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dame la respuesta con paso a paso, sustituyendo 30 n con 45n

Let's solve the problem step by step, assuming that the tension in the vertical rope (cuerda 1) is now 45 N instead of 30 N.

### Step 1: Identify the Forces
There are three forces acting on the object:
1. The weight \( W \) acting vertically downward.
2. The tension \( T_1 = 45 \, \text{N} \) in the vertical rope (cuerda 1).
3. The tension \( T_2 = 45 \, \text{N} \) in the horizontal rope (cuerda 2).
4. The tension \( T_3 \) in the rope at a \( 40^\circ \) angle to the horizontal.

### Step 2: Set Up the Equilibrium Equations
Since the object is in equilibrium, the sum of the forces in both the horizontal and vertical directions must be zero.

#### Horizontal Direction:
\[ T_2 = T_3 \cdot \cos(40^\circ) \]

Substituting the given values:
\[ 45 \, \text{N} = T_3 \cdot \cos(40^\circ) \]

Solving for \( T_3 \):
\[ T_3 = \frac{45 \, \text{N}}{\cos(40^\circ)} \]

#### Vertical Direction:
\[ T_1 + T_3 \cdot \sin(40^\circ) = W \]

Substituting the value of \( T_3 \) from the horizontal equation:
\[ W = 45 \, \text{N} + \left(\frac{45 \, \text{N}}{\cos(40^\circ)}\right) \cdot \sin(40^\circ) \]

### Step 3: Calculate the Values
1. First, calculate \( \cos(40^\circ) \) and \( \sin(40^\circ) \):

\[ \cos(40^\circ) \approx 0.7660 \]
\[ \sin(40^\circ) \approx 0.6428 \]

2. Calculate \( T_3 \):

\[ T_3 = \frac{45 \, \text{N}}{0.7660} \approx 58.75 \, \text{N} \]

3. Finally, calculate the weight \( W \):

\[ W = 45 \, \text{N} + (58.75 \, \text{N} \cdot 0.6428) \]
\[ W \approx 45 \, \text{N} + 37.76 \, \text{N} \]
\[ W \approx 82.76 \, \text{N} \]

### Final Answer:
The weight of the object is approximately **82.76 N**.

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Would you like a more detailed explanation or have any specific questions?

Here are five related questions to explore further:
1. How would the solution change if the tension in the vertical rope was different?
2. What if the horizontal force was applied at a different angle?
3. How can we check the equilibrium condition in such systems?
4. What are the potential errors when calculating trigonometric functions?
5. How do changes in tension affect the stability of the system?

**Tip:** In equilibrium problems, always double-check the angles and tensions before substituting values into equations to ensure accurate results.

Explore how to solve an equilibrium problem with adjusted tensions. Step-by-step guide includes trigonometric calculations and equilibrium equations. Learn how changes in tension affect stability. Suitable for high school students.

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