Question

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?

Answer 1

`~~9.82497..."degrees East North"`

Magnitude `5.7944912bar(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 `rarr` as positive thus `larr` is negative

Set `uarr` as positive thus `darr` is negative

`color(brown)("Consider the horizontal")`

`AB_h=+[7.4xxsin(45^o)]~~+5.23159..`
`BC_h=+[2.8xxcos(30^o)]~~+2.42487..`
`CD_h=-[5.2xxsin(22^o)]~~-1.94795..`

Sum `~~+5.70950....`

`color(brown)("Consider the vertical")`

`AB_h=+[7.4xxcos(45^o)]~~-5.23259.....`
`BC_h=+[2.8xxsin(30^o)]=+1.4" "...`
`CD_h=-[5.2xxcos(22^o)]~~+4.82135...`

Sum `~~+0.98876....`

So we end up with:

`beta =tan^(-1)(0.988765...)/(5.709507..)~~9.82497..."degrees East North"`

`"Resultant "r ~~ sqrt((0.988765..)^2+(5.709507..)^2)`

`r=5.794491212bar(12)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("This is a repeating decimal so a rational number")`

`color(brown)("Converting to an exact fractional answer")`

Set `x_1=5.794491212bar(12)`

Set`x_2=0.794491212bar(12)`

`10000000x_2=7944912.1212bar(12)`
`ul(color(white)("d0")100000x_2=color(white)("00")79449.1212bar(12)larr" Subtract")`
`color(white)("d")9900000x_2=7865463`

`x_2=7865463/9900000`

`r=x_1 = 5 color(white)("d")7865463/9900000 color(red)(larr" What an awful number!!!")`

`color(red)("I'm sticking with the decimal!!")`