A boat sails 4km on a bearing of 038 degree and then 5kmon a bearing of 067 degree.
How far is the boat from its starting point?
Calculate the bearing of the boat from is starting point
you sail on a heading; you take a bearing.
4cis52° + 5cis23° = 8.7cis36°
so the boat is 8.7 km on a bearing of 54°
To find the distance the boat is from its starting point, we can use the Law of Cosines.
Step 1: Draw a diagram and label the given information.
- Let A be the starting point of the boat's journey.
- Let B be the ending point of the boat's journey after sailing 4km on a bearing of 038 degrees.
- Let C be the ending point of the boat's journey after sailing an additional 5km on a bearing of 067 degrees.
- Let a be the distance from A to C (the distance we want to find).
- Let b be the distance from A to B (4km).
- Let c be the distance from B to C (5km).
Step 2: Use the Law of Cosines to find side a.
The Law of Cosines states that:
c^2 = a^2 + b^2 - 2ab * cos(C)
Plugging in the given values:
5^2 = a^2 + 4^2 - 2 * a * 4 * cos(112 degrees)
Step 3: Solve for a.
25 = a^2 + 16 - 8a * cos(112 degrees)
a^2 - 8a * cos(112 degrees) + 9 = 0
Using the quadratic formula:
a = (8 * cos(112 degrees) ± √((8 * cos(112 degrees))^2 - 4 * 1 * 9)) / (2 * 1)
a ≈ (8 * cos(112 degrees) ± √(64 * cos^2(112 degrees) - 36)) / 2
a ≈ (8 * cos(112 degrees) ± √(36 * (cos^2(112 degrees) - 1))) / 2
Since cos^2(112 degrees) - 1 = -sin^2(112 degrees), we know that:
a ≈ (8 * cos(112 degrees) ± 6 * sin(112 degrees)) / 2
a ≈ 4 * (2 * cos(112 degrees) ± 3 * sin(112 degrees))
Using a calculator, we can find the values of cos(112 degrees) and sin(112 degrees):
cos(112 degrees) ≈ -0.42262
sin(112 degrees) ≈ 0.90631
Plugging in these values:
a ≈ 4 * (2 * (-0.42262) ± 3 * 0.90631)
a ≈ 4 * (-0.84524 ± 2.71893)
Taking both values of a:
a ≈ 4 * (-3.56417) ≈ -14.25667 (negative value and does not make sense in this context)
a ≈ 4 * (1.87369) ≈ 7.49476
Therefore, the distance the boat is from its starting point is approximately 7.495km.
To calculate the bearing of the boat from its starting point, we can use the Law of Sines.
Step 1: Draw a diagram and label the given information.
- Let A be the starting point of the boat's journey.
- Let B be the ending point of the boat's journey after sailing 4km on a bearing of 038 degrees.
- Let C be the ending point of the boat's journey after sailing an additional 5km on a bearing of 067 degrees.
- Let α be the angle at A (the bearing we want to find).
- Let b be the distance from A to B (4km).
- Let c be the distance from B to C (5km).
Step 2: Use the Law of Sines to find angle α.
The Law of Sines states that:
sin(α) / b = sin(C) / c
Plugging in the given values:
sin(α) / 4 = sin(67 degrees) / 5
Step 3: Solve for α.
sin(α) = 4 * sin(67 degrees) / 5
α = arcsin(4 * sin(67 degrees) / 5)
Using a calculator, we can find the value of arcsin(4 * sin(67 degrees) / 5):
α ≈ 33.102 degrees
Therefore, the bearing of the boat from its starting point is approximately 33.102 degrees.
To find the distance and bearing of the boat from its starting point, we can use trigonometry and vector addition.
Step 1: Convert the bearings to compass angles
The bearing of 038 degrees can be converted to a compass angle by adding 360 degrees, giving us 398 degrees. Similarly, the bearing of 067 degrees becomes 067 degrees.
Step 2: Convert the compass angles to radians
To use trigonometric functions, we need to convert the angles from degrees to radians. To do this, we multiply the angle in degrees by π/180.
In radians, 398 degrees becomes (398 * π)/180, and 067 degrees becomes (067 * π)/180.
Step 3: Calculate the x and y components of the boat's displacement
Since the boat travels on different bearings in two different stages, we need to calculate the x and y components of displacement for each stage.
For the first stage (4 km on a bearing of 398 degrees), the x component is given by:
x1 = 4 * cos((398 * π)/180)
The y component is given by:
y1 = 4 * sin((398 * π)/180)
For the second stage (5 km on a bearing of 067 degrees), the x component is given by:
x2 = 5 * cos((067 * π)/180)
The y component is given by:
y2 = 5 * sin((067 * π)/180)
Step 4: Calculate the total x and y components of displacement
To find the total displacement, we add the x and y components calculated in step 3 for each stage.
The total x component is:
x_total = x1 + x2
The total y component is:
y_total = y1 + y2
Step 5: Calculate the distance from the starting point
The distance from the starting point can be found using the Pythagorean theorem. We square the x_total and y_total, sum them up, and then find the square root.
distance = √(x_total^2 + y_total^2)
Step 6: Calculate the bearing of the boat from the starting point
The bearing of the boat from the starting point can be found using the inverse tangent function. We take the ratio of the y_total to x_total and calculate its angle in degrees.
bearing = arctan(y_total / x_total) * (180/π)
Now that we know the steps, let's calculate the distance and bearing.
Step 1: Convert the bearings to compass angles
398 degrees and 067 degrees.
Step 2: Convert the compass angles to radians
(398 * π)/180 and (067 * π)/180
Step 3: Calculate the x and y components of the boat's displacement
For the first stage:
x1 = 4 * cos((398 * π)/180)
y1 = 4 * sin((398 * π)/180)
For the second stage:
x2 = 5 * cos((067 * π)/180)
y2 = 5 * sin((067 * π)/180)
Step 4: Calculate the total x and y components of displacement
x_total = x1 + x2
y_total = y1 + y2
Step 5: Calculate the distance from the starting point
distance = √(x_total^2 + y_total^2)
Step 6: Calculate the bearing of the boat from the starting point
bearing = arctan(y_total / x_total) * (180/π)
Plugging in the values and calculating each step will give us the final answers.