826 CHAPTER 11 Systems of Equations and Inequalities Running a Long-distance Race In a 50-mile race, the winner crosses the finish line 1 mile ahead of the secondplace runner and 4 miles ahead of the third-place runner.Assuming that each runner maintains a constant speed throughout the race, by how many miles does the secondplace runner beat the third-place runner? EXAMPLE 6 Let v v , , 1 2 and v3 denote the speeds of the first-, second-, and third-place runners, respectively. Let t1 and t2 denote the times (in hours) required for the first-place runner and the second-place runner to finish the race. Then the following system of equations results: = = = = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ v t v t v t v t 50 49 46 50 1 1 2 1 3 1 2 2 We want the distance d of the third-place runner from the finish at time t .2 At time t ,2 the third-place runner has gone a distance of v t3 2 miles, so the distance d remaining is −v t 50 . 3 2 Now = − = − ⋅ = − ⋅ = − ⋅ = − ⋅ ≈ d v t v t t t v v v v 50 50 50 46 50 50 50 46 50 46 50 49 3.06 miles 3 2 3 1 2 1 2 1 1 2 3 miles 1 mile Solution (1) First-place runner goes 50 miles in t1 hours. (2) Second-place runner goes 49 miles in t1 hours. (3) Third-place runner goes 46 miles in t1 hours. (4) Second-place runner goes 50 miles in t2 hours. ( ) ( ) ( ) = = = ⎧ ⎨ ⎪⎪ ⎪⎪ ⎪⎪ ⎪ ⎩ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ v t t v t v Fromequation 3, 46 Fromequation 4, 50 Fromequation 1, 50 3 1 2 2 1 1 Multiply and divide by t .1 From the quotient of equations (1) and (2) The next example illustrates an imaginative solution to a system of nonlinear equations.
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