A plane flies 430 miles at a bearing of N27°E, then turns and flies N63°E for 135 miles. Find the distance and the bearing from the starting point to the ending point ?
I found distance to be 450.69 mi by using A^2+B^2=C^2 because it is a right triangle. I am confused where to place the bearing angle therefore how to solve for the bearing
these are headings, not bearings.
I do not get a right triangle. I get an angle of 27+90+27= 144 at the turning point.
To get the final location, convert each side to its x- and y-components.
Add them up, and you can then get the angle θ using tanθ = y/x
and then the bearing is N(90-θ)E as usual
To determine the bearing from the starting point to the ending point, we can use the concept of vector addition.
First, let's break down the initial flight into its north and east components. The bearing of N27°E means that the plane is flying 27 degrees east of north.
The north component can be found by calculating the sine of the angle: sin(27°) = (opposite side) / (hypotenuse) = N / 430.
Thus, the north component of the initial flight is N = 430 * sin(27°) = 199.17 miles.
The east component can be found by calculating the cosine of the angle: cos(27°) = (adjacent side) / (hypotenuse) = E / 430.
Thus, the east component of the initial flight is E = 430 * cos(27°) = 366.48 miles.
Now, let's move on to the second leg of the flight, which travels N63°E for 135 miles.
To find the north and east components for this leg, we can follow the same process.
The north component can be found by calculating the sine of the angle: sin(63°) = (opposite side) / (hypotenuse) = N / 135.
Thus, the north component of the second leg is N = 135 * sin(63°) = 122.19 miles.
The east component can be found by calculating the cosine of the angle: cos(63°) = (adjacent side) / (hypotenuse) = E / 135.
Thus, the east component of the second leg is E = 135 * cos(63°) = 64.48 miles.
To find the total north and east components, we can add the corresponding components from both legs.
The total north component is 199.17 + 122.19 = 321.36 miles.
The total east component is 366.48 + 64.48 = 430.96 miles.
Now, we have the total north and east components, we can calculate the distance and bearing from the starting point to the ending point using the formula:
Distance = √(north^2 + east^2)
Distance = √(321.36^2 + 430.96^2) = 539.63 miles (rounded to two decimal places)
To find the bearing, we can use the inverse tangent function to calculate the angle.
Bearing = tan^(-1)(east / north)
Bearing = tan^(-1)(430.96 / 321.36) = 53.25° (rounded to two decimal places)
Therefore, the distance from the starting point to the ending point is approximately 539.63 miles, and the bearing is N53.25°E.
To solve for the bearing, we can use trigonometry and some principles of vector addition. Let's break down the problem step by step:
1. Start by drawing a diagram to visualize the situation. Draw a coordinate grid and add the starting point, indicated as "S", and the ending point, indicated as "E". Label the coordinates for each point.
2. Calculate the x and y components of the first leg of the flight. The bearing N27°E can be interpreted as moving 27° eastward from North. In other words, it is moving in the direction of the positive x-axis by 27°.
The x-component of the first leg can be calculated as follows:
430 miles * cos(27°)
The y-component of the first leg can be calculated as follows:
430 miles * sin(27°)
Note: Make sure your calculator is set to degrees mode.
3. Next, calculate the x and y components of the second leg of the flight. The bearing N63°E indicates that the plane is moving 63° eastward from North. Therefore, the angle between the second leg and the positive x-axis will be (90 - 63) = 27°.
The x-component of the second leg can be calculated as follows:
135 miles * cos(27°)
The y-component of the second leg can be calculated as follows:
135 miles * sin(27°)
4. Now, we can add the x-components and y-components separately to find the total displacement in the x and y directions.
- Add the x-components: (x1 + x2)
- Add the y-components: (y1 + y2)
5. To find the distance from the starting point to the ending point, we can use the Pythagorean theorem. The distance (d) can be calculated as follows:
d = √((x1 + x2)^2 + (y1 + y2)^2)
From your calculations, it seems like you have correctly used the Pythagorean theorem and found the distance to be 450.69 miles. Now, to solve for the bearing:
6. We can use trigonometry to determine the bearing angle. The bearing angle (θ) can be calculated as follows:
θ = arctan((y1 + y2) / (x1 + x2))
7. Once you have calculated the bearing angle, you can express it in the standard form, such as N45°E or N45°W, depending on the direction from the starting point.
Now you should have the distance and the bearing from the starting point to the ending point.