A girl walks 50m on a bearing 025° and then 200m due east.How far is she from her starting point
she walks on a heading, not a bearing.
50@25° = (50sin25°,50cos25°) = (21.13,45.32)
add to that (200,0)
and you end up at (221.13,45.32)
the distance moved is thus 225.73m
To find out how far the girl is from her starting point, we can use trigonometry and vector addition.
First, let's break down the girl's movements into components.
The girl walks 50m on a bearing of 025°. This means she walks in the northeast direction. We can find the horizontal component and vertical component of this movement using trigonometry.
The horizontal component can be found by multiplying the distance by the cosine of the angle:
Horizontal component = 50m * cos(25°) = 45.02m
The vertical component can be found by multiplying the distance by the sine of the angle:
Vertical component = 50m * sin(25°) = 21.79m
Next, the girl walks 200m due east. Since this movement is in a straight line and does not involve any angles, the entire 200m contributes to the horizontal component.
Now, let's calculate the final horizontal and vertical components:
Final horizontal component = 45.02m + 200m = 245.02m
Final vertical component = 21.79m
To find the total distance from the starting point, we can use the Pythagorean theorem:
Total distance = square root of [(Final horizontal component)^2 + (Final vertical component)^2]
Total distance ≈ square root of [(245.02m)^2 + (21.79m)^2]
Total distance ≈ square root of [60025.01m^2 + 475m^2]
Total distance ≈ square root of 60500.01m^2
Total distance ≈ 246.23m
Therefore, the girl is approximately 246.23m away from her starting point.
To find the distance from her starting point, we can apply trigonometry and treat the girl's movements as displacement vectors along two axes: north-south (y-axis) and east-west (x-axis).
1. Convert the bearing angle to the x and y components:
The given bearing is 025°, which means the angle is measured clockwise from the north.
The x-axis represents east, and the y-axis represents north.
For the first part of her walk, the girl walks 50m on a bearing 025°:
- The x-component can be found using cosine:
x1 = 50 * cos(025°) ≈ 49.29m
- The y-component can be found using sine:
y1 = 50 * sin(025°) ≈ 21.59m
For the second part of her walk, the girl walks 200m due east:
- The x-component is 200m.
- The y-component is 0 since she doesn't move along the y-axis.
2. Add the respective x and y components together:
x_total = x1 + x2 = 49.29m + 200m = 249.29m
y_total = y1 + y2 = 21.59m + 0m = 21.59m
3. Use the Pythagorean theorem to find the distance from the starting point:
distance = √(x_total² + y_total²)
distance = √(249.29m² + 21.59m²)
distance ≈ √(62176.44m² + 466.49m²)
distance ≈ √(62642.93m²)
distance ≈ 250.28m
Therefore, the girl is approximately 250.28 meters from her starting point.