two boats have a port at the same time. the first travels at 15km/h on a bearing 135° while the second travels at 20km/h on a bearing of 063°. if after 2hrs, the second boat is directly north of the first boat, calculate their distance apart
for each boat, distance = speed * time
Now draw a diagram, and note the angle θ between the two headings.
Finally, use the law of cosines:
c^2 = a^2 + b^2 - 2ab cosθ
To calculate the distance between the two boats, we can use the formulas for calculating the coordinates of a point after a given time and speed.
Let's start by calculating the coordinates of the first boat after 2 hours.
Speed of the first boat = 15 km/h
Time = 2 hours
To calculate the displacement (change in position), we can use the formula:
Displacement = Speed * Time
Displacement = 15 km/h * 2 hours = 30 km
Next, we need to calculate the change in x and y coordinates for the first boat. Since the bearing is given in degrees, we need to convert it to radians.
Bearing = 135°
Angle in radians = (Bearing * pi) / 180°
Angle in radians = (135 * pi) / 180° = 2.3562 rad
Now, we can use trigonometry to calculate the change in x and y coordinates.
Change in x = Displacement * cos(Angle in radians)
Change in x = 30 km * cos(2.3562) ≈ -21.2132 km
Change in y = Displacement * sin(Angle in radians)
Change in y = 30 km * sin(2.3562) ≈ 21.2132 km
Therefore, after 2 hours, the first boat has moved approximately -21.2132 km in the x direction and 21.2132 km in the y direction.
Now, let's calculate the coordinates of the second boat after 2 hours.
Speed of the second boat = 20 km/h
Time = 2 hours
Displacement = 20 km/h * 2 hours = 40 km
Bearing = 63°
Angle in radians = (Bearing * pi) / 180°
Angle in radians = (63 * pi) / 180° ≈ 1.0996 rad
Change in x = Displacement * cos(Angle in radians)
Change in x = 40 km * cos(1.0996) ≈ 19.2388 km
Change in y = Displacement * sin(Angle in radians)
Change in y = 40 km * sin(1.0996) ≈ 35.2372 km
Therefore, after 2 hours, the second boat has moved approximately 19.2388 km in the x direction and 35.2372 km in the y direction.
Now, we can calculate the distance between the two boats using the distance formula:
Distance = sqrt((Change in x)^2 + (Change in y)^2)
Distance = sqrt((-21.2132)^2 + (21.2132 + 35.2372)^2)
Distance ≈ sqrt(450 + 2170)
Distance ≈ sqrt(2620)
Distance ≈ 51.18 km
Therefore, the two boats are approximately 51.18 km apart after 2 hours.
To calculate the distance between the two boats, we can use the concept of relative velocity. We need to find the components of the velocities of both boats in the same direction, which in this case will be north.
Let's start by finding the components of the velocities for each boat.
Boat 1:
Velocity: 15 km/h
Bearing: 135°
To find the northward component of the velocity, we can use the equation: Velocity * sin(Bearing).
Northward component of Boat 1's velocity = 15 km/h * sin(135°)
Now, let's calculate it:
Northward component of Boat 1's velocity = 15 km/h * (-0.707)
= -10.606 km/h (negative value indicates southward direction)
Boat 2:
Velocity: 20 km/h
Bearing: 063°
Similarly, to find the northward component of the velocity for Boat 2, we can use the equation: Velocity * sin(Bearing).
Northward component of Boat 2's velocity = 20 km/h * sin(63°)
Now, let's calculate it:
Northward component of Boat 2's velocity = 20 km/h * (0.891)
= 17.82 km/h (positive value indicates northward direction)
After 2 hours, we need to calculate the distance between the two boats. Since the northward component of Boat 2's velocity is directly north of Boat 1, their distance apart is equal to the northward component of Boat 2's velocity multiplied by the time (2 hours).
Distance apart = Northward component of Boat 2's velocity * Time
Now, let's calculate it:
Distance apart = 17.82 km/h * 2 hours
= 35.64 km
Therefore, the distance between the two boats after 2 hours is 35.64 km.