A canoeist wishes to cross a river 0.5km wide from A to B. If a current of 3 m/s flows and he is capable of paddling at 5m/s in still water,how long does it take him to cross the river?

First, we need to find the velocity of the canoeist relative to the water.

Let x be the angle between the direction the canoeist paddles and the direction of the current. The velocity of the canoeist relative to the water is given by the vector sum of the velocity of the canoeist in still water and the velocity of the current.

Let V be the velocity of the canoeist relative to the water. Then, using the cosine rule, we have:

V^2 = (5^2 + 3^2 - 2*5*3*cos(x))

V^2 = 25 + 9 - 30cos(x)

V^2 = 34 - 30cos(x)

Since cos(x) = adjacent/hypotenuse = 3/5 = 0.6, we have:

V^2 = 34 - 30*0.6

V^2 = 34 - 18

V^2 = 16

V = 4 m/s

Now, we can calculate the time it takes for the canoeist to cross the river using the formula:

Time = Distance / Velocity

Time = 500m / 4m/s

Time = 125 seconds

Therefore, it takes the canoeist 125 seconds to cross the river.

Step 1: Determine the effective speed of the canoe in the river.

The effective speed of the canoe is the speed of the canoe relative to the water. This is calculated by subtracting the speed of the current from the speed of the canoe in still water.

Effective speed = Speed of canoe in still water - Speed of current
Effective speed = 5 m/s - 3 m/s
Effective speed = 2 m/s

Step 2: Calculate the distance the canoe must travel.

The distance the canoe must travel is the width of the river. This is given as 0.5 km or 500 meters.

Step 3: Calculate the time taken to cross the river.

Time = Distance / Speed
Time = 500 m / 2 m/s
Time = 250 seconds

Therefore, it takes the canoeist 250 seconds or approximately 4 minutes and 10 seconds to cross the river.