Question

(2)Please help with both questions

Answer 1

Answer:

See below for answers and explanations

Step-by-step explanation:

Problem 1

Let's think of the boat and wind as vectors:

Boat Vector --> $\langle28cos36^\circ,28sin36^\circ\rangle$

Wind Vector --> $\langle10cos22^\circ,10sin22^\circ\rangle$

Now, let's add the vectors:

$\langle28cos36^\circ+10cos22^\circ,28sin36^\circ+10sin22^\circ\rangle$

Find the magnitude (the true velocity):

$\sqrt{(28cos36^\circ+10cos22^\circ)^2+(28sin36^\circ+10sin22^\circ)}\approx37.78\approx38$

Find the direction (angle):

$\theta=tan^{-1}(\frac{28sin36^\circ+10sin22^\circ}{28cos36^\circ+10cos22^\circ})\approx32.32^\circ\approx32^\circ$

Thus, D is the best answer

Problem 2

Recall that the angle between two vectors is $\theta=cos^{-1}(\frac{u\cdot v}{||u||*||v||})$ where $u\cdot v$ is the dot product of the vectors and $||u||*||v||$ is the product of each vector's magnitude:

$\theta=cos^{-1}(\frac{u\cdot v}{||u||*||v||})\\\\\theta=cos^{-1}(\frac{\langle-82,47\rangle\cdot\langle92,80\rangle}{\sqrt{(-82)^2+(47)^2}*\sqrt{(92)^2+(80)^2}})\\\\\theta=cos^{-1}(\frac{(-82)(92)+(47)(80)}{\sqrt{6724+2209}*\sqrt{8464+6400}})\\\\\theta=cos^{-1}(\frac{(-7544)+3760}{\sqrt{8933}*\sqrt{14864}})\\\\\theta=cos^{-1}(\frac{-3784}{\sqrt{132780112}})\\\\\theta\approx109.17^\circ\approx109^\circ$

Therefore, C is the best answer