A boat sails 4km on a bearing 038 degrees and then 5km on a bearing of 067 degree how far is the boat from its starting point and calculate the bearing of the boat from its starting point
To find the distance between the boat's starting point and its current position, we can use the Law of Cosines.
Let's call the starting point O, the point after sailing 4km A, and the point after sailing 5km B.
Using the given bearings, we can draw a triangle with side lengths 4km, 5km, and an unknown side x (the distance from the starting point to the current position of the boat).
By drawing a sketch, we can also observe that the angle between the known sides of length 4km and 5km is (67 degrees - 38 degrees) = 29 degrees.
Applying the Law of Cosines:
x² = 4² + 5² - 2 * 4 * 5 * cos(29°)
x² = 16 + 25 - 40 * cos(29°)
x² = 41 - 40 * cos(29°)
x² = 41 - 40 * 0.875
x² = 41 - 35
x² = 6
Taking the square root of both sides:
x = √6
Therefore, the boat is approximately √6 km away from its starting point.
To calculate the bearing of the boat from its starting point, we can use the Law of Sines.
Using the same triangle formed by points O, A, and B:
sin(29°) / x = sin(α) / 5
Where α is the angle between the known side of length 5km and the unknown side of length x.
Simplifying the equation:
sin(α) = (5 * sin(29°)) / √6
sin(α) = 3.532 / 2.449
sin(α) ≈ 1.441
Since sin(α) is positive, the angle α is in the first quadrant. Thus, the bearing of the boat from its starting point is given by:
α ≈ arcsin(1.441) = 88.169 degrees
Therefore, the boat is approximately √6 km away from its starting point, and its bearing from the starting point is approximately 88.169 degrees.
To find the distance the boat is from its starting point, we can use the Pythagorean theorem.
1. Start by drawing a diagram with the starting point as the origin (0,0).
2. The boat sails 4km on a bearing of 038 degrees. To represent this in the diagram, draw a line segment that is 4 units long and makes a 38-degree angle with the positive x-axis.
3. Next, the boat sails 5km on a bearing of 067 degrees. To represent this in the diagram, draw another line segment that starts at the end of the previous line segment and is 5 units long, making a 67-degree angle with the positive x-axis.
4. Now, we have two line segments that form a right triangle with the origin as one of the vertices. The length of the hypotenuse (the line connecting the starting point to the endpoint of the second line segment) is the distance the boat is from its starting point.
5. To find the length of the hypotenuse, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
6. From the diagram, we have the lengths of the two sides: one side is 4km, and the other side is 5km. Let's label the sides as follows:
- Side 1: 4km
- Side 2: 5km
- Hypotenuse: unknown (let's call it "d", representing the distance from the starting point)
7. Applying the Pythagorean theorem, we can write the equation:
d^2 = 4^2 + 5^2
8. Evaluating the equation:
d^2 = 16 + 25
d^2 = 41
9. Taking the square root of both sides to solve for "d":
d = √41
Thus, the boat is approximately √41 km from its starting point.
To calculate the bearing of the boat from its starting point, we can use trigonometric functions.
10. Look at the right triangle formed in the diagram. The angle between the positive x-axis and the line connecting the starting point to the endpoint of the second line segment is the angle we need to find.
11. To find this angle, we can use the inverse tangent (arctan) function. Let's call this angle "θ".
12. Applying the inverse tangent function:
θ = arctan(Side 2 / Side 1)
θ = arctan(5 / 4)
13. Using a scientific calculator, we can find the approximate value of θ to be:
θ ≈ 51.3 degrees
Hence, the bearing of the boat from its starting point is approximately 051 degrees.
To find the distance and bearing of the boat from its starting point, we can break down the problem into two components: the horizontal (east-west) displacement and the vertical (north-south) displacement.
1. Horizontal Displacement:
The boat sails 4km on a bearing of 038 degrees. This means it moves 4km in the direction 38 degrees east of north.
To find the horizontal displacement, we need to calculate the east-west component of this movement. We can use the formula:
horizontal displacement = distance * sin(bearing)
horizontal displacement = 4km * sin(38 degrees)
horizontal displacement = 4km * 0.6157 ≈ 2.4628 km
2. Vertical Displacement:
The boat then sails 5km on a bearing of 067 degrees. This means it moves 5km in the direction 67 degrees east of north.
To find the vertical displacement, we need to calculate the north-south component of this movement. We can use the formula:
vertical displacement = distance * cos(bearing)
vertical displacement = 5km * cos(67 degrees)
vertical displacement = 5km * 0.3988 ≈ 1.9940 km
3. Distance from Starting Point:
The distance from the starting point to the current position of the boat is the hypotenuse of a right-angled triangle formed by the horizontal and vertical displacements. We can calculate this distance using the Pythagorean theorem:
distance = √(horizontal displacement² + vertical displacement²)
distance = √(2.4628 km² + 1.9940 km²)
distance = √(6.070 km²)
distance ≈ 2.466 km
4. Bearing from Starting Point:
To find the bearing of the boat from its starting point, we need to calculate the angle between the horizontal displacement and the reference meridian (north). We can use the formula:
bearing = arctan(horizontal displacement / vertical displacement)
bearing = arctan(2.4628 km / 1.9940 km)
bearing ≈ 51.35 degrees
Therefore, the boat is approximately 2.466 km from its starting point, and its bearing from the starting point is approximately 51.35 degrees.