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When the bodies move in the same direction, the rate at which the distance between them is changing is the di erence between their rates A cyclist starts at a certain point and rides at a rate of 10 mph Twelve minutes later, another cyclist starts from the same point in the same direction and rides at 16 mph How long will it take for the second cyclist to catch up with the rst When the second cyclist begins, the rst has traveled 10 12 2 miles 60 The 10 is the rate and the 12 is the twelve minutes converted to 60 hours Because the cyclists are moving in the same direction, the rate at which the distance between them is shrinking is 16 10 6 mph Then, the question boils down to How long will it take for something traveling 6 mph to cover 2 miles
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Let t represent the number of hours the second cyclist is traveling d rt 2 6t 2 t 6 1 t 3
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It will take the second cyclist 1 of an hour or 20 minutes to catch up the 3 rst cyclist A car passes an intersection heading north at 40 mph Another car passes the same intersection 15 minutes later heading north traveling at 45 mph How long will it take for the second car to overtake the rst In 15 minutes, the rst car has traveled 40 15 10 miles The second 60 car is gaining on the rst at a rate of 45 40 5 mph So the question becomes How long will it take a body traveling 5 mph to cover 10 miles Let t represent the number of hours the second car has traveled after passing the intersection d rt 10 5t 10 t 5 2 t It will take the second car two hours to overtake the rst
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1 Lori starts jogging from a certain point and runs 5 mph Je rey jogs from the same point 15 minutes later at a rate of 8 mph How long will it take Je rey to catch up to Lori 2 A truck driving east at 50 mph passes a certain mile marker A motorcyclist also driving east passes that same mile marker 45 minutes later If the motorcyclist is driving 65 mph, how long will it take for the motorcyclist to pass the truck
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1 Lori has jogged r t d 5 15 5 60 4 miles before Je rey began Je rey is catching up to Lori at the rate
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of 8 5 3 mph How long will it take a body traveling 3 mph to cover 5 miles 4 Let t represent the number of hours Je rey jogs 5 3t 4 1 5 t 3 4 5 t 12 Je rey will catch up to Lori in
5 12 5 hours or 12 60 25 minutes
2 The truck traveled 50 45 75 miles The motorcyclist is catching up 60 2 to the truck at a rate of 65 50 15 mph How long will it take a body moving at a rate of 15 mph to cover 75 miles 2 Let t represent the number of hours the motorcyclist has been driving since passing the mile marker 75 15t 2 1 75 t 15 2 5 t 2 21 t 2 The motorcyclist will overtake the truck in 21 hours or 2 hours 30 2 minutes When two bodies are moving in opposite directions, whether towards each other or away from each other, the rate at which the distance between them is changing, whether growing larger or smaller, is the sum of their individual rates
Examples
Two cars meet at an intersection, one heading north; the other, south If the northbound driver drives at 30 mph and the southbound driver at 40 mph, when will they be 35 miles apart