Question

(2)Please help with both questions

Answer 1

See below for answers and explanations

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):

`√((28cos36^\circ+10cos22^\circ)^2+(28sin36^\circ+10sin22^\circ))\approx37.78\approx38`

Find the direction (angle):

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

Therefore, C is the best answer