a boat sails 150km on a bearing of 42 degree and then 250km due east.how far is the boat from the starting point.
d = 150km @ 42o + 250km @ 0o.
X = 150*cos42 + 250 = 361.5 km.
Y = 150*sin42 = 111.5 km.
D^2 = X^2 + Y^2 = 143,114.5
D = 378.3 km
Please what is the meaning and the diagram
Well, it seems like the boat took quite a detour to the east! If we draw a triangle with the starting point, the end point, and the eastern detour, we find that we have a right-angled triangle. The boat sailed 150km on a bearing of 42 degrees, forming one side of the triangle, and then sailed 250km due east, forming another side. To find the distance from the starting point, we can use good ol' Pythagoras' theorem! So, grab your math goggles, and let's dive in!
Using Pythagoras' theorem, we can say that the square of the hypotenuse (the distance from the starting point) is equal to the sum of the squares of the other two sides. In this case, it's 150km squared plus 250km squared. Time to get your calculators out!
150km squared plus 250km squared equals 62,500km squared. Now, to find the length of the hypotenuse, or the distance from the starting point, we need to take the square root of that. Drumroll, please... 🥁
The square root of 62,500km squared equals 250km. Ta-da!
So, the boat is approximately 250 kilometers away from the starting point. Now, that's quite the adventure just to end up right where you began!
To find the total distance the boat is from its starting point, we first need to break down the boat's movement into components.
Step 1: Calculate the horizontal and vertical distances covered by the boat.
The initial distance covered by the boat sailing on a bearing of 42 degrees can be broken down into two components: the horizontal distance (adjacent side) and the vertical distance (opposite side) of a right-angled triangle.
The horizontal distance covered is given by: horizontal distance = initial distance * cos(bearing)
horizontal distance = 150 km * cos(42°)
The vertical distance covered is given by: vertical distance = initial distance * sin(bearing)
vertical distance = 150 km * sin(42°)
Step 2: Calculate the final position of the boat after moving 250 km due east.
Since the boat sailed 250 km due east, there is no vertical displacement in this movement. Only the horizontal distance needs to be added to the previous horizontal position.
Final horizontal position = horizontal distance + 250 km
Step 3: Calculate the distance of the boat from the starting point (the magnitude of the displacement).
To find the distance of the boat from its starting point, we can use the Pythagorean theorem:
Distance from starting point = √((final horizontal position)² + (vertical distance)²)
Now, let's calculate the distances:
horizontal distance = 150 km * cos(42°)
horizontal distance = 150 km * 0.7431
horizontal distance ≈ 111.465 km
vertical distance = 150 km * sin(42°)
vertical distance = 150 km * 0.6691
vertical distance ≈ 100.365 km
Final horizontal position = horizontal distance + 250 km
Final horizontal position ≈ 111.465 km + 250 km
Final horizontal position ≈ 361.465 km
Distance from starting point = √((final horizontal position)² + (vertical distance)²)
Distance from starting point = √((361.465 km)² + (100.365 km)²)
Distance from starting point ≈ √((130582.694225 km²) + (10073.134225 km²))
Distance from starting point ≈ √(140655.828450225 km²)
Distance from starting point ≈ 374.989 km
Therefore, the boat is approximately 374.989 km away from its starting point.