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The Pythagorean theorem is a fundamental principle in mathematics and physics, essential for problems involving right triangles. It states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it is expressed as:\[ c^2 = a^2 + b^2 \]Here, \(c\) represents the hypotenuse, while \(a\) and \(b\) represent the other two sides. In our problem, we apply this theorem to determine the vertical displacement of the mail bag when it is displaced sideways by 2.0 m. The rope forms the hypotenuse (3.5 m), the horizontal displacement is one leg (2.0 m), and the vertical displacement is the unknown side. By rewriting the equation to solve for this missing height (h), we get:\[ h = \sqrt{3.5^2 - 2.0^2} \]This calculation allows us to find how much the mail bag is lifted vertically, which helps us in understanding the mechanics of the entire setup.

Trigonometry plays a crucial role in solving physics problems, especially those involving forces and angles. Our problem requires an understanding of how trigonometric functions like sine, cosine, and tangent relate to the components of forces in a right triangle setup. When the mail bag is displaced, it forms a triangle where:- The rope acts as the hypotenuse.- The vertical and horizontal displacements are the other sides.To find the angle \(\theta\) that the rope makes with the vertical, we use the tangent function:\[ \tan(\theta) = \frac{2.0}{h} \]Calculating \(\theta\) lets us break down the tension in the rope into its vertical and horizontal components. This is essential because:- Vertical component is balanced by the gravitational force \(mg\).- Horizontal component is what the postal worker needs to apply to keep the bag still.These trigonometric calculations simplify how we can analyze and solve the problem using forces and angles.

The horizontal force calculation is crucial to ascertain how much force the worker needs to exert to hold the mail bag in its displaced position. Once we have the angle \(\theta\), we resolve the tension in the rope into its components. The tension \(T\) in the rope is related to gravity as it's the force countering the weight of the bag:\[ T = \frac{mg}{\cos(\theta)}\]Where:- \(m\) is the mass of the mail bag (120 kg).- \(g\) is the acceleration due to gravity, approximately 9.8 \(\text{m/s}^2\).Next, since the mail bag is at rest horizontally, the horizontal component of the tension must balance the horizontal force the worker applies. Using the sine function, we have:\[ F = T \cdot \sin(\theta)\]Thus, this horizontal force \(F\) needed to maintain the position is crucial for understanding equilibrium in this statics problem. This approach highlights the blend of physics and trigonometry to solve real-world problems efficiently.