A plane flies 120km on a bearing of 045 degree and then flies 150km due east. How far east of the starting point is the plane?
In my sketch the angle between their flights is 135°
So wouldn't we just have the cosine law??
x^2 = 120^2 + 150^2 - 2(120)(150)cos135°
carry on, take care with the cos135, since it will be negative
d = 120*sin45 + 150 =
since all we're interested in is the eastward distance, just add up the x-components of the two legs of the trip.
The first leg puts you 120 sin45° = 120/√2 km east
then add the next leg, which puts you another 150 km east
...
To find the distance the plane is east of the starting point, we can use trigonometry and vector addition.
First, let's break down the plane's movement into two components - north/south and east/west.
The plane flies 120km on a bearing of 045 degrees. This means that it travels 120km towards the northeast direction (45 degrees clockwise from due north). We can find the north/south and east/west components of this movement using trigonometry.
The north/south component can be found by calculating the cosine of the angle and multiplying it by the total distance traveled:
North/South component = 120km * cos(45 degrees)
Similarly, the east/west component can be found by calculating the sine of the angle and multiplying it by the total distance traveled:
East/West component = 120km * sin(45 degrees)
Next, the plane flies an additional 150km due east, which means it only increases its east/west position.
To find the final east/west distance, we can add the east/west component from the first leg and the distance traveled due east:
Final East/West distance = East/West component (from first leg) + 150km
Note: The north/south component is not relevant to finding the east/west distance as the plane only traveled east during the second leg.
Now, let's calculate it step-by-step:
North/South component = 120km * cos(45 degrees)
East/West component = 120km * sin(45 degrees)
Final East/West distance = East/West component + 150km
Using a calculator:
North/South component = 120km * cos(45 degrees) ≈ 84.85km
East/West component = 120km * sin(45 degrees) ≈ 84.85km
Final East/West distance = 84.85km + 150km = 234.85km
Therefore, the plane is approximately 234.85km east of the starting point.