Question 373689: A man with a boat is located at point on the shore of a circular lake of radius 6 miles. He wants to reach the point on the shore diametrically opposed to as quickly as possible. He plans to paddle his boat at an angle () to to some point on the shore, then walk along the shore to his . If he can paddle 3 miles per hour and walk at 2.7 miles per hour, what is the shortest possible time it will take him to reach ?
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Something is very fishy with your numbers. The way you have stated the problem it is a no-brainer -- he just paddles straight across the lake and gets there in 4 hours. Any alternate path takes longer.
The length of a chord given the subtended angle and the radius is given by:
)
and for this circle:
)
Hence the elapsed time to traverse that chord is:
}{3}\ =\ 4\sin\left(\frac{\theta}{2}\right))
The remaining distance to be traveled is that proportion of the semi-circumference represented by:
The semi-circumference for this circle being  , the distance walked has to be:
)
And the elapsed time for this part of the trip must be:
}{2.7}\ =\ \frac{20}{9}(\pi\ -\ \theta))
And finally, the entire function for Elapsed Time as a function of the angle subtended by the chord of the circle is:
\ =\ 4\sin\left(\frac{\theta}{2}\right)\ +\ \frac{20}{9}(\pi\ -\ \theta))
Taking the first derivative:
\ -\ \frac{20}{9})
Setting the first derivative equal to zero we get:
\ =\ \frac{10}{9})
But since  is outside the range of the cos function, there is no solution on the interval  . Therefore, one of the endpoints of the interval must be the local minimum and the other end the local maximum. A little mental arithmetic should tell you which is which.
Now, if you reverse the speeds you gave, so that the paddling speed is 2.7 and the walking speed is 3, THEN there is an optimum solution somewhere between walking all the way and paddling all the way. However, you should be able to figure that out with the information I've already given you.
John
My calculator said it, I believe it, that settles it
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