SOLUTION: A small island is 4 miles from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row a boat 2 miles per hour and can walk 3 miles per hour

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Question 758089: A small island is 4 miles from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row a boat 2 miles per hour and can walk 3 miles per hour, where should the boat be landed in order to arrive at a town 12 miles down the shore from P in the least time? Let x be the distance between point P and where the boat lands on the lake shore.
(A) Enter a function T(x) that describes the total amount of time the trip takes as a function of the distance x.
T(x) =
(B) What is the distance x=c that minimizes the travel time? Note: The answer to this problem requires that you enter the correct units.
c = .
(C) What is the least travel time? Note: The answer to this problem requires that you enter the correct units.
The least travel time is .
(D) Recall that the second derivative test says that if T′(c)=0 and T′′(c)>0, then T has a local minimum at c. What is T′′(c)?
T′′(c) =

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The distance walked would be (12-x) miles.
The distance rowed would be sqrt%2816%2Bx%5E2%29 miles.
(A) highlight%28T%28x%29=sqrt%2816%2Bx%5E2%29%2F2%2B%2812-x%29%2F3%29 , measured in hours.

(B) T'(x) = %281%2F2%29%281%2F2%2Fsqrt%2816%2Bx%5E2%29%29%2A2x-1%2F3=x%2F2%2Fsqrt%2816%2Bx%5E2%29-1%2F3 is zero for x=c so
c%2F2sqrt%2816%2Bc%5E2%29-1%2F3=0 --> c%2F2sqrt%2816%2Bc%5E2%29=1%2F3 --> c%2Fsqrt%2816%2Bc%5E2%29=2%2F3 --> c%5E2%2F%2816%2Bc%5E2%29=4%2F9 --> 9c%5E2=4%2816%2Bc%5E2%29 --> 9c%5E2=64%2B4c%5E2 --> 5c%5E2=64 --> c%5E2=64%2F5 --> c=sqrt%2864%2F5%29=8%2Fsqrt%285%29=8sqrt%285%29%2F5
The approximate value is highlight%28c=3.5777miles%29
Since you are entering an answer, the expected answer may be "3.6 miles", but I would not know what format will be accepted. Maybe "miles" must be abreviated. Maybe spaces are not accepted, and the computer wants "c=3.58mi" for an answer. Maybe an answer starting with "c=" is not accepted.

(C) The least travel time is T%28c%29=sqrt%2816%2Bc%5E2%29%2F2%2B%2812-c%29%2F3%29
I calculated T%28c%29=approx5.49hours

(D) T"(x) = 1%2F2%2Fsqrt%2816%2Bx%5E2%29-x%5E2%2F2%2F%2816%2Bx%5E2%29%5E%283%2F2%29=8%2F%2816%2Bx%5E2%29%5E%283%2F2%29
I calculated T"(c) = approx0.052hour%2Fmile%5E2
I see no reason to make us calculate that. Finding that T(c)=0, while T(x)<0 for x < c and T(x)>0 for x > c proves that T(x) decreases for x < c and increases for x > c, so there is a minimum at x=c.