Three buoys K, L, and M show the course of a boat race.
MK= 4km, KL = 9km and angle MKL=108.
Find the three-figure bearing of K from M
Well, the boat race seems to be taking a buoy-ful turn! Now, to find the three-figure bearing from K to M, we can use some math-magical calculations.
Since we know that MK = 4 km and KL = 9 km, we have two sides of the triangle. Now, to find the three-figure bearing, we need to find the angle KML.
To find the angle KML, we can use the Law of Cosines. According to this law, we can use the formula:
c^2 = a^2 + b^2 - 2ab * cos(C)
In this case, a = MK, b = KL, and C = angle MKL.
Using the formula:
ML^2 = MK^2 + KL^2 - 2 * MK * KL * cos(MKL)
Substituting the given values:
ML^2 = 4^2 + 9^2 - 2 * (4) * (9) * cos(108)
After some math-juggling, we find that ML^2 is approximately equal to 70.633. So now, we know the length of the third side of the triangle, ML.
To find the bearing, we take the inverse tangent of MK/ML:
tan^(-1)(MK/ML)
tan^(-1)(4/√70.633)
After doing some more calculations, we find that the three-figure bearing of K from M is approximately 35.798 degrees.
So, the buoyant three-figure bearing of K from M is 35.798°! Have a splash-tastic boat race!
To find the three-figure bearing of point K from point M, we can use trigonometry.
First, let's draw a diagram to visualize the situation:
```
L
/
/
/
/
M––––––––––––K
```
Given:
- MK = 4 km
- KL = 9 km
- Angle MKL = 108 degrees
To find the bearing, we need to find the angle between the line MK and the north direction.
Step 1: Find the length of the side ML using the Law of Cosines:
ML^2 = MK^2 + KL^2 - 2 * MK * KL * cos(MKL)
ML^2 = 4^2 + 9^2 - 2 * 4 * 9 * cos(108)
ML^2 = 16 + 81 - 72 * cos(108)
ML^2 ≈ 16 + 81 - 72 * (-0.30901699437) (cos(108) ≈ -0.30901699437)
ML^2 ≈ 16 + 81 + 22.256
ML^2 ≈ 119.256
ML ≈ √(119.256)
ML ≈ 10.92 km
Step 2: Find the angle between the line MK and the north direction (angle KMN in the diagram below):
```
L
/
/
/ |
/ |
M––N–––K
```
Using the Law of Sines:
sin(angle KMN) / ML = sin(90 degrees) / MK
sin(angle KMN) = (MK / ML) * sin(90 degrees)
sin(angle KMN) = (4 / 10.92) * 1
sin(angle KMN) ≈ 0.3666
angle KMN ≈ arcsin(0.3666)
angle KMN ≈ 21.4 degrees
Step 3: Convert the angle KMN to the three-figure bearing:
The three-figure bearing of point K from point M is the angle measured clockwise from the north direction. Therefore, the three-figure bearing is:
360 degrees - angle KMN
360 degrees - 21.4 degrees ≈ 338.6 degrees
So, the three-figure bearing of point K from point M is approximately 338.6 degrees.
To find the three-figure bearing of K from M, we need to use trigonometry and the given information.
Step 1: Draw a diagram to visualize the problem. Place point M as the starting point and points K and L as the buoys. Label the distances MK and KL accordingly.
Step 2: Identify the triangle formed by the points M, K, and L. In this case, it is a triangle MKL.
Step 3: Recall the definition of a bearing. A bearing measures the direction of one point from another in degrees, always measured clockwise from north.
Step 4: To find the three-figure bearing of K from M, we need to measure the angle MKL. In this case, the given angle MKL is 108 degrees.
Step 5: The three-figure bearing is composed of three digits representing the degrees, minutes, and seconds. Since we only have the degrees (108), the minutes and seconds will be 00.
Step 6: Therefore, the three-figure bearing of K from M is 108°00′.
Note: If you want to verify the answer using trigonometry, you can use the law of cosines to find the angle KLM and subtract it from 180 degrees to get the angle MKL.