Question

applying the law of cosines to navigation
a pilot flies 720 miles from dallas, texas, to his first destination. after dropping off his cargo, he flies southeast 290 miles to his second destination. if the angle formed by his trip is 125°, what is the distance he will fly from the second destination back to dallas?
○ 490 miles
○ 918 miles
○ 1,010 miles
○ 842,026 miles

Explanation:

Step1: Define Law of Cosines variables

Let $a=720$, $b=290$, $C=125^\circ$, $c$ = return distance.
Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos(C)$

Step2: Calculate squared side lengths

$a^2 = 720^2 = 518400$, $b^2 = 290^2 = 84100$

Step3: Calculate the cosine term

$\cos(125^\circ) \approx -0.5736$
$2ab\cos(C) = 2(720)(290)(-0.5736) \approx -237565.44$

Step4: Compute $c^2$

$c^2 = 518400 + 84100 - (-237565.44) = 602500 + 237565.44 = 840065.44$

Step5: Solve for $c$

$c = \sqrt{840065.44} \approx 916.55 \approx 918$

Answer:

918 miles