SOLUTION: A small island is 3km from the nearest point P on the straight shoreline of a large lake. A village is 12km down the shore from P. If a man can row a boat at a rate of 2.5 km per h

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Question 1162238: A small island is 3km from the nearest point P on the straight shoreline of a large lake. A village is 12km down the shore from P. If a man can row a boat at a rate of 2.5 km per hour and walk at 6.5km per hour, where should he land the boat to minimize the time to town?
Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


Let x be the distance from P where the man reaches shore. Then the distance from there to the village is 12-x.

The distance the man rows, by the Pythagorean Theorem, is sqrt%28x%5E2%2B9%29.

The time the man takes to row to the shore is sqrt%28x%5E2%2B9%29%2F2.5

The time he takes to walk from there to town is %2812-x%29%2F6.5

We want to find x that makes the total time a minimum:

y+=+sqrt%28x%5E2%2B9%29%2F2.5+%2B+%2812-x%29%2F6.5

dy%2Fdt+=+%280.5%282x%29%29%2F%282.5sqrt%28x%5E2%2B9%29%29-1%2F6.5

Set the derivative equal to 0 and solve for t:

%280.5%282x%29%29%2F%282.5sqrt%28x%5E2%2B9%29%29+=+1%2F6.5

x%2F%282.5sqrt%28x%5E2%2B9%29%29+=+1%2F6.5

6.5x+=+%282.5sqrt%28x%5E2%2B9%29%29

%2813%2F5%29x+=+sqrt%28x%5E2%2B9%29

%28169%2F25%29x%5E2+=+x%5E2%2B9

%28144%2F25%29x%5E2+=+9

x%5E2+=+225%2F144

x+=+15%2F12+=+5%2F4+=+1.25

ANSWER: The man should land the boat 1.25 miles from P.

The answer can be confirmed by finding the minimum value of the time y using a graphing calculator.