SECTION A.8 Problem Solving: Interest, Mixture, Uniform Motion, Constant Rate Job Applications A71 Now Work problem 29 5 Solve Constant Rate Job Problems Here we look at jobs that are performed at a constant rate. The assumption is that if a job can be done in t units of time, then t 1 of the job is done in 1 unit of time. In other words, if a job takes 4 hours, then 1 4 of the job is done in 1 hour. Physics: Uniform Motion A motorboat heads upstream a distance of 24 miles on a river whose current is running at 3 miles per hour (mi/h). The trip up and back takes 6 hours. Assuming that the motorboat maintains a constant speed relative to the water, what is its speed? Solution EXAMPLE 6 See Figure 30. Use r to represent the constant speed of the motorboat relative to the water. Then the true speed going upstream is −r 3 mi h, and the true speed going downstream is +r 3 mi h.Since = ⋅ Distance Rate Time, then = Time Distance Rate . Set up a table. Rate (mi/h) Distance (mi) ( ) = Time h Distance Rate Upstream − r 3 24 − r 24 3 Downstream + r 3 24 + r 24 3 The total time up and back is 6 hours, which gives the equation 24 miles r - 3 mi/h r + 3 mi/h Figure 30 Add the quotients on the left. Simplify. Multiply both sides by − r 9. 2 Write in standard form. Divide by 6. Factor. Use the Zero-Product Property and solve. r r r r r r r r r r r r r r r r r r 24 3 24 3 6 24 3 24 3 3 3 6 48 9 6 48 6 9 6 48 54 0 8 9 0 9 1 0 9 or 1 2 2 2 2 ( ) ( ) ( ) ( )( ) ( )( ) − + + = + + − − + = − = = − − − = − − = − + = = = − Discard the solution = − r 1 mi h and conclude that the speed of the motorboat relative to the water is 9 mi/h. Working Together to Do a Job At 10 am Danny is asked by his father to weed the garden. From past experience, Danny knows that this will take him 4 hours, working alone. His older brother Mike, when it is his turn to do this job, requires 6 hours. Since Mike wants to go golfing with Danny and has a reservation for 1 pm, he agrees to help Danny. Assuming no gain or loss of efficiency, when will they finish if they work together? Can they make the golf date? EXAMPLE 7 (continued)
RkJQdWJsaXNoZXIy NjM5ODQ=