A fisherman is 7 miles west and 3 miles north of the marina. To head directly back to the marina, what distance and direction must he travel? Include both angle value and how it is measured in the answer. Give both distance and direction to the nearest tenth.

Now we can see that a right-angle triangle has been made. So to find h, which represents the hypotenuse of the right-angle above, we use Pytagoras theorem. we have

[tex]\begin{gathered} \text{hyp}^2=opp^2+adj^2 \\ h^2=3^2+7^2 \\ h^2=9+49 \\ h^2=58 \\ h=\sqrt[]{58} \\ h=7.615773 \\ h=7.6\text{ miles } \end{gathered}[/tex]

Hence his distance from the marina is 7.6 miles to the nearest tenth.

To find his direction, let us find the acute angle

From SOHCAHTOA, we have that

[tex]\begin{gathered} \tan \theta=\frac{opposite}{\text{adjacent}} \\ \tan \theta=\frac{3}{7} \\ \theta=\tan ^{-1}\frac{3}{7} \\ \theta=23.19859 \\ \theta=23.2\text{ } \end{gathered}[/tex]

So, from the diagram, his bearing is 90 degree + the angle theta. So we have

[tex]\begin{gathered} 90+23.2 \\ =113.2\degree \end{gathered}[/tex]

So his distance is 7.6 miles on a bearing of 113.2 degrees

You can also subtract 23.2 from 90 degree to get his direction in north-west, we have

[tex]\begin{gathered} 90-23.2=66.8\degree \\ =N66.8\degree W \end{gathered}[/tex]

So we can say his distance is 7.6 miles on a bearing of

[tex]N66.8\degree W[/tex]