a man walk first 70m in the direction of 37 degree north of east,then walk then walk 82 m in the20 degree of south east and walk in 28 inin the 30 degree of west of north ,how far his walk? in what direction he return?
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To find the total distance the man walked, we need to find the vector sum of his three displacements.
Let's break down the displacements into their x and y components.
1. The first displacement is 70m in the direction of 37 degrees north of east. This can be broken down into x and y components as follows:
- x component = 70m * cos(37°)
- y component = 70m * sin(37°)
Calculate the x and y components:
- x = 70m * cos(37°) = 70m * 0.7986 ≈ 55.90m (rounded to two decimal places)
- y = 70m * sin(37°) = 70m * 0.6018 ≈ 42.13m (rounded to two decimal places)
2. The second displacement is 82m in the direction of 20 degrees south-east. To break this displacement into x and y components, we need to consider that south-east is a combination of south and east directions:
- x component = 82m * cos(45°) (since 45 degrees represents south-east)
- y component = -82m * sin(45°) (we multiply by -1 since it's in the opposite direction of the y-axis)
Calculate the x and y components:
- x = 82m * cos(45°) = 82m * 0.7071 ≈ 58.60m (rounded to two decimal places)
- y = -82m * sin(45°) = -82m * -0.7071 ≈ 58.60m (rounded to two decimal places)
3. The third displacement is 28m in the direction of 30 degrees west of north. To break it down into x and y components:
- x component = 28m * sin(30°)
- y component = 28m * cos(30°)
Calculate the x and y components:
- x = 28m * sin(30°) = 28m * 0.5 = 14m
- y = 28m * cos(30°) = 28m * 0.866 ≈ 24.25m (rounded to two decimal places)
Now, we can find the total x and y components by adding the individual components:
Total x component = 55.90m + 58.60m + 14m = 128.50m (rounded to two decimal places)
Total y component = 42.13m + 58.60m + 24.25m = 125.98m (rounded to two decimal places)
To find the total distance traveled, we use the Pythagorean theorem:
Total distance = √((Total x component)^2 + (Total y component)^2)
Substituting the values:
Total distance = √((128.50m)^2 + (125.98m)^2) ≈ √(16545.25m^2 + 15856.40m^2) ≈ √(32401.65m^2) ≈ 180.00m (rounded to two decimal places)
Therefore, the man walked approximately 180.00 meters in total.
To find the direction he returns, we need to find the angle from the origin (starting point) to the final position. We can use the inverse tangent function to find this angle:
Angle = arctan((Total y component) / (Total x component))
Substituting the values:
Angle = arctan(125.98m / 128.50m) ≈ 44.42° (rounded to two decimal places)
Since the angle is measured counterclockwise from the positive x-axis, we need to calculate the direction from the positive x-axis (east) to the direction of 44.42° counterclockwise.
Therefore, the man returns in the direction of approximately 44.42° counterclockwise from the positive x-axis, which would be east of north.
i want nw
geez, impatient much?
Just get the x-y components of each part of the journey, and add them up to get the final location. Then apply the usual distance formula. If he started at (0,0) and ended up at (x,y) then his final direction to get home will be
SθW where tanθ = x/y