ANSWER
EXPLANATION
Given that;
The speed of the pilot = 670 miles per hour
The first path took her 1 hour 30 minutes
The second path took her 2 hours
Included angle = 170 degrees
To find the distance from the starting position, follow the steps below
Step 1; Find the distance of the first and second path
Recall, that
[tex]\text{ Distance }=\text{ speed }\times\text{ time}[/tex]For the first leg
time = 1 hour 30 minutes
speed = 670 miles per hour
[tex]\begin{gathered} \text{ Distance }=\text{ 1.5}\times\text{ 670} \\ \text{ Distance }=\text{ 1005 miles} \end{gathered}[/tex]The first leg = 1005 miles
Second leg
time = 2 hours
speed = 670 miles per hour
[tex]\begin{gathered} \text{ Distance }=\text{ 2 }\times\text{ 670} \\ \text{ Distance }=\text{ 1340 miles} \end{gathered}[/tex]The second leg = 1340 miles
apply the cosine rule to find the distance from the starting position
[tex]\text{ a}^2\text{ }=\text{ b}^2\text{ }+\text{ c}^2\text{ - 2bc cos A}[/tex]substitute the given data into the cosine formula
[tex]\begin{gathered} \text{ a}^2\text{ }=\text{ \lparen1005\rparen}^2+\text{ \lparen1340\rparen}^2\text{ - 2\lparen1005}\times\text{ 1340\rparen cos 170} \\ \text{ a}^2=\text{ 1010025 }+\text{ 1795600 - 2693400 }\times(-0.9848) \\ \text{ a}^2\text{ }=\text{ 2805625 }+2652460.32 \\ \text{ a}^2\text{ }=\text{ 5458085.32} \\ \text{ take the square roots of both sides} \\ \sqrt{a^2}\text{ }=\text{ }\sqrt{5458085.32} \\ \text{ a }=\text{ 2336.3 miles} \end{gathered}[/tex]Therefore, distance from the starting position is 2336.3 miles
