.1514$, then $c=\frac{43\times0.9945}{0.1514}\approx281.87$ km.

Step3: Find height $h$

We know that $\sin\alpha=\frac{h}{c}$.
So, $h = c\sin\alpha$.
Substituting $c\approx281.87$ km and $\alpha = 87.2^{\circ}$ (and $\sin(87.2^{\circ})\approx0.9988$), we get $h=281.87\times0.9988\approx281.33$ km.

Answer:

distance from \(A = 281.87\) km
\(h = 281.33\) km