Final answer:
The boat's speed relative to the river bottom is the resulting velocity, calculated using the Pythagorean theorem, which gives approximately 10.44 km/h.
Step-by-step explanation:
The speed of the boat relative to the river bottom can be found by using the Pythagorean theorem to account for the fact that the river's flow is perpendicular to the motion of the boat. The boat's speed relative to the water is given as 10.0 km/h, and the speed of the river is given as 3.00 km/h. To find the boat's speed relative to the river bottom, we calculate the resultant velocity by adding the two velocities vectorially:
-
- Vboat = 10.0 km/h (forward, y-direction)
-
- Vriver = 3.00 km/h (right, x-direction)
We use the Pythagorean theorem because the river flow is perpendicular to the boat's motion:
Vtotal = √(Vboat² + Vriver²)
= √(10.0² + 3.00²)
= √(100 + 9)
= √(109)
≈ 10.44 km/h
Therefore, the magnitude of the boat's velocity relative to the river bottom is approximately 10.44 km/h.