two forces act simultaneously on a box. one pulls with a force of 12 pounds due south. the other pulls with a force of 8 pounds in a direction 20 degrees north od east. what is the resultant of these two forces?

Lets do them in N, E vectors

R=-12N+8cos20 N +8sin20 E
figure that out.

then magnitude will be
R^2=N^2+ E^2
and the angle will be arctan(E/N) (E of S) (in the SE quadrant)

To find the resultant of two forces, we can use vector addition. We'll break down each force into its horizontal and vertical components and then add them up.

First, let's calculate the components of the force pulling due south. Since the force is acting directly south, its horizontal component will be zero, and its vertical component will be the magnitude of the force itself, which is 12 pounds.

Next, let's calculate the components of the force pulling 20 degrees north of east. We know that the magnitude of this force is 8 pounds. To find the horizontal and vertical components, we can use trigonometry.

The horizontal component can be calculated using the cosine function:
Horizontal component = magnitude of the force * cos(angle)
Horizontal component = 8 pounds * cos(20 degrees) ≈ 7.452 pounds

The vertical component can be calculated using the sine function:
Vertical component = magnitude of the force * sin(angle)
Vertical component = 8 pounds * sin(20 degrees) ≈ 2.72 pounds

Now, let's add up the horizontal and vertical components of the two forces separately to find the resultant components.

Horizontal component of the resultant = 0 + 7.452 pounds ≈ 7.452 pounds (east direction)
Vertical component of the resultant = 12 pounds + 2.72 pounds ≈ 14.72 pounds (north direction)

Using the Pythagorean theorem, we can calculate the magnitude of the resultant force:
Magnitude of the resultant = sqrt((Horizontal component)^2 + (Vertical component)^2)
Magnitude of the resultant = sqrt((7.452 pounds)^2 + (14.72 pounds)^2) ≈ 16.53 pounds

Lastly, we can find the direction of the resultant force by calculating its angle from the positive x-axis using the inverse tangent function:
θ = arctan(Vertical component / Horizontal component)
θ = arctan(14.72 pounds / 7.452 pounds) ≈ 61.7 degrees north of east

Thus, the resultant of the two forces is approximately 16.53 pounds in the direction 61.7 degrees north of east.