Thanks in advance, these problems have been stumping me. Please include answers though, I have tried to work it out, but need answers to see where I am going wrong/where I need to go.

1. Determine all solutions for X^4+2-3i

2. Forces of 60 and 100 pounds have a resultant force of 125 pounds, what is the angle between the vectors?

3. Consider a truck hat weighs 5000 pounds at an incline of 20 degrees. What is the braking force required to keep the truck from rolling down the hill?

4. Two forces with a direction of 30 degrees and 60 degrees and magnitudes of 5 and 13 respectively act on a point. Find the direction and magnitude of the resultant force.

1. You have not written an equation. Algebraic expressions by themselves do not have solutions.

2. Use the law of cosines.

z^2 = x^2 + y^2 + 2xy cosZ
z is the resultant and Z is the angle between the two components.
z = 125; x = 60 and y = 100
Solve for the angle Z by getting its cosine.

3. The required braking force is the component of 5000# down the slope.

4. Add the x and y components of the two forces separately. Then get the magnitude of the resultant with the Pythagorean theorem. Get the angle from the arctangent.

Show your work if you have problems, then we can tell you where you are making mistakes.

Alright, the first one is equal to zero.

I figured out 2,

for three I got 171 pounds of force

Im currently working on 4

Sure! Here are the explanations and answers to the four problems:

1. To find the solutions for X^4 + 2 - 3i, we can use the quadratic formula. First, let's rewrite the equation: X^4 + 2 = 3i. Subtracting 2 from both sides gives us X^4 = 3i - 2. Next, we'll take the fourth root of both sides to solve for X. Remember that complex numbers have four distinct fourth roots. Thus, we have:

X₁ = (3i - 2)^(1/4)
X₂ = (-3i - 2)^(1/4)
X₃ = (2 - 3i)^(1/4)
X₄ = (-2 + 3i)^(1/4)

These are the four solutions for X.

2. To find the angle between two forces, we can use the law of cosines. Let's call the angle we want to find θ. The formula is:

c^2 = a^2 + b^2 - 2*a*b*cos(θ),

where a and b are the magnitudes of the forces and c is the magnitude of the resultant force. In this case, a = 60 pounds, b = 100 pounds, and c = 125 pounds (given). Plugging these values into the formula, we get:

125^2 = 60^2 + 100^2 - 2*60*100*cos(θ).

Now we can solve for cos(θ):

cos(θ) = (125^2 - 60^2 - 100^2) / (-2*60*100).
cos(θ) = -23495 / 12000.
θ ≈ cos^(-1)(-23495 / 12000).

To find the angle, we use a calculator to calculate the inverse cosine of -23495 / 12000. The answer will be in radians.

3. To find the braking force required to keep the truck from rolling down the hill, we need to consider the forces acting on the truck. The gravitational force pulling the truck down the incline can be calculated using the formula:

Force = mass * acceleration.

The acceleration down the incline can be found using the equation:

acceleration = gravitational acceleration * sin(θ),

where θ is the incline angle. The gravitational acceleration is approximately 32.2 ft/s^2.

In this case, the mass of the truck is given as 5000 pounds. To find the gravitational force, multiply the mass by the acceleration:

Force = 5000 pounds * 32.2 ft/s^2 * sin(20 degrees).

Note that the angle should be converted to radians before plugging into the equation.

4. To find the direction and magnitude of the resultant force when two forces are applied at angles of 30 degrees and 60 degrees, respectively, you can break the two forces down into their respective x and y components using the trigonometric identities:

For the force at 30 degrees:
Fx₁ = 5 * cos(30 degrees)
Fy₁ = 5 * sin(30 degrees)

For the force at 60 degrees:
Fx₂ = 13 * cos(60 degrees)
Fy₂ = 13 * sin(60 degrees)

Next, find the resultant forces in the x and y directions by summing up the components:
Rx = Fx₁ + Fx₂
Ry = Fy₁ + Fy₂

The magnitude of the resultant force can be calculated using the Pythagorean theorem:
Resultant Magnitude (R) = sqrt(Rx^2 + Ry^2)

Finally, the direction (θ) can be calculated using the inverse tangent function:
θ = arctan(Ry / Rx)

Solving these equations will provide the direction and magnitude of the resultant force.