A ship moved from point B 12km away on a bearing of N035E.it then moved westward to another point c20km which is on the bearing of N052w.calculate the distance /bc/
Call A the point 12km from B, where the ship turned west. Then the angle BAC is 65°. Now just use the law of cosines to find BC:
BC^2 = 12^2 + 20^2 - 2*12*20*cos65°
To calculate the distance between points B and C, we can use the law of cosines.
Let's first visualize the movement of the ship:
Point B is 12 km away on a bearing of N035E.
Point C is 20 km away on a bearing of N052W.
Now, let's break down the movement of the ship into components:
1. Movement from Point B to Point C:
- North-South component: Since the bearing is N035E, we can find the vertical component using trigonometry:
Vertical component = 12 km * sin(35°)
- East-West component: The ship moved westward, so the horizontal component will be negative (-20 km).
Now, we can use the law of cosines to find the distance between points B and C:
Distance^2 = (Vertical component)^2 + (Horizontal component)^2
Distance^2 = (12 km * sin(35°))^2 + (-20 km)^2
Distance^2 = (12 km * 0.5736)^2 + 400 km^2
Distance^2 = 8.71 km^2 + 400 km^2
Distance^2 = 408.71 km^2
Distance ≈ √408.71 km^2
Distance ≈ 20.21 km
Therefore, the distance between points B and C is approximately 20.21 km.
To calculate the distance BC, we can use the concept of vector addition.
1. Draw a diagram: Start by drawing a diagram with points B and C. Label the distance from B to C as BC.
2. Calculate the horizontal and vertical components: The bearing N035E means that the ship moved 12 km in the direction 35 degrees east of north. This can be broken down into horizontal and vertical components.
Horizontal Component:
- The ship moved 12 km in the east direction, so the horizontal component of the displacement is 12 km * cos(35°).
Vertical Component:
- The ship moved 12 km in the north direction, so the vertical component of the displacement is 12 km * sin(35°).
3. Calculate the position of point C: Point C is located 20 km west of point B on a bearing of N052W. This means that the horizontal component of the displacement is -20 km and the vertical component is 0 km.
4. Add the horizontal and vertical components: To find the total horizontal and vertical components of BC, add the horizontal and vertical components of the two displacements.
Horizontal Component of BC = Horizontal Component of B to C + Horizontal Component of C
Vertical Component of BC = Vertical Component of B to C + Vertical Component of C
5. Use the Pythagorean theorem: The distance BC can be calculated using the Pythagorean theorem, which states that the square of the hypotenuse (BC) is equal to the sum of the squares of the other two sides (horizontal and vertical components).
BC^2 = (Horizontal Component of BC)^2 + (Vertical Component of BC)^2
6. Calculate BC: Take the square root of both sides of the equation to find the distance BC.
BC = √[(Horizontal Component of BC)^2 + (Vertical Component of BC)^2]
Plug in the values and calculate the distance BC using the above steps. It's important to note that the calculations involve trigonometric functions, so a calculator may be needed.