It takes 3 hours to paddle a kayak 12 miles downstream and 4 hours for the return trip upstream. Find the rate of the kayak in still water. Let k = the rate of the kayak in still w... It takes 3 hours to paddle a kayak 12 miles downstream and 4 hours for the return trip upstream. Find the rate of the kayak in still water. Let k = the rate of the kayak in still water and c = the rate of the current.
Understand the Problem
The question asks to find the rate of the kayak in still water given the distance and time to paddle downstream and upstream. The variables k and c are given so we need to solve for k.
Answer
$k = 3.5$ miles per hour
Answer for screen readers
$k = 3.5$ miles per hour
Steps to Solve
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Write the equations based on the given information We know that distance = rate * time. Downstream: $3(k+c) = 12$ Upstream: $4(k-c) = 12$
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Simplify the equations Divide the first equation by 3 and the second equation by 4: $k + c = 4$ $k - c = 3$
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Solve for k by eliminating c Add the two equations together: $(k + c) + (k - c) = 4 + 3$ $2k = 7$
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Solve for k Divide both sides by 2: $k = \frac{7}{2} = 3.5$
$k = 3.5$ miles per hour
More Information
The speed of the kayak in still water is 3.5 miles per hour. To find the rate of the current, plug k into one of the equations above e.g. $3.5 + c = 4$, so $c = 0.5$. This means the current is 0.5 miles per hour.
Tips
A common mistake is to incorrectly set up the equations. For example, some may subtract the current when going downstream instead of adding it. Also, not simplifying the equations before solving can make the algebra more difficult.
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