Erica bought a boat and wants to test it. She wants to travel 3141 miles Northwest of her current position forming an angle of 130. The boat will be traveling at 150 mph. She gets ready and fuels her boat for 3150 miles then realizes that there is a strong 40 miles/hour wind blowing from the Southwest at 40 degrees(positive x axis is 0 degrees). She needs to employ advanced mathematics to help her fuel her boat appropriately. You need to calculate the total distance she will travel assuming the wind continues to blow at 40 mph and the time for the trip. Use Vectors for the solutions and then use the law of sines/cosines as another solution. I need both the workings.
Using the law of sines/cosines I'm getting ~4300 and with vectors, I'm getting ~76000 so there is a big disparity between the solutions even though they should be the same. Working so far: https://i.stack.imgur.com/M43Jc.jpg
First of all, a comment on your law of sines solution. I have redrawn (and corrected) your picture using GeoGebra, to be able to talk about it more comprehensively. Please note that it is only a sketch.
As you can see, the black angle I drew inside the triangle, differs from your black angle on the same spot, with mine being the correct one because of Z-angles. Thus you must first determine the non-black part of angle C before you can apply your method, which was fine otherwise, except that the value of the green angle is also wrong: what angle does AB make with the x-axis, viewed counterclockwise (thus the non-green angle)? Substract that from $180$ degrees to get the proper green angle.
Now let's take a look at your vector solution. You start by constructing the vectors from their length and their angle, which is good. Note that you know the green vector, not the blue one, since the green one represents the direction the person has to steer in to compensate for the blowing wind and in the question, it says that the person is traveling at 150 mph. Now let's take a look at the construction of the red vector. The construction of the green one is similar, just make sure to use the proper angle! I have provided a visualisation of the construction with the following picture:
Note that $\cos(40^\circ)=\frac{x}{40}$ and $\sin(40^\circ)=\frac{y}{40}$, so the red vector is given by $( 40\cdot\cos(40^\circ),40\cdot\sin(40^\circ) )$ instead of $(40+\cos(40^\circ),40+\sin(40^\circ))$, as in your answer. I notice that, for the computation of the blue vector, you use the distance traveled, not the speed, which explains the large difference in your answers (even though the way you used the blue vector is wrong in itself).
Now all you have to do is to compute the green vector using the proper speed; the rest of your method seems to be correct. I'm not quite sure I understand the "(21)" part in your answer, so I'll say this: your final answer should have the form $\sqrt{(x_{green}+x_{red})^2+(y_{green}+y_{red})^2}$.
I hope this helps. If some parts are still unclear to you, or if you notice a mistake, leave a comment on this answer and I will edit it accordinly.