A small island is 5 miles from the nearest point P on the straight shoreline of

Question

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 6 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 .

T(x)=_____ (include units)

(b) What is the distance x= c that minimizes the travel time?

c=______ (include units)

(c) What is the least travel time?

The least travel time is_____ (include units)

Explanation / Answer

Let x be the distance down the shore where the boat is landed.

t1 = time to row to the shore (5^2 + x^2)1/2/2
t2 = time to walk from the point where the boat is landed = (6 - x)/3

t = total time = t1 + t2 = T(x)=(5^2 + x^2)1/2/2 + (6 - x)/3

t = (25 + x^2)1/2/2 + (6 - x)/3
dt/dx = x/2[sqrt(25 + x^2)] - 1/3 = 0

x=c= (20)^0.5

least time= put value of c in T(x). we get. t= (45)^0.5/2+2-(20)^0.5/3
So the distance is (18/7)*sqrt(7

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