Question

A boat leaves a dock and goes 240 miles in a direction of 290 degrees. It then turns and goes 50 miles in a direction on 30 degrees. How far is the boat from the dock and in what direction?

Answer 1

By Pythagorean theorem and trigonometric functions, the magnitude and direction of the boat are approximately 284.403 miles and 295.043° with respect to the east.

What is the direction and magnitude of the resulting displacement with respect to the origin

In this problem we assume that the first angle is measured with respect to the east side and that the second angle is counterclockwise and measured with respect to the direction of the first vector. First, we need to determine the resulting vector by using trigonometric and vectorial formulas:

(x, y) = (240 · cos 290°, 240 · sin 290°) + (50 · cos 320°, 50 · sin 320°)

(x, y) = (120.387, - 257.666) [mi]

The magnitude is found by Pythagorean theorem:

$r = \sqrt{(120.387\,mi)^{2}+(-257.666\,mi)^{2}}$

r ≈ 284.403 mi

The direction of the boat is obtained by inverse trigonometric functions:

θ = tan⁻¹ (- 257.666/120.387)

θ ≈ 295.043°

By Pythagorean theorem and trigonometric functions, the magnitude and direction of the boat are approximately 284.403 miles and 295.043° with respect to the east.

To learn more on vectors: brainly.com/question/13322477

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