Two boat leave a port at the same time.the first boat travel at 15km/hr on a bearing of 135°while the second travel at 20km/hr on a bearing 63°.if after 2hours ,the second boat is directly north of the first boat , calculate their distance apart
draw a diagram, then use the law of cosines
To solve this problem, we can use the concept of vectors and vector addition.
First, let's find the displacement vectors for each boat after 2 hours.
For the first boat:
Distance traveled = speed × time = 15 km/hr × 2 hrs
Displacement vector = 15 km/hr × 2 hrs × (cos(135°), sin(135°))
= 30 × (-0.707, 0.707) km
For the second boat:
Distance traveled = speed × time = 20 km/hr × 2 hrs
Displacement vector = 20 km/hr × 2 hrs × (cos(63°), sin(63°))
= 40 × (0.448, 0.894) km
Next, we can find the position of the second boat relative to the first boat by subtracting their displacement vectors.
Relative displacement vector = displacement vector of the second boat - displacement vector of the first boat
= (40 × 0.448, 40 × 0.894) km - (30 × -0.707, 30 × 0.707) km
Simplifying this gives:
Relative displacement vector = (17.92, 35.68) km + (21.21, -21.21) km
= (17.92 + 21.21, 35.68 - 21.21) km
= (39.13, 14.47) km
Finally, we can calculate the distance between the two boats using the Pythagorean Theorem.
Distance = √(x^2 + y^2)
= √((39.13)^2 + (14.47)^2)
≈ √(1530.35 + 209.60)
≈ √1739.95
≈ 41.68 km
Therefore, the two boats are approximately 41.68 km apart after 2 hours.
To calculate the distance between the two boats after 2 hours, we need to find the coordinates of each boat after that time.
First, let's find the coordinates of the first boat using its bearing and speed.
The bearing of 135° means that the boat is moving in the southeast direction. Using trigonometry, we can find the displacement along the x-axis (horizontal) and y-axis (vertical) after 2 hours.
Since the boat is traveling at 15 km/hr for 2 hours, the displacement along the x-axis is:
Displacement_x1 = speed * cos(bearing) * time = 15 * cos(135°) * 2
And the displacement along the y-axis is:
Displacement_y1 = speed * sin(bearing) * time = 15 * sin(135°) * 2
Next, let's find the coordinates of the second boat. The bearing of 63° means that the boat is moving in the northeast direction.
Similarly, using trigonometry, we can find the displacement along the x-axis and y-axis after 2 hours.
Displacement_x2 = speed * cos(bearing) * time = 20 * cos(63°) * 2
Displacement_y2 = speed * sin(bearing) * time = 20 * sin(63°) * 2
Now that we have the coordinates of both boats, we can calculate the distance between them using the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Substituting the respective values and calculating the distance will give us the answer.