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A hunter wishes to cross a river that is 1.5 $\mathrm{km}$ wide and flows with a speed of 5.0 $\mathrm{km} / \mathrm{h}$ parallel to its banks. The hunter uses a small powerboat that moves at a maximum speed of 12 $\mathrm{km} / \mathrm{h}$ with respect to the water. What is the minimum time necessary for crossing?

$t = 0.125 \mathrm { h }$

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Numerade Educator

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Hope College

University of Winnipeg

So we have the velocity of you have this triangle essentially and here this would be the velocity of the water relative to the shore or the current. This would be the velocity of the boat relative to the shore and then the velocity of the boat relative to the water. And so we can then say that this angle here would be essentially Fada and we can say that then the velocity, the magnitude rather of the velocity of the boat relative to the shore, would be equaling the square root of the magnitude of the velocity of the boat relative to the water squared, plus the velocity of the water relative to the shore squared. Of course, this is due to Pythagorean theorem, and we have that this is equaling a square root of 12 kilometers per hour squared, quantity squared, plus 5.0 kilometers per hour quantity squared. And we find that the magnitude of the velocity of the boat relative to the shore is equaling approximately 13 exactly rather 13 kilometers per hour. Now, at this point, we can say that this is gonna be directed in some, uh, direction. So this would be Fada arc, 10 of the why component in this case, it would be the velocity of the boat relative to the water, divided by the velocity of the water relative to the shore, the X component. And so this is equaling arc Tanne of 12 divided by five and this is giving us 67 degrees. So this would be our ah direction north of east, and this would be our magnitude. Now we convince say that the minimum time to cross the river T would be equaling the width of the river, divided by the velocity of the boat relative to the water. And so this is equaling 1.5 kilometers divided by 12 kilometers per hour. We can multiply this by 60 minutes for every one hour, and this is giving us 7.5 minutes now during that time, so this would be our answer for the minimum time to cross the river. Now, during this time, the boat drifts downstream and this distance downstream would be equaling the velocity of the water relative to the shore multiplied by t. And so this should be equaling 5.0 kilometers per hour, multiplied by 7.5 minutes multiplied by one hour for every 60 minutes. Multiplied by 10 to the third meters for every one kilometer. And we find that the distance downstream is going to be approximately 630 meters. This would be our final answer. That is the end of the solution. Thank you for watching.