Question

Rachel rides her bicycle due east at 12 kilometers per hour. Amos rides his

bicycle due north at 16 kilometers per hour. If they left from the same point at

the same time, how far apart will they be after 3 hours?

Answer 1

To do the question we first draw a diagram such as

Both Rachel and Amos left from the same point $O.$

Rachel rides her bycycle due east at $12km/hour$ and after $3$ hours he will be at the point $A.$

Therefore Rachel rides along the path $\overrightarrow{OA}$ and her distance is $|\overrightarrow{OA}|$ .

$\therefore |\overrightarrow{OA}|=(12×3)=36$

Similarly Amos rides his bycycle due north at $16km/hour$ and after $3$ hours he will be at the point $B.$

Therefore Amos rides along the path $\overrightarrow{OB}$ and her distance is $|\overrightarrow{OB}|$

$\therefore |\overrightarrow{OB}|=(16×3)=48$

By vector triangle law,

$|\overrightarrow {AB}|^2=|\overrightarrow {OA}|^2+|\overrightarrow {OB}|^2+|\overrightarrow {OA}||\overrightarrow {OB}|cos\theta$ [ $\theta$ is the angle between $\overrightarrow {OA}$ and $\overrightarrow {OB}$

$=(36)^2+(48)^2$ [ here $\theta =90°$]

$=3600$

$\therefore |\overrightarrow {AB}|=\sqrt{3600}=60$

So after $3$ hours they will be $60km$ apart.