Final answer:
The ant's journey involves moving east, south, and then west up a plane. The resulting distance from the starting point to the endpoint is approximately 4.29 meters, calculated by resolving the motion into horizontal and vertical components and applying the Pythagorean theorem.
Step-by-step explanation:
The problem describes an ant's journey that consists of three parts: walking 5 m east, 2 m south, and then 4 m west up a plane with 60° inclination. To calculate the distance between the starting point and the endpoint, we need to consider the journey in parts and use vector addition for 2D motion. Here's how to solve it:
- First, the ant moves 5 m east.
- Secondly, it goes 2 m south.
- Finally, the ant climbs 4 m in the direction which is 60° from the horizontal. This part of the trip will have both vertical (upward) and a horizontal component (towards the west).
To find the vertical component, we use the cosine function (cos(60°) = 0.5). So, the vertical movement is 4 m * cos(60°) = 2 m.
The horizontal component will be calculated using the sine function (sin(60°) = √3/2). Therefore, the horizontal movement is 4 m * sin(60°) ≈ 3.46 m to the west.
Now we can combine these vectors. The ant ends up 5 m - 3.46 m = 1.54 m east and 2 m + 2 m = 4 m south relative to the starting point.
Using the Pythagorean theorem, we calculate the hypotenuse of this right triangle to find the straight-line distance:
Distance = √(1.542 + 42)
≈ √(2.3716 + 16)
≈ √18.3716
≈ 4.29 m