) In flying the 76 miles from Champaign to Peoria, a student pilot sets a heading that is 11° off course and maintains an average speed of 112 miles per hour. After 15 minutes, the instructor notices the course error and tells the student to correct his heading. Through what angle will the plane move to correct the heading and how many miles away is Peoria when the plane turns?

Answer:48.8 miles17.3°Step-by-step explanation:The law of cosines can be used to find the remaining distance (c). It is given by ...   c² = a² +b² -2ab·cos(C)where we have a=76, b=112/4 = 28, C=11°.   c² = 76² +28² -2·76·28·cos(11°) ≈ 2382.195   c ≈ √(2382.195) ≈ 48.8 . . . . milesBy the law of sines, ...   sin(A)/a = sin(C)/c   sin(A) = (a/c)sin(C)If you draw a diagram of the problem, you realize that angle A is obtuse. The arcsin function does not return obtuse angles, so an adjustment must be made.   A = arcsin(a/c·sin(C)) = arcsin(76/48.8·sin(11°)) ≈ 180° -17.3°The angle through which the plane must turn is the supplement of angle A, so is 17.3°.