A straight river flows at a speed of 20km/h. A boater starts at the south shore and heads in ths direction of 70degrees from the shore. The boat has a speed of 20km/h in still water. Find the true speed and direction of the boat.

Let the river flow in the +x direction.

Across the river is then +y.

river velocity = (20,0)
boat velocity relative to the water
= (20cos70°,20sin70°) = (6.84,18.79)

resultant velocity relative to land
= (26.84,18.79)

speed: √(26.8^2+18.79^2) = 32.73
direction:
tanθ = y/x = 18.79/26.84 = .70
θ = 35°

To find the true speed and direction of the boat, we need to determine the vector components of the boat's velocity.

Let's break down the boat's velocity into its components. We have the river's velocity, which is 20 km/h, and the boat's velocity in still water, also 20 km/h.

The velocity of the boat relative to the shore can be found using vector addition. We combine the two velocities, taking into account both magnitude and direction.

First, let's convert the angle from degrees to radians. We know that 1 radian is equal to 180/π degrees.

Angle in radians = 70 degrees × (π/180) ≈ 1.22 radians

Next, we can calculate the horizontal and vertical components of the boat's velocity.

Horizontal component of boat's velocity = velocity in still water × cos(angle)
Vertical component of boat's velocity = velocity in still water × sin(angle)

Horizontal component of boat's velocity = 20 km/h × cos(1.22) ≈ 7.08 km/h
Vertical component of boat's velocity = 20 km/h × sin(1.22) ≈ 2.17 km/h

Now, we can find the true speed and direction of the boat.

True speed of the boat = square root of [(horizontal component of boat's velocity)^2 + (vertical component of boat's velocity)^2]

True speed of the boat = sqrt((7.08 km/h)^2 + (2.17 km/h)^2) ≈ 7.41 km/h

To find the direction of the boat, we can use the inverse tangent function.

Direction of the boat = arctan(vertical component of boat's velocity / horizontal component of boat's velocity)

Direction of the boat = arctan(2.17 km/h / 7.08 km/h) ≈ 0.29 radians

Converting the direction from radians to degrees:

Direction of the boat in degrees = (Direction in radians) × (180/π) ≈ 16.6 degrees

Therefore, the true speed of the boat is approximately 7.41 km/h, and the direction of the boat is approximately 16.6 degrees from the south shore.