a student weighing 160 pounds hangs for dear life from a cable tied to two other cables fastened to a support as shown above. the upper cables make angles of 39 and 55 with the horizontal. these upper cables are not as strong as the vertical cable and will break if the tension in them exceeds 500N

M = 160Lbs * 0.454kg/Lb. = 72.64 kg.

M*g = 72.64 * 9.8 = 711.9 N., Downward = -711.9 N.

T1*Cos(180-39) = -T2*Cos55.
T1 = 0.738T2.

T1*sin(180-39) + T2*sin55 = 711.9.
0.629T1 + 0.819T2 = 711.9,
Replace T1 with 0.738T2:
0.629*0.738T2 + 0.819T2 = 711.9,
1.28T2 = 711.9, T2 = 556 N.

T1 = 0.738T2 = 0.738*556 = 410 N.

To solve this problem, we need to analyze the forces acting on the student while hanging from the cable. We can start by breaking down the forces into horizontal and vertical components.

1. Identify the forces:
- Weight of the student: 160 pounds (converted to Newtons)
- Tension in the upper cables: Unknown (let's call it T)
- Tension in the vertical cable: Unknown (let's call it T_v)

2. Resolve the forces:
- Horizontal forces: There are no horizontal forces acting on the student since we only have vertical cables.
- Vertical forces:
- Weight of the student: This force acts downward with a magnitude of 160 pounds or 712 Newtons (1 pound = 4.45 Newtons).
- Tension in the vertical cable: This force acts upward and opposes the weight of the student. We need to find this force.

3. Calculate the tension in the vertical cable:
Since the student is in equilibrium, the sum of the vertical forces must be zero. This means that the tension in the vertical cable should balance out the weight of the student.

T_v - 712 N = 0

Therefore, T_v = 712 N

The tension in the vertical cable is equal to the weight of the student, which is 712 Newtons.

4. Calculate the tension in the upper cables:
Now we need to determine the tension in the upper cables to ensure it doesn't exceed 500 Newtons.

Since the tension in the vertical cable (T_v) acts as a support, it will contribute to the tension in the upper cables.

For the upper cable making an angle of 39 degrees:
- Resolve the tension into horizontal and vertical components.
- The vertical component of the tension will add to the vertical force balance equation.

Let's calculate the vertical component of tension (T_v1) using the given angle:

T_v1 = T_v * sin(angle)

Using the given angle of 39 degrees:
T_v1 = 712 N * sin(39°)

For the upper cable making an angle of 55 degrees:
- Resolve the tension into horizontal and vertical components.
- The vertical component of the tension will add to the vertical force balance equation.

Let's calculate the vertical component of tension (T_v2) using the given angle:

T_v2 = T_v * sin(angle)

Using the given angle of 55 degrees:
T_v2 = 712 N * sin(55°)

If the sum of the vertical components of tension in the upper cables (T_v1 and T_v2) exceeds 500 Newtons, the upper cables will break.

Remember to convert all units to the same system (either metric or imperial) before performing calculations.