A fisherman leaves his home port and heads in the direction N 70 degree W. He travels 30 mi and reaches Egg Island. The next day he sails N 10 degree E for 50 mi, reaching Forrest Island.
a) Find the distance between the fisherman's home port and Forrest Island.
b) Find the bearing from Forrest Island back to his home port.
I got part (a) right. It is 62.6 miles. But I don't understand how to find the bearing. Please help. Thanks.
Compute separately the total distance travelled in the x (east) and y (north) directions. Call them X and Y
The total distance travelled is sqrt(X^2+Y^2)
The bearing (neasured east from north) is
arctan X/Y
X = 30*cos(70) = -15.8
Y = 30*sin(70) + 50*sin(10) = 58.2
Distance = sqrt(15.8^2 + 58.2^2) = 62.6 miles
Bearing = arctan(-15.8/58.2) = -25.3 degrees (measured east from north)
To find the bearing from Forrest Island back to the fisherman's home port, you first need to determine the total distance traveled in the east (x) and north (y) directions.
From the given information, the fisherman traveled N 70 degrees W for 30 miles and N 10 degrees E for 50 miles.
To calculate the x and y distances, you can use trigonometry:
For the first leg of the journey (N 70 degrees W), the x-distance traveled (in the west direction) can be found by calculating 30 * sin(70 degrees).
The y-distance traveled (in the north direction) can be found by calculating 30 * cos(70 degrees).
For the second leg of the journey (N 10 degrees E), the x-distance traveled (in the east direction) can be found by calculating 50 * cos(10 degrees).
The y-distance traveled (in the north direction) can be found by calculating 50 * sin(10 degrees).
Now, with the x and y distances determined, you can find the total distance traveled using the Pythagorean theorem:
Total distance traveled = sqrt((x-distance)^2 + (y-distance)^2)
Finally, to find the bearing (measured east from north), you can use the arctan function:
Bearing = arctan(x-distance / y-distance)
By plugging in the appropriate values, you can calculate the bearing from Forrest Island back to the fisherman's home port.
To find the bearing from Forrest Island back to the fisherman's home port, you first need to understand the concept of bearings. Bearings are usually measured in degrees clockwise from true north. In this case, the bearing is measured in degrees east from north.
To calculate the bearing, you need to determine the values of X and Y, which represent the total distances travelled in the east (x) and north (y) directions, respectively. Let's calculate X and Y based on the given information.
In the first leg of the journey, the fisherman travels N 70 degrees W for 30 miles. To break this down into east and north components, we can use trigonometry. The angle 70 degrees can be viewed as the angle between the north direction and a line segment going towards the west. This creates a right triangle, with the 30-mile leg as the hypotenuse.
To find the value of X (east component), you can calculate the adjacent side of the triangle using cosine. Since cosine(angle) = adjacent/hypotenuse, you have:
cos(70 degrees) = X/30 miles
Solving for X, you get:
X = 30 miles * cos(70 degrees)
Next, to find the value of Y (north component), you can calculate the opposite side of the triangle using sine. Since sine(angle) = opposite/hypotenuse, you have:
sin(70 degrees) = Y/30 miles
Solving for Y, you get:
Y = 30 miles * sin(70 degrees)
Now that you have the values of X and Y, you can calculate the total distance traveled using the Pythagorean theorem:
Total distance = sqrt(X^2 + Y^2)
Plug in the calculated values of X and Y to find the total distance traveled, which you have correctly calculated as 62.6 miles.
Finally, to find the bearing from Forrest Island back to the fisherman's home port, you can use the arctan function. The arctan function gives you the angle whose tangent is a given number. In this case, you want to find the angle that gives you X/Y.
Bearing = arctan(X/Y)
Plug in the values of X and Y calculated earlier to find the bearing.