Please check ans: A light plane has a cruising speed of....

[val1]

Please can you check this?

A light aeroplane has a cruising speed in still air of $\displaystyle 40{\rm ms}^{ - 1}$. It is pointed in the direction $\displaystyle {\rm S21}^ \circ W$, but flies in a wind of speed $\displaystyle {\rm 18ms}^{{\rm - 1}}$ from the direction $\displaystyle {\rm N73}^ \circ W$.

Take i to be $\displaystyle 1{\rm ms}^{ - 1}$ due east and j to be $\displaystyle 1{\rm ms}^{ - 1}$ due north. Take $\displaystyle v$ to be the resultant velocity of the aeroplane.

Through previous calculations it has been shown that the resultant velocity $\displaystyle v$ of the aeroplane is given in component form by

$\displaystyle v = 2.8788i - 42.6059j$

a) Find the overall speed $\displaystyle \left| v \right|$ of the aeroplane (to 4 s.f) and its direction of travel, as a bearing (with the angle to 1 d.p)

b) How long does it take the aeroplane to travel 1 kilometre? How far east does it travel in this time?. Give you answers to 3 s.f.

Here is my working:

a) The overall speed=$\displaystyle \left| v \right| = \sqrt {2.8722^2 + 42.6059^2 } = 42.70{\rm (4 s}{\rm .f)}$

The direction of travel $\displaystyle \approx \arctan \left| {\frac{{42.6059}}{{2.8788}}} \right| \approx 86.1{\rm (1 d}{\rm .p)}$

This is the 4th quadrant so is -86.1, which corresponds to bearing $\displaystyle {\rm S 3.9}^ \circ E$

b) The j component is the rate of progress. It travels 42.6059 ms/s so to travel 1000m takes $\displaystyle \frac{{{\rm 1000}}}{{{\rm 42}{\rm .6059}}} \approx 23.4709{\rm secs}$

In that time it would have travelled 23.4709 x 2.8788 = 67.6m east or approx 70.0m (3 s.f)

Please can you check my calculations, especially for part b) and I'm not sure about how to express the significant figures?

Thank you