A ship starts at port P, sails 226 deg (T) for a distance of 120 km. 1. How far south of P is it?
2. How far west of P is it?
south: 120 cos46°
west: 120 sin46°
To determine how far south and west the ship is from port P, we need to break down the vector movement based on the given information.
Step 1: Convert the given angle to the reference angle
To get the reference angle, subtract the given angle from 360 degrees.
Reference angle = 360 - 226 = 134 degrees
Step 2: Calculate the distance travelled south
Using trigonometry, we can determine the distance travelled south using the reference angle.
Distance south = distance * sin(reference angle)
Distance south = 120 km * sin(134 degrees) ≈ 100 km
Step 3: Calculate the distance travelled west
Using trigonometry, we can determine the distance travelled west using the reference angle.
Distance west = distance * cos(reference angle)
Distance west = 120 km * cos(134 degrees) ≈ -61 km
Note: The negative sign indicates that the ship is west of the starting point.
Therefore, the answers to the questions are:
1. The ship is approximately 100 km south of port P.
2. The ship is approximately 61 km west of port P.
To find the distance south of port P, we need to calculate the component of the distance traveled in the southward direction. To do this, we can use trigonometric functions, specifically the sine function, as it relates the angle and the ratio of the opposite side (distance south) to the hypotenuse (total distance traveled).
1. To find how far south of P the ship is located, we use the following formula:
Distance south = Total distance * sin(angle)
In this case, the total distance traveled is 120 km and the angle is 226 degrees. However, trigonometric functions usually take angles in radians rather than degrees, so we need to convert 226 degrees to radians by using the conversion factor:
1 radian = 180 degrees / π
So, 226 degrees can be converted to radians as:
226 degrees * (π / 180 degrees) = 226π / 180 radians
Now we can substitute the values into the formula:
Distance south = 120 km * sin(226π / 180 radians)
To find the distance west of port P, we can use the cosine function, as it relates the angle and the ratio of the adjacent side (distance west) to the hypotenuse (total distance traveled).
2. To find how far west of P the ship is located, we use the following formula:
Distance west = Total distance * cos(angle)
Again, we need to convert the angle to radians:
226 degrees * (π / 180 degrees) = 226π / 180 radians
Now we can substitute the values into the formula:
Distance west = 120 km * cos(226π / 180 radians)
Calculating the values above will give you the specific distances south and west of port P.