In the figure below, the distances between 2 pairs of cities are shown, as well as the angle formed at Ewing, which has a measure of \(127^\circ\). Which of the following values is closest to the distance, in miles, from Deerborn to Fergus?

(Note: \(\cos{127^\circ}\approx -0.6\); \(\sin{127^\circ}\approx 0.8\))

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We can use law of cosines here.

$$ c^2=a^2+b^2-2ab\cos{C} $$

Treat angle \(C\) as Ewing. The side opposite, which we are to find, is \(c\).

$$ c^2=80^2+120^2-2(80)(120)\cos{127^\circ} $$ $$ c^2 = 32{,}320 $$ $$ c\approx \boxed{180} $$