To find the tension in each supporting cable, we can analyze the forces acting on the traffic light. Let's denote the tension in one cable as T1 and the tension in the other cable as T2.
Considering the equilibrium of forces, the sum of the tensions in the two cables should balance the weight of the traffic light. The weight of the traffic light can be calculated using the formula: weight = mass * acceleration due to gravity.
Given that the mass of the traffic light is 13.0 kg and the acceleration due to gravity is approximately 9.8 m/s^2, we can find the weight of the traffic light:
weight = 13.0 kg * 9.8 m/s^2 = 127.4 N
Since the light hangs 0.786 m below the level of the supports, we can consider this as the vertical distance between the traffic light and the supports.
Now, let's focus on one of the supporting cables with a length of 11.806 m. Using the concept of similar triangles, we can set up the following proportion:
(0.786 m) / (11.806 m) = (weight - T1) / T1
Simplifying the equation, we have:
0.06667 = (weight - T1) / T1
Rearranging the equation, we get:
T1 = weight / (1 + 0.06667)
T1 = 127.4 N / 1.06667
T1 ≈ 119.44 N
Using the same approach for the other supporting cable with a length of 7.461 m, we can set up the following proportion:
(0.786 m) / (7.461 m) = (weight - T2) / T2
Simplifying the equation:
0.10554 = (weight - T2) / T2
Rearranging the equation, we have:
T2 = weight / (1 + 0.10554)
T2 = 127.4 N / 1.10554
T2 ≈ 115.06 N
Therefore, the tension in each supporting cable is approximately 119.44 N and 115.06 N, respectively.