To find the course the boat must travel to offset the flow of the current, you can use vector addition.
Let:
- \( \vec{V_b \) = velocity of the boat relative to the water
- \( \vec{V_c} \) = velocity of the current
- \( \vec{V_r} \) = resultant velocity of the boat (the intended path from A to B)
Given:
- The bearing from A to B is 045 degrees, which translates to \( \theta_r = 45^\circ \)
- The speed of the boat \( |V_b| = 30 , \text{km/h} \)
- The current speed \( |V_c| = 5 , \text{km/h} \)
- The current bearing is 315 degrees, translating to \( \theta_c = 315^\circ \)
You can express the velocities in terms of their components using trigonometric functions:
\[ \vec{V_b} = |V_b| \cdot (\cos(\theta_b), \sin(\theta_b)) = (30 \cdot \cos \theta_b, 30 \cdot \sin \theta_b) \] \[ \vec{V_c} = |V_c| \cdot (\cos(315^\circ), \sin(315^\circ)) = (5 \cdot \cos(315^\circ), 5 \cdot \sin(315^\circ)) \]
Since \( \cos(315^\circ) = \frac{\sqrt{2}}{2} \) and \( \sin(315^\circ) = -\frac{\sqrt{2}}{2} \):
\[ \vec{V_c} = (5 \cdot \frac{\sqrt{2}}{2}, 5 \cdot -\frac{\sqrt{2}}{2}) = \left(\frac{5\sqrt{2}}{2}, -\frac{5\sqrt{2}}{2}\right) \]
For the resultant velocity of the boat to go from A to B, we want:
\[ \vec{V_r} = \vec{V_b} + \vec{V_c} \]
The resultant velocity should have a direction of 045 degrees:
\[ \vec{V_r} = |V_r| \cdot (\cos(45^\circ), \sin(45^\circ)) = \left(\frac{50\sqrt{2}}{2}, \frac{50\sqrt{2}}{2}\right) \]
Thus, separating into components we get the following equations for the x and y components:
- \( 30 \cdot \cos \theta_b + \frac{5\sqrt{2}}{2} = \frac{50\sqrt{2}}{2} \)
- \( 30 \cdot \sin \theta_b - \frac{5\sqrt{2}}{2} = \frac{50\sqrt{2}}{2} \)
These two equations will allow you to solve for \( \theta_b \), the required course angle.
To encapsulate all that in one equation that represents \( \theta_b \) directly, we can express it in the following form:
\[ \tan \theta_b = \frac{\left( \frac{50\sqrt{2}}{2} + \frac{5\sqrt{2}}{2} \right)}{30 - \frac{5\sqrt{2}}{2}} \]
This equation allows you to solve directly for the course angle of the boat considering the offset from the current.