Final answer:
The direction of the fourth displacement is approximately 87.9° with respect to the positive x-axis.
Step-by-step explanation:
To determine the direction of the fourth displacement, we need to add up all the given displacements and find their resultant. Let's start by considering the first displacement, which is 180 m straight west. Since it's straight west, the displacement is in the negative x-direction. Next, we have a displacement of 290 m in a direction 45° east of south. This means the displacement is in the 4th quadrant, which is a negative y value and a positive x value. Lastly, we have a displacement of 280 m at 30° east of north, which is in the 1st quadrant, meaning both the x and y values are positive. Now, let's add up all these displacements to find the resultant displacement.
180 m west = -180i
290 m at 45° east of south = 290(cos(45)i - sin(45)j)
280 m at 30° east of north = 280(cos(30)i + sin(30)j)
Adding up all these vectors, we get the result:
-180i + 290(cos(45)i - sin(45)j) + 280(cos(30)i + sin(30)j)
Simplifying further, we have:
-180i + 290(cos(45)i - sin(45)j) + 280(cos(30)i + sin(30)j) = (-180 + 290cos(45) + 280cos(30))i + (290sin(45) - 280sin(30))j
After simplifying, we find that the resultant displacement is approximately (-15.8i + 464.3j) m. To determine the direction of the fourth displacement, we can find the angle it makes with the positive x-axis using the inverse tangent function:
θ = arctan(464.3/15.8)
θ ≈ 87.9°
Therefore, the direction of the fourth displacement is approximately 87.9° with respect to the positive x-axis.