You walk 50 m to the north, then turn 60° to your right and walk another 45 m. How far are you from where you originally started?
To find the distance from where you originally started, we can use the concept of vector addition.
First, let's represent your initial position as the origin (0,0) on a coordinate plane.
Next, we'll analyze the given information step by step:
1. You walk 50 m to the north. This means you move straight up on the y-axis by 50 units. So, your new position is (0, 50).
2. Then, you turn 60° to your right. This rotation changes the direction of your movement from the positive y-axis to some other direction.
3. Lastly, you walk another 45 m. Since your direction has changed, you'll need to break down your total distance into horizontal and vertical components.
To calculate these components, we'll use trigonometry. In a right triangle, the vertical component (opposite side) can be found using the sine function, and the horizontal component (adjacent side) can be found using the cosine function.
The triangle formed by your final position, the origin, and the x-axis is a right triangle with a hypotenuse of length 45 m. The angle between the hypotenuse and the x-axis is 60°.
4. Using trigonometry, we can find the vertical and horizontal components:
Vertical component: sin(60°) = opposite/hypotenuse = y/45
Horizontal component: cos(60°) = adjacent/hypotenuse = x/45
Simplifying these equations gives us:
y = 45 * sin(60°)
x = 45 * cos(60°)
Now, we have the coordinates of your new position, in terms of the vector components:
x = 45 * cos(60°) = 45 * 0.5 = 22.5
y = 45 * sin(60°) = 45 * (√3/2) = 38.91 (rounded to two decimal places)
To find the distance from where you originally started, we need to calculate the distance between your final position (x, y) and the origin (0, 0) using the distance formula:
Distance = sqrt((x - 0)^2 + (y - 0)^2)
Plugging in the values:
Distance = sqrt((22.5 - 0)^2 + (38.91 - 0)^2)
Distance = sqrt(506.25 + 1513.25)
Distance = sqrt(2019.5)
Distance ≈ 44.91 (rounded to two decimal places)
Therefore, you are approximately 44.91 meters away from where you originally started.