We have been given that a ship is sailing through the water in the English Channel with a velocity of 22 knots along a bearing of 157°.
Further we have been given that current has a velocity of 5 knots along a bearing of 213°.
Therefore, angle between the direction of ship and direction of current will be
[tex]\theta = 213 - 157 = 56^{0}[/tex]
We can find the magnitude of resultant by using parallelogram law of vectors.
[tex]R=\sqrt{P^{2}+Q^{2}+2PQcos(\theta)}[/tex]
Upon substituting [tex]P=22, Q = 5 \text{ and }\theta = 56[/tex] in this formula, we get
[tex]R=\sqrt{22^{2}+5^{2}+2\cdot 22\cdot 5cos(56)}\\
R=\sqrt{484+25+220\cdot0.55919}\\
R=\sqrt{632.0224}\\
R=25.14 \text{ knots}[/tex]
Therefore, resultant velocity of the ship is 25.14 knots.
We find the angle of resultant from P, that direction of ship using the formula
[tex]\alpha = arctan(\frac{Qsin(\theta)}{P+Qcos(\theta)})[/tex]
Upon substituting the values, we get
[tex]\alpha = arctan(\frac{5sin(56)}{22+5cos(56)})\\
\alpha = arctan(\frac{4.14518}{24.79596})\\
\alpha = arctan(0.16717)\\
\alpha = 9.49^{0}[/tex]
Therefore, bearing of the resultant is [tex]157+9.49 = 166.49^{0}[/tex]
Hence, option (A) is the correct choice!
