Final answer:
The student is solving for the current Io using mesh analysis. After establishing equations for the currents in each mesh, substituting, and rearranging, values for I1, I2, and I3 can be found. The problem might contain two separate questions as the last part with an equation involving Io and cosine seems unrelated.
Step-by-step explanation:
Mesh Analysis to Find Io
The student appears to be working through a problem involving mesh analysis, a technique commonly used in electrical engineering to determine the currents that flow through a circuit. By applying mesh analysis, we can establish equations for each mesh (loop) in the circuit. In solving these equations, the student has reached a step where current I3 can be expressed in terms of I1: I3 = 22.5 - 3I1. After substituting I3 and another expression for I2 into the first equation, the student finds that I1 can be described as: I1 = I2 + I3 = (6 - 2I1) + (22.5 - 3I1) = 28.5 - 5I1. By rearranging and solving, we can determine the value of I1 and, subsequently, I2 and I3. It's important to note that the values for I2 and I3 given later indicate that I2 flows in the opposite direction, denoted by a negative value. Finally, these currents can be used to check against other equations, like the one noted as 'abcdefgha loop', for verification of the answers.
From the given information, we can further solve the equation I = Io cos²(0) to find the value of Io related to the intensity I, if this is part of the original problem the student is working with. However, with the information provided, it seems there might be an inconsistency or missing parts since the mesh analysis doesn't directly relate to the given equation involving cosines. It's possible there are two separate questions here, or further clarification is needed from the assignment.