Question

Solve each problem. See Examples 1 and 2. Distance between Two Cities The bearing from Atlanta to Macon is S 27° E, and the bearing from Macon to Augusta is N 63° E. An automobile traveling at 62 mph needs 1¼ hr to go from Atlanta to Macon and 1¾ hr to go from Macon to Augusta. Find the distance from Atlanta to Augusta.

Accepted Answer

Identify the distances traveled from Atlanta to Macon and from Macon to Augusta using the formula $$	ext{distance} = 	ext{speed} 	imes 	ext{time}. $$For Atlanta to Macon, calculate $$d_1 = 62$$ 	imes 1.25$$, and for Macon to Augusta, calculate d_2$ = 62 	imes 1.75. $$Draw a diagram representing the positions of Atlanta, Macon, and Augusta, including the bearings. The bearing from Atlanta to Macon is $$S 27^\circ E$$, which means starting from south, rotate 27 degrees towards east. The bearing from Macon to Augusta is $$N 63^\circ E$$, starting from north, rotate 63 degrees towards east. Determine the angle between the two paths (Atlanta to Macon and Macon to Augusta) at point Macon. Since the bearings are given relative to the cardinal directions, find the interior angle between the two directions by adding or subtracting the given angles appropriately. Use the Law of Cosines to find the distance from Atlanta to Augusta. Label the triangle with sides $$d_1$$, $$d_2$$, and the unknown side $$d$$, and the included angle $$	heta$$ found in the previous step. The Law of Cosines formula is:  $$d^2$$ = $$d_1^2$$ + $$d_2^2 - 2$$ $$d_1$$ $$d_2$$ \cos(	heta)$$. Solve for d$ by taking the square root of both sides: d$$ = \sqrt{$$d_1^2$$ + $$d_2^2 - 2$$ $$d_1$$ $$d_2$$ \cos(	heta)}$$. This will give the distance from Atlanta to Augusta.

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Concetti chiave

Bearing and Direction in Navigation

Distance, Speed, and Time Relationship

Law of Cosines for Non-Right Triangles