SOLUTION: Can someone help me with this? Thank you. A man can row his boat at a rate of 4 mph and he can run at a rate of 7 mph. He is 1 mile directly west of a point on an north-south sho

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Question 1100776: Can someone help me with this? Thank you.
A man can row his boat at a rate of 4 mph and he can run at a rate of 7 mph. He is 1 mile directly west of a point on an north-south shoreline, and the camp he needs to reach is 2 miles south of that same point on the shoreline.
1. How far down the shoreline should he row to minimize the amount of time it will take him to get to the camp?
2. What is the minimum time?
Answer by Boreal(15235) (Show Source):
You can put this solution on YOUR website!
Helps to draw this
If he rows directly to the site, the distance is sqrt (1+4)=sqrt(5) miles
At 4 mph, that is 33.5 minutes or .559 hours
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If he rows directly to shore, the distance is 1 mile and the time 1/4 hr 15 min. He then runs down the shore 2 miles at 7 miles per hour, and that is 2/7 of an hour or 120/7 min or 17.1 min. The total time is 32.1 min.
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The distance rowed is sqrt (1+x^2), where x is the distance south of the point. The time is sqrt (1+x^2)/4 hours.
The distance down the shore line is 2-x miles, and the time is (2-x)/7 hours
Take the first derivative of the sum
(1/4)*(1/2)(2x)(1/sqrt(1+x^2))-(1/7)=0
This is x/4sqrt(1+x^2)-(1/7)=0
so (1/7)=x/4sqrt(1+x^2)
4sqrt(1+x^2)=7x
16(x^2+1)=49x^2
16x^2+16=49x^2
33x^2=16
x=0.696 miles
distance rowed is sqrt(1+0.485^2)=1.22 miles, and that takes .3046 hours or 18.28 min
The distance run is 1.304 miles, and that takes 0.186 hours or 11.18 minutes
Total time is 29.46 minutes