Andre W. answered • 11/25/13

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In this problem, you are supposed to use the Law of Cosines, a generalization of the Pythagorean theorem. Pythagorean gives you the third side of a

*right*triangle; the Law of Cosines does the same for*any*triangle, if you know the angles. It statesc

^{2}= a^{2}+ b^{2}-2ab cos θ,where θ is the angle between sides a and b. Notice that if θ=90°, cos θ=0, and you get back the Pythagorean theorem.

In this problem, the three sides of the triangle are the distances between A and B (=a), B and C (=b), and C and A (=c, unknown). We have a= 54*2.1=113.4 miles and b = 54*1=54 miles. The angle between cities A and C, as measured at B, is θ=180-40-25=115°. Therefore,

c

^{2}= 113.4^{2}+54^{2}-2(113.4)(54)cos(115) = 20951,c = 145 miles.

Scott J.

03/03/18