Miguel is driving his motorboat across a river. The speed of the boat in still water is 10 mi/h. The river flows directly south at 3 mi/h. If Miguel heads directly west, what are the boat’s resultant speed and direction? Round answers to the nearest tenth.

you have a right triangle with legs 3 and 10.

The resultant speed is the hypotenuse: √(36+100) = √136 = 11.7

direction is W θ S

where tanθ = 3/10, so that's W16.7°S

the resultant speed is the hypotenuse

9+100=109
√109=10.4

To find the boat's resultant speed and direction, we need to use vector addition.

The boat's speed in still water, which is directly west, is 10 mi/h.

The river flows directly south at 3 mi/h. This velocity acts as a "current" or "crosswind" on the boat.

To add the vectors, we can use the Pythagorean theorem and trigonometry.

The resulting speed is found by squaring the boat's speed and the river's velocity, adding them together, and then taking the square root:

Resultant Speed = √(10^2 + 3^2) ≈ √(100 + 9) ≈ √109 ≈ 10.4 mi/h

The resultant direction is found by taking the inverse tangent (tan^(-1)) of the river's velocity divided by the boat's velocity:

Resultant Direction = tan^(-1)(3/10) ≈ 17.6° south of west

Therefore, the boat's resultant speed is approximately 10.4 mi/h and its resultant direction is approximately 17.6° south of west.