direction needed to head to cross river, and how long will it take.

Question

Vanessa is on one side of the river at camp and wants to go straight across to the island. The current flows to the vest and has a speed of 3 km/h. she can paddle the canoe with a speed of 5 km/h (in still water). If the river is 1.5 km wide, in what direction does Vanessa need to head in order to cross the river? and how long will it take her to cross over to the island. Solve algebraically.

Answer 1

Set up the coordinate system with North pointing up and East to the right.  Then the current flowing to the west can be represented by the vector <-3,0>.  Since Vanessa wants to go due North, she wants to align her heading so that when her paddling vector is added to the current vector the x-components will cancel out.  So her heading must be of the form <3,x>.  Since the magnitude of her paddling vector is 5, the Pythagorean Theorem gives 3^2 + x^2 = 5^2 which implies that x^2 = 16 so x = +4 (since she wants to go North, not South).  So she needs to paddle in the direction given by the vector <3,4>.  This is Northeast at a angle theta from due East, where theta = arctan(4/3) ~ 53.13 degrees.  Since the Northerly component of the vector sum is 4 km/h, it will take Vanessa (1.5km) / (4 km/h) = 0.375 hours or 22.5 minutes to cross.