To determine the direction of the velocity of the boat, we need to calculate the angle [tex]\(\theta\)[/tex] that the resultant velocity vector makes with the y-axis.
### Step-by-Step Solution:
1. Identify the velocity components:
- The boat's velocity in the y-direction: [tex]\( v_y = 9.00 \, \text{m/s} \)[/tex]
- The current's velocity in the x-direction: [tex]\( v_x = 2.50 \, \text{m/s} \)[/tex]
2. Use trigonometry to find the angle [tex]\(\theta\)[/tex]:
- The angle [tex]\(\theta\)[/tex] can be found using the tangent function, which relates the opposite side (the velocity component in the x-direction) to the adjacent side (the velocity component in the y-direction).
[tex]\[
\tan(\theta) = \frac{v_x}{v_y}
\][/tex]
3. Calculate [tex]\(\theta\)[/tex]:
- Take the arctangent (inverse tangent) of the ratio of these velocities to find the angle [tex]\(\theta\)[/tex].
[tex]\[
\theta = \arctan\left(\frac{2.50 \, \text{m/s}}{9.00 \, \text{m/s}}\right)
\][/tex]
4. Convert [tex]\(\theta\)[/tex] from radians to degrees:
- Since the angle is typically desired in degrees, we convert from radians to degrees using the fact that [tex]\(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\)[/tex].
[tex]\[
\theta \approx 15.5241^{\circ}
\][/tex]
Therefore, the direction of the velocity of the boat is approximately [tex]\( \theta = 15.5241^{\circ} \)[/tex] from the y-axis, towards the direction of the current (x-axis).
