question #11 reference q.46960 a boat steers west straight across a river that has a current of 11.1 m/s…

Explanation:

Step1: Identify velocity components

The velocity of the boat in still - water is $v_{bw}=9.4$ m/s west and the velocity of the river current is $v_{r}=11.1$ m/s south. These are two perpendicular components of the boat's velocity in the river.

Step2: Use the Pythagorean theorem

The magnitude of the resultant velocity $v$ of the boat in the river is given by $v = \sqrt{v_{bw}^{2}+v_{r}^{2}}$. Substitute $v_{bw}=9.4$ m/s and $v_{r}=11.1$ m/s into the formula: [ \begin{align*} v&=\sqrt{(9.4)^{2}+(11.1)^{2}}\ &=\sqrt{88.36 + 123.21}\ &=\sqrt{211.57}\ &\approx14.6\text{ m/s} \end{align*} ]

Step3: Find the direction

The direction $\theta$ of the boat's motion relative to the west - direction can be found using the tangent function. $\tan\theta=\frac{v_{r}}{v_{bw}}$. Substitute $v_{bw}=9.4$ m/s and $v_{r}=11.1$ m/s: [ \begin{align*} \tan\theta&=\frac{11.1}{9.4}\ \theta&=\arctan\left(\frac{11.1}{9.4}\right)\ \theta&\approx49.7^{\circ}\text{ south of west} \end{align*} ]

Answer:

The magnitude of the boat's velocity in the river is approximately $14.6$ m/s and the direction is approximately $49.7^{\circ}$ south of west.