A boat sails 4km on a bearing of 038 and then 5km on a bearing of 067(a)how far is the boat from its starting point (b)calculate the bearing of the boat from its starting point
A boat sails 4km on a HEADING of 038 and then 5km on a HEADING of 067(a)how far is the boat from its starting point (b)calculate the bearing (Yes, used correctly) of the boat from its starting point .
Compass angles measured clockwise from North(Y axis)
NORTH ( Y ) direction: 4 cos 38 + 5 cos 67 = 3.15 + 1.95 = 5.10
EAST (X) direction 4 sin 38 + 5 sin 67 = 2.46 + 4.60 = 7.06
distance = sqrt (5.10^2 + 7.06^2)
bearing from origin; tan angle = (East/North) = 7.06/5.10 = 1.38
tan^-1 (1.38) = 54 deg East of North
(a) Well, if we draw a diagram, it's clear that we have a right-angled triangle formed by the boat's path. So, we can use the Pythagorean theorem to figure out the distance from the starting point:
Distance^2 = (4km)^2 + (5km)^2
Distance^2 = 16km^2 + 25km^2
Distance^2 = 41km^2
Distance = √41km (approximately 6.4km)
So, the boat is approximately 6.4km away from its starting point.
(b) To calculate the bearing of the boat from its starting point, we can use some trigonometry. The angle θ formed by the boat's path can be found using the inverse tangent function:
θ = tan^(-1)(opposite/adjacent)
θ = tan^(-1)(4km/5km)
θ = tan^(-1)(4/5)
θ ≈ 38.66 degrees
However, this is not the bearing we need. The bearing is measured clockwise from due north. So, we need to subtract this angle from 360 degrees to get the bearing:
Bearing = 360 degrees - θ
Bearing ≈ 360 degrees - 38.66 degrees
Bearing ≈ 321.34 degrees
So, the boat's bearing from its starting point is approximately 321.34 degrees.
To solve this problem, we can break it down into two components: the horizontal and vertical displacements.
Let's first calculate the horizontal displacement:
Horizontal displacement = (4 km * sin(90 - 38)) + (5 km * sin(90 - 67))
Horizontal displacement = (4 km * sin(52)) + (5 km * sin(23))
Horizontal displacement = (4 km * 0.788) + (5 km * 0.390)
Horizontal displacement = 3.152 km + 1.95 km
Horizontal displacement = 5.102 km
Next, let's calculate the vertical displacement:
Vertical displacement = (4 km * cos(90 - 38)) + (5 km * cos(90 - 67))
Vertical displacement = (4 km * cos(52)) + (5 km * cos(23))
Vertical displacement = (4 km * 0.618) + (5 km * 0.924)
Vertical displacement = 2.472 km + 4.62 km
Vertical displacement = 7.092 km
Now, we can calculate the total displacement using the Pythagorean theorem:
Total displacement = √(horizontal displacement^2 + vertical displacement^2)
Total displacement = √(5.102 km^2 + 7.092 km^2)
Total displacement = √(25.98 km^2 + 50.165 km^2)
Total displacement = √(76.145 km^2)
Total displacement = 8.733 km
(a) The boat is approximately 8.733 km from its starting point.
To calculate the bearing of the boat from its starting point, we will use trigonometric functions:
Bearing = arctan(vertical displacement / horizontal displacement)
Bearing = arctan(7.092 km / 5.102 km)
Bearing ≈ 54.03 degrees
(b) The bearing of the boat from its starting point is approximately 54.03 degrees.
To solve this problem, we can break it down into two main steps:
Step 1: Calculate the displacement from the starting point to the ending point.
Step 2: Calculate the distance and bearing from the starting point to the ending point.
Step 1: Calculate the displacement
To find the displacement, we need to calculate the x and y components of each leg of the journey. We can use basic trigonometry to find these components.
For the first leg:
Distance = 4 km
Bearing = 038
To calculate the x-component, we can use the cosine of the bearing:
Cos(38°) = x / 4 km
x = 4 km * Cos(38°)
To calculate the y-component, we can use the sine of the bearing:
Sin(38°) = y / 4 km
y = 4 km * Sin(38°)
For the second leg:
Distance = 5 km
Bearing = 067
To calculate the x-component, we can use the cosine of the bearing:
Cos(67°) = x / 5 km
x = 5 km * Cos(67°)
To calculate the y-component, we can use the sine of the bearing:
Sin(67°) = y / 5 km
y = 5 km * Sin(67°)
Step 2: Calculate the distance and bearing
To find the distance between the starting point and the ending point, we can use the Pythagorean theorem:
Distance = √(x^2 + y^2)
The bearing can be calculated using the inverse tangent function:
Bearing = atan(y / x)
Let's calculate the displacements and then the distance and bearing.
For the first leg:
x1 = 4 km * Cos(38°)
y1 = 4 km * Sin(38°)
For the second leg:
x2 = 5 km * Cos(67°)
y2 = 5 km * Sin(67°)
Now let's calculate the total displacement:
Total x displacement = x1 + x2
Total y displacement = y1 + y2
For the distance:
Distance = √((Total x displacement)^2 + (Total y displacement)^2)
For the bearing:
Bearing = atan(Total y displacement / Total x displacement)
Now you can calculate the distance and bearing of the boat from its starting point.