A ship leaves port and travels 21km on A bearing of 032 and then 45km on a bearing of 287. calculate it distance from the port with the bearing diagram
To solve this problem, we can use the cosine and sine rules and plot the points on a bearing diagram.
First, we need to plot the starting point (port) at the origin (0,0) on the bearing diagram. Then, we can plot the first leg of the journey by drawing a line at a bearing of 032 for a distance of 21km. This takes us to point A.
Next, we need to plot the second leg of the journey by drawing a line at a bearing of 287 for a distance of 45km. This takes us to point B.
Now, we can use the cosine rule to find the length of the line connecting points A and B (the distance of the ship from the port). The cosine rule states that:
c^2 = a^2 + b^2 - 2ab*cos(C), where c is the length of the line connecting A and B, a is the distance traveled on the first leg (21km), b is the distance traveled on the second leg (45km), and C is the angle between them.
To find C, we need to subtract the bearing of the second leg (287) from the bearing of the first leg (032). However, bearings are usually measured clockwise from north, so we need to convert them to angles measured counterclockwise from the x-axis (east). This means that:
- The bearing of the first leg (032) corresponds to an angle of 90-32=58 degrees counterclockwise from the x-axis.
- The bearing of the second leg (287) corresponds to an angle of 360-287=73 degrees counterclockwise from the x-axis.
Therefore, the angle between the two legs is:
C = 73 - 58 = 15 degrees.
Substituting the values into the cosine rule, we get:
c^2 = 21^2 + 45^2 - 2*21*45*cos(15)
c^2 = 672.9
c ≈ 25.9
Therefore, the ship is about 25.9km from the port, and we can use the sine rule to find its bearing from the port. The sine rule states that:
sin(A)/a = sin(B)/b = sin(C)/c, where A and B are the angles opposite the sides a and b, respectively.
To find the bearing, we need to find the angle that the line connecting the port to the ship makes with the x-axis (east). This is simply the angle that the line connecting A and B makes with the x-axis, plus 180 degrees (since bearings are measured clockwise from north). Therefore, we have:
sin(A)/25.9 = sin(15)/(45)
sin(A) ≈ 0.174
A ≈ 10 degrees (measured counterclockwise from the x-axis)
Therefore, the ship is about 25.9km from the port at a bearing of 010 (measured clockwise from north). This is shown in the following bearing diagram:
```
N
|
|
|
032 /--------+-------\ 287
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
/ | \
+-----------------+-----------------> E
PORT
```
To calculate the distance from the port and the bearing, we can use the concept of vector addition.
Step 1: Start by plotting the initial position of the ship at the port on the bearing diagram. Label it as point A.
Step 2: Draw a line segment from point A in the direction of a bearing of 032. The length of this line segment represents the distance traveled on that bearing (21 km in this case). Label the end point of this line segment as point B.
Step 3: From point B, draw another line segment in the direction of a bearing of 287. The length of this line segment represents the distance traveled on that bearing (45 km in this case). Label the end point of this line segment as point C.
Step 4: Measure the distance from point A to point C. This represents the final distance of the ship from the port.
Step 5: Measure the angle between the direction from point A to point C and the north direction. This represents the bearing of the ship from the port.
I hope this helps! Let me know if you have any other questions.