Three astronauts, propelled by jet backpacks, push and guide a 118 kg asteroid toward a processing dock, exerting the forces shown in the figure, with F1 = 32 N, F2 = 56 N, F3 = 38 N, θ1 = 30°, and θ3 = 60°. What is the (a) magnitude and (b) angle (measured relative to the positive direction of the x axis in the range of (-180°, 180°]) of the asteroid's acceleration?

F = 32N[30o] + 56N[0o]? + 38N[60o] = 27.7+16i + 56 + 19+32.9i = 102.7 + 48.9i = 118N{25.5o].

F = M*a, a = F/M = 118[25.5o]/118 = 1 m/s^2[25.5o].

To find the magnitude and angle of the asteroid's acceleration, we first need to find the net force acting on the asteroid.

Step 1: Resolve the forces into their x and y components.

F1x = F1 * cos(θ1)
F1y = F1 * sin(θ1)

F2x = F2 * cos(θ2)
F2y = F2 * sin(θ2)

F3x = F3 * cos(θ3)
F3y = F3 * sin(θ3)

Step 2: Calculate the net force in the x and y directions.

Net force in the x direction (Fnet_x) = F1x + F2x + F3x
Net force in the y direction (Fnet_y) = F1y + F2y + F3y

Step 3: Calculate the magnitude and angle of the net force.

Magnitude of the net force (Fnet) = √(Fnet_x^2 + Fnet_y^2)

Angle of the net force (θ) = arctan(Fnet_y/Fnet_x)

Let's calculate the net force, magnitude, and angle step by step:

Step 1: Resolve the forces into their x and y components.

F1x = 32 N * cos(30°) = 27.712 N
F1y = 32 N * sin(30°) = 16 N

F2x = 56 N * cos(180°) = -56 N
F2y = 56 N * sin(180°) = 0 N

F3x = 38 N * cos(60°) = 19 N
F3y = 38 N * sin(60°) = 32.864 N

Step 2: Calculate the net force in the x and y directions.

Fnet_x = F1x + F2x + F3x = 27.712 N - 56 N + 19 N = -9.288 N
Fnet_y = F1y + F2y + F3y = 16 N + 0 N + 32.864 N = 48.864 N

Step 3: Calculate the magnitude and angle of the net force.

Magnitude of the net force (Fnet) = √((-9.288 N)^2 + (48.864 N)^2) = √(86.148 N^2 + 2393.429 N^2) = √(2479.577 N^2) = 49.8 N (approx.)

Angle of the net force (θ) = arctan(48.864 N / -9.288 N) = arctan(-5.254) = -79.278° (approx.)

Therefore, the (a) magnitude of the asteroid's acceleration is approximately 49.8 N, and the (b) angle is approximately -79.278° (measured relative to the positive direction of the x-axis in the range of (-180°, 180°]).

To find the magnitude and angle of the asteroid's acceleration, we can use vector addition.

Step 1: Resolve the forces into their x and y components.
Let's start by finding the x and y components of each force. For each force, we can use trigonometry to split it into horizontal (x) and vertical (y) components.

Resolve F1:
Fx1 = F1 * cos(θ1)
Fy1 = F1 * sin(θ1)

Substitute the given values:
Fx1 = 32 N * cos(30°)
Fy1 = 32 N * sin(30°)

Resolve F2:
Fx2 = F2 * cos(0°) [cos(0°) = 1]
Fy2 = F2 * sin(0°) [sin(0°) = 0]

Substitute the given values:
Fx2 = 56 N * cos(0°)
Fy2 = 56 N * sin(0°)

Resolve F3:
Fx3 = F3 * cos(θ3)
Fy3 = F3 * sin(θ3)

Substitute the given values:
Fx3 = 38 N * cos(60°)
Fy3 = 38 N * sin(60°)

Step 2: Determine the net force in the x and y directions.
The net force in the x and y directions can be found by summing the x and y components of all the forces, respectively.

Net Fx = Fx1 + Fx2 + Fx3
Net Fy = Fy1 + Fy2 + Fy3

Substitute the resolved values:
Net Fx = (32 N * cos(30°)) + (56 N * cos(0°)) + (38 N * cos(60°))
Net Fy = (32 N * sin(30°)) + (56 N * sin(0°)) + (38 N * sin(60°))

Step 3: Find the magnitude and angle of the net force.
The magnitude of the net force can be calculated using the Pythagorean theorem:

|Net F| = √(Net Fx^2 + Net Fy^2)

The angle, θ, of the net force can be found using arctan:

θ = arctan(Net Fy / Net Fx)

Substitute the calculated values into these formulas to find the answers.