# A skier skis 7.40 km in the direction 45.0 east of south, then 2.80 km in the direction 30.0° north of east. and finally 5.20 km in the direction 22.0° west of north. Show these displacements on a diagram? How far is the skier from the starting point?

Jun 24, 2018

$\approx 9.82497 \ldots \text{degrees East North}$

Magnitude $5.7944912 \overline{12}$

#### Explanation:

Often a good idea to draw a very quick and rough sketch to give you an idea of what you are having to deal with.

Set $\rightarrow$ as positive thus $\leftarrow$ is negative

Set $\uparrow$ as positive thus $\downarrow$ is negative

$\textcolor{b r o w n}{\text{Consider the horizontal}}$

$A {B}_{h} = + \left[7.4 \times \sin \left({45}^{o}\right)\right] \approx + 5.23159 . .$
$B {C}_{h} = + \left[2.8 \times \cos \left({30}^{o}\right)\right] \approx + 2.42487 . .$
$C {D}_{h} = - \left[5.2 \times \sin \left({22}^{o}\right)\right] \approx - 1.94795 . .$

Sum $\approx + 5.70950 \ldots .$

$\textcolor{b r o w n}{\text{Consider the vertical}}$

$A {B}_{h} = + \left[7.4 \times \cos \left({45}^{o}\right)\right] \approx - 5.23259 \ldots . .$
$B {C}_{h} = + \left[2.8 \times \sin \left({30}^{o}\right)\right] = + 1.4 \text{ } \ldots$
$C {D}_{h} = - \left[5.2 \times \cos \left({22}^{o}\right)\right] \approx + 4.82135 \ldots$

Sum $\approx + 0.98876 \ldots .$

So we end up with:

$\beta = {\tan}^{- 1} \frac{0.988765 \ldots}{5.709507 . .} \approx 9.82497 \ldots \text{degrees East North}$

$\text{Resultant } r \approx \sqrt{{\left(0.988765 . .\right)}^{2} + {\left(5.709507 . .\right)}^{2}}$

$r = 5.794491212 \overline{12}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{This is a repeating decimal so a rational number}}$

$\textcolor{b r o w n}{\text{Converting to an exact fractional answer}}$

Set ${x}_{1} = 5.794491212 \overline{12}$

Set${x}_{2} = 0.794491212 \overline{12}$

$10000000 {x}_{2} = 7944912.1212 \overline{12}$
ul(color(white)("d0")100000x_2=color(white)("00")79449.1212bar(12)larr" Subtract")
$\textcolor{w h i t e}{\text{d}} 9900000 {x}_{2} = 7865463$

${x}_{2} = \frac{7865463}{9900000}$

$r = {x}_{1} = 5 \textcolor{w h i t e}{\text{d")7865463/9900000 color(red)(larr" What an awful number!!!}}$

$\textcolor{red}{\text{I'm sticking with the decimal!!}}$