SOLUTION: I just do not understand optimization every problem confuses me please explain this problem and its solution.
An island is 2mi due north of its closest point along a straight sh
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Question 1157615: I just do not understand optimization every problem confuses me please explain this problem and its solution.
An island is 2mi due north of its closest point along a straight shoreline. A visitor is staying at a cabin on the shore that is 6mi west of that point. The visitor is planning to go from the cabin to the island. Suppose the visitor runs at a rate of 8mph and swims at a rate of 3mph. How far should the visitor run before swimming to minimize the time it takes to reach the island?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
In this problem, we are given distances and rates and we are looking to minimize time. We know that distance = rate times time, so time is distance divided by rate.
Here is the particular situation graphically:
The total time is the sum of the Running time and the Swimming time, so
\ =\ \frac{6\,-\,x}{8}\ +\ \frac{sqrt{x^2\,+\,4}}{3})
And note that the domain of the function for this particular problem is 
Figure 2 is the graph of the time function:
Note that the graph has a minimum value somewhere in the neighborhood of 
(the vertical grid lines are spaced 0.5 apart)
The exact point is where the slope of the tangent line to the curve is zero.
Calculate:
)
Then set the derivative function equal to zero and solve for 
. The answer to the question as posed is then 
John
My calculator said it, I believe it, that settles it
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