a ferry shuttles people from one side of a river to the other. the speed of the ferry is still water is 25 mi/h. the river flows directly north at 9 mi/h. if the ferry heads directly west, what are the ferrys resultant speed and direction

To determine the ferry's resultant speed and direction, we must consider both the speed of the ferry in still water and the speed of the river current as vector quantities. The speed of the ferry in still water is 25 mi/h; this is the ferry's velocity vector heading directly west. The current of the river flows directly north at 9 mi/h, and this is the river's velocity vector. To find the resultant speed and direction, we'll use vector addition.

The ferry's westward speed and current's northward speed form a right-angled triangle. The ferry's speed is the horizontal leg of the triangle, and the river's current is the vertical leg. The resultant speed (the ferry's actual speed across the ground) will be the hypotenuse of this right-angled triangle.

We use the Pythagorean theorem to find the length of the hypotenuse:

resultant speed = √(ferry's speed² + current's speed²) resultant speed = √(25² + 9²) resultant speed = √(625 + 81) resultant speed = √706 resultant speed ≈ 26.57 mi/h

The direction of the ferry is calculated by finding the angle of the resultant vector with respect to the westward direction. We can use trigonometry for this—the tangent of the angle (θ) is the ratio of the opposite side (current's speed) to the adjacent side (ferry's speed):

tan(θ) = current's speed / ferry's speed tan(θ) = 9 / 25 θ = arctan(9/25) θ ≈ arctan(0.36) θ ≈ 20 degrees north of west

Therefore, the ferry's resultant speed is approximately 26.57 mi/h, and it is traveling at an angle of approximately 20 degrees north of west.