Sure! Let's solve the problem step by step.
### A. Find the distance travelled by each boat after 3 hours
1. Distance travelled by boat A:
- Speed of boat A: 35 kmph
- Time travelled: 3 hours
[tex]\[
\text{Distance travelled by boat A} = \text{Speed of boat A} \times \text{Time} = 35 \, \text{kmph} \times 3 \, \text{hours} = 105 \, \text{km}
\][/tex]
2. Distance travelled by boat B:
- Speed of boat B: 22 kmph
- Time travelled: 3 hours
[tex]\[
\text{Distance travelled by boat B} = \text{Speed of boat B} \times \text{Time} = 22 \, \text{kmph} \times 3 \, \text{hours} = 66 \, \text{km}
\][/tex]
So, after 3 hours:
- Boat A has travelled 105 km
- Boat B has travelled 66 km
3. Position calculation
- The bearing is given in degrees:
- Bearing of Boat A: 335 degrees
- Bearing of Boat B: 65 degrees
### Calculate the positions of each boat (x, y coordinates):
1. Converting bearings to radians:
[tex]\[
\text{Bearing A in radians} = \frac{335 \times \pi}{180}
\][/tex]
[tex]\[
\text{Bearing B in radians} = \frac{65 \times \pi}{180}
\][/tex]
2. Position of Boat A:
[tex]\[
x_A = 105 \times \cos \left(\frac{335 \times \pi}{180}\right)
\][/tex]
[tex]\[
y_A = 105 \times \sin \left(\frac{335 \times \pi}{180}\right)
\][/tex]
3. Position of Boat B:
[tex]\[
x_B = 66 \times \cos \left(\frac{65 \times \pi}{180}\right)
\][/tex]
[tex]\[
y_B = 66 \times \sin \left(\frac{65 \times \pi}{180}\right)
\][/tex]
4. Calculate the distance between the two boats after 3 hours:
[tex]\[
\text{Distance between boats} = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}
\][/tex]
The distance between the two boats after 3 hours is approximately 124.02 km.
### B. Calculate the bearing of A from B
To find the bearing of boat A from boat B, we calculate the angle relative to the north:
1. Calculate the change in coordinates:
[tex]\[
\Delta x = x_A - x_B
\][/tex]
[tex]\[
\Delta y = y_A - y_B
\][/tex]
2. Calculate the bearing:
[tex]\[
\text{Bearing of A from B} = \left(\text{arctan2}\left(\Delta y, \Delta x\right) \times \frac{180}{\pi} + 360\right) \mod 360
\][/tex]
The bearing of A from B is approximately 302.85 degrees.
Summarizing our results:
- After 3 hours, the distance between the two boats is approximately 124.02 km.
- The bearing of A from B is approximately 302.85 degrees.
