Two planes leave the airport at the same time. In one hour, the plane flying north has traveled 400 miles, and the plane flying southwest has traveled 250 miles. How far apart are the planes?
Use the Law of Cosines.
Answer (2)
a 2 = b 2 + c 2 − 2 ⋅ b ⋅ c ⋅ cos α − − − − − − − − − − − − − − − x 2 = 25 0 2 + 40 0 2 − 2 ⋅ 250 ⋅ 400 ⋅ cos 13 5 0 x 2 = 62 , 500 + 160 , 000 − 200 , 000 ⋅ cos ( 18 0 0 − 4 5 0 ) x 2 = 222 , 500 − 200 , 000 ⋅ ( − cos 4 5 0 ) x 2 = 222 , 500 − 200 , 000 ⋅ 2 2 x 2 = 222 , 500 + 100 , 000 ⋅ 2 x 2 = 2500 ⋅ ( 89 + 40 2 )
x = 50 ⋅ 89 + 40 2 ⇒ x ≈ 603.26 [ mi l es ] A n s . t h e d i s t an ce i s ab o u t 603.26 mi l es
To find the distance between the two planes, we applied the Law of Cosines. After calculating the values, we determined that the distance between the planes is approximately 603.26 miles. The process involved determining the angle between their paths and using the formula to find the distance.
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