A small island is 5 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 11 miles down the shore from P in the least time? Let x be the distance (in miles) between point P and where the boat lands on the lakeshore.
a. Enter a function T(x) that describes the total amount of time the trip takes as a function of the distance x.
b. What is the distance x = c that minimizes travel time?
c. What is the least travel time.
I know that x is between 0 and 11, the velocity on the shore is 3 mi/hr, that the velocity on water is 2mi/hr. For x = 0, the total time would be about 6 hours. For x = 11, the time would be 6.16666 hours. The function should therefore be some quadratic. But how do I find this function? If I can find the function, I can use calculus to find the distance and the time that it would take. I would be grateful if someone explained the needed steps.
Best Answer
Let x be the landing point between P and nearest town such as $$0\le x\le11$$ Then the path travelled by boat $$P_b=\sqrt{5^2+x^2}$$ and by walking $$P_w=11-x$$ Then the total time for travel $$T(x)=\frac{\sqrt{5^2+x^2}}{2}+\frac{11-x}{3}$$
Full Solution
To minimize travel time differentitate wrt x and set to zero $$\frac{dT}{dx}=-\frac{1}{3}+\frac{x}{2\sqrt{5^2+x^2}}=0$$ which has the solution at $x=2\sqrt{5}$. The minimum travel time is
$$T(2\sqrt{5})=\frac{\sqrt{5^2+2^2 5}}{2}+\frac{11-2\sqrt{5}}{3}=\frac{9\sqrt{5}+22-4\sqrt{5}}{6}=\frac{5\sqrt 5 + 22}{6}$$