Problem
A kayaker travels two miles upstream and two miles downstream in a total of 5 hours. In still water, the kayaker travels at an average of two miles per hour. The goal of this paper will be to determine the average speed of the river’s current.
Variables
Let x represent the speed of the current of the river. Let t1 represent the time it takes to travel upstream. Let t2 represent the time it takes to travel downstream. Let r1 represent the rate of the kayaker when moving upstream. Let r2 represent the rate of the kayaker when moving downstream. Let d1 represent the distance of the journey upstream. Let d2 represent the distance of the journey downstream. f(x), will represent time. Variables without subscripts will have the same meaning
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The sum of the time upstream and the time downstream must be equal to 5 hours. t_1+t_2=5 The sum of the upstream and downstream rates must be equal to 4 mph. r_1+r_(2 )=4 r1 must be greater than 2, the average rate for the entire journey. r2 must be less than two. In the given scenario, going upstream, the current of the river will be working against the kayaker; thus, the rate of the rivers current will be less than the average speed the kayaker can travel and vice versa when the kayaker is traveling downstream.
The rates of the kayaker can be determined through modified guess and check. The correct rates, when plugged into the equations of; r=d/t Should yield results for t that when added are equal to five. The rates and times that appear to most closely work are; r_(1 )=0.45mph; r_(2 )=3.55mph; t_1=4.444444444; t_2=0.56338
The speed of the current of the river will be the absolute value of the distance from the average speed to the individual rates. The average speed of the current of the river as deduced by the first algebraic method is 1.55 miles per hour.
Algebraic Solution
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The asymptotes of the rational function are at -2 and 2. There are two points at which the functions intersect. The points are at, -1.54919333848296 and 1.549193338482967 from left to right respectively. These values represent the possible speeds of the current of the river when the time of travel is equal to five hours. The negative solution is extraneous because in the context of this problem, a negative value of x does not make sense as a river current could not move at a negative
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