If the airplane had to return it would have to travel 254.3 miles
The correct option is D. 254.3 miles
From the diagram, if the airplane had to return to the airport, it will travel on a straight line. The route the airplane will take back to the airport is shown in the attached image as /PA/.
From the diagram in the image
Let /PA/ = b, /PB/ = a and /BA/ = p
∴ a = 150 mi, p = 160 mi and b is the distance the airplane would have to travel.
Consider Δ BPA
Using the Cosine rule
[tex]b^{2} = a^{2} + p^{2} -2ap(cosB)[/tex]'
[tex]b^{2} = 150^{2} + 160^{2} -2\times150\times160(cos110.167^{o} )[/tex]
[tex]b^{2} = 22500 + 25600 - 48000(cos110.167^{o} )[/tex]
[tex]b^{2} = 48100 - (-16548.3652)[/tex]
[tex]b^{2} = 48100 +16548.3652[/tex]
[tex]b^{2} = 64648.3652[/tex]
∴ [tex]b =\sqrt{64648.3652}[/tex]
[tex]b = 254.2604[/tex] mi
b ≅ 254.3 miles
∴ /PA/ ≅ 254.3 miles
Hence, if the airplane had to return, it would have to travel 254.3 miles
Learn more on bearing and distances here: https://brainly.com/question/22518031
