a ship travels 60 miles due east, then adjusts its course northward. after travelling 80 miles in the new direction, the ship is 139 miles from its point of departure. calculate the ship's bearing.
To calculate the ship's bearing, we can use the concept of right triangles and trigonometric functions.
First, let's define the distance traveled in the northward direction as "y."
Using the Pythagorean theorem, we can write the following equation:
60^2 + y^2 = 80^2
Simplifying this equation, we get:
3600 + y^2 = 6400
Subtracting 3600 from both sides:
y^2 = 2800
Taking the square root of both sides:
y = √2800
y ≈ 52.92 miles
Now, let's find the bearing of the ship. We can use inverse tangent to calculate it.
tan(θ) = opposite / adjacent
tan(θ) = y / 60
Taking the inverse tangent (arctan) of both sides:
θ ≈ arctan(y / 60)
θ ≈ arctan(52.92 / 60)
Using a calculator, we find:
θ ≈ arctan(0.882)
The approximate bearing of the ship is θ ≈ 44.68°.
To calculate the ship's bearing, we can use trigonometry and the given information about the ship's travel distance. The bearing refers to the angle measured clockwise from the north direction to the ship's course.
Let's break down the problem step by step:
1. Draw a diagram: Visualizing the problem is helpful. Draw a coordinate plane with a starting point, and then plot the movements of the ship.
2. Analyze the given information: The ship initially travels 60 miles due east. This means the ship moves straight along the positive x-axis. The ship then adjusts its course northward and travels 80 miles in the new direction. The resulting position is 139 miles away from the starting point.
3. Calculate the x and y components: Determine the change in x and y coordinates from the starting point to the resulting position using the given information.
Since the ship initially travels east, the x-component of the resulting position is +60 miles. Then, the ship adjusts its course northward, resulting in a change in the y-component. The y-component of the resulting position is +80 miles.
4. Use the Pythagorean theorem: Since we have the x and y components, we can use the Pythagorean theorem to calculate the distance from the starting point to the resulting position.
The distance d can be calculated as:
d = sqrt(x^2 + y^2)
Substituting the values:
d = sqrt(60^2 + 80^2)
Calculating the distance:
d = sqrt(3600 + 6400)
d = sqrt(10000)
d = 100 miles
5. Calculate the ship's bearing: To determine the ship's bearing, we need to find the angle between the north direction and the line connecting the starting point to the resulting position.
Using trigonometry, we can determine the angle θ:
θ = arctan(y / x)
Substituting the values:
θ = arctan(80 / 60)
Calculating the angle:
θ = arctan(4/3)
θ ≈ 53.13 degrees
Therefore, the ship's bearing would be approximately 53.13 degrees from the north direction.
we want the ship's heading, not bearing.
using the law of cosines,
60^2 + 139^2 - 2(60)(139)cosθ = 80^2
now just find θ (that is the direction relative to due east, not the ship's heading)