Alright, here we have a problem. The hanging mass from some rope and two separate angles. We're told that we have a maximum tension in the rope and we want to find out what the maximum weight of any mass would be before they would break the rope. So, before we solve this problem, we need to think about what we're going to need to find. We need to understand the components involved. We know we're going to need tension one, tension two, and weight. We'll also need angle one and angle two. However, we shouldn't just write these down and start using them. We need to create a free body diagram first to avoid mistakes. The formula we will use is the sum of the forces equals mass times acceleration. In this case, since the system is not moving, the acceleration is zero. So, we have f1 plus f2 equals zero. Now, we need to find out what our angles are and what the tensions are. We also need to find the weights, although for this problem we won't need to consider gravity. Next, we need to find the components of the problem. We can recognize that the weight is going straight down, so it's in the negative y-direction. Therefore, the x-axis is perpendicular to the y-direction. Through trigonometry, we can determine that the angles with respect to the x-axis are 60 degrees and 40 degrees. With these angles, we can create a free body diagram. We can draw tension one at a 60-degree angle and tension two at a 40-degree angle. To make calculations easier, we can subtract 60 degrees from 180 degrees and use 120 degrees instead. This allows us to avoid worrying about negative signs. Breaking up the forces, we have tension in the x-direction for...