A small island is 3 kilometers from the nearest point p on the straight shoreline of a large lake. if a woman on the island can
row her boat 2.5 kilometers per hour and can walk 4 kilometers per hour where should she land her boat in order to arrive in the shortest time at a town 12 kilometers down the shore from p
The distance walked will be (12-x) km the distance rowed will be (9+x²) km A] The function T(x) will be given by: Time=distance/speed thus we shall have: T(x)=[√(9+x²)]/2.5+(12-x)/4
B] To get the distance x=c that minimizes the time travel, we differentiate the above. T'(x)=(1/2.5)[1/(2.5√9+x²)*2x-1/4] this should give us 0 for x=c, thus c/[2.5*√(9+c²)]-1/4=0 ⇒c/[2.5*√(9+c²)]=1/4 c/√(9+c²)=2.5/4 squaring both sides we get: c²/(9+c²)=5/8 8c²=5(9+c²) 8c²=45+5c² 3c²=45 c²=15 c=3.87 km
c] The least travel time is T(c)=[√(9+c²)]/2.5+(12-c)/4 this will give us: T(c)=[√(9+3.87²)]/2.5+(12-3.87)/4 T(c)=3.9999209~4 hours
d] The second derivative will be: T"(x)=1/[2.5√(9+x²)]-x²/[2.5(9+x²)^(3/2)] but x=c=3.87 T"(x)=0.01668 hours/ mile² Given 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
