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Calculus I Optimization problem?
A traveler must reach a town on the other side of a river. The river (assumed straight) flows 5 miles North of the starting point of the traveler and is 1 mile wide. The town is 6 miles North and 5 miles West of the starting point. The traveler speed is of 5 miles/hour on the ground and 1 miles/hour in water.
How far West of the initial position should the traveler get to the river to minimize the time of travel?
-This was a problem on my most recent exam, for which out of 25 points I received only 5 for drawing a diagram. I'm unsure how to create the function necessary to then use to optimize a minimal time of travel.
Thanks in advance for any help.
1 Answer
- ?Lv 68 years agoFavorite Answer
Let the traveller be x miles west of starting point when he reaches the river.
Then, he travels 5 miles north and x miles west on the land:
Total distance on land = sqrt(25 + x^2)
Total time on land = sqrt(25 + x^2) / 5
He travels 1 mile north and (5-x) miles west in the river:
Total distance in river = sqrt[ 1 + (5-x)^2 ]
Total time in river = sqrt[ 1 + (5-x)^2 ] / 1
Add the times to get the function for total time.
Hope this helps.
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