Surya is competing in a triathlon that includes swimming, bicycling and running. For the first two segments of the race, Surya swam at an average rate of 4 kilometers per hour and bicycled at an average rate of 40 kilometers per hour. It took 81.75 minutes for Surya to swim and cycle a combined total of 43.29 kilometers, completing the first two segments of the race. How long, in minutes, did it take him to complete the swimming portion of the race? Express your answer to the nearest hundredth.
Let d1 be the distance Surya swam and d2 be the distance he bicycled. We know that d1 + d2 = 43.29 km. Now let t1 and t2 be the time, in hours, it took Surya to complete the swimming and bicycling portions of the race, respectively. That means t1 + t2 = 81.75/60 = 1.3625 hours, and t2 = 1.3625 – t1. Using the distance formula, we have d1 = 4t1 and d2 = 40t2 = 40(1.3625 – t1) = 54.5 – 40t1. Substituting these expressions for d1 and d2 in our first equation yields 4t1 + 54.5 – 40t1 = 43.29. Solving for t1 we see that −36t1 = −11.21, meaning it took Surya t1 = 0.311388888 hours to complete the swimming portion of the race. This is equivalent to 0.311388888(60) = 18.6833333 ≈ 18.68 minutes.
If an athlete who completes this triathlon swims, bicycles and runs a combined total of 52.95 kilometers, the distance the athlete swam is what percent of the total race distance? Express your answer as a percent to the nearest hundredth.
Let’s first determine the distance the athlete swam. From the previous problem it follows that the distance swam is d1 = 4(0.311388888) = 1.2455555 km. So, the distance swam is 1.2455555/52.95 = 0.02352324 ≈ 2.35% of the total race distance.
At what average rate, in kilometers per hour, must Surya run the third, and final, portion of the race if he wishes to complete the entire race in no more than 140 minutes? Express your answer to the nearest hundredth.
In the previous problems we were told that it took Surya a combined 81.75 minutes to complete the swimming and bicycling segments of the race. So, he must complete the running portion in 140 – 81.75 = 58.25 minutes, or 58.25/60 = 0.970833333 hrs. The distance run is equal to the total race distance less the combined distance for the swimming and bicycling segments. Using information from the previous problems, 52.95 – 43.29 = 9.66 km. So, the average rate at which Surya must run to finish 9.66 km in 0.970833333 hours is 9.66/0.970833333 = 9.950214592 ≈ 9.95 kilometers per hour.
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