https://file.aiquickdraw.com/mathbot/file/1727424600674i3s8qm50.jpg

To solve for the reaction forces at points **A** and **B**, we will apply the static equilibrium equations for a truss system.

The truss system in the image appears to have the following:
- A roller support at **B** (provides only a vertical reaction \( R_B \)),
- A pinned support at **A** (provides both vertical \( R_A \) and horizontal \( H_A \) reactions),
- Distributed forces acting vertically at three points (marked 3160 lb, 8400 lb, and 10,000 lb),
- The overall length is given as 72 ft, and distances between each force are provided.

We use the following static equilibrium equations:
1. \(\sum F_x = 0\): Sum of horizontal forces equals zero.
2. \(\sum F_y = 0\): Sum of vertical forces equals zero.
3. \(\sum M_A = 0\): Sum of moments about point \( A \) equals zero.

### Step 1: Sum of Moments about Point A
Taking moments about point A to eliminate \( R_A \):

\[
\sum M_A = 0
\]

We'll consider counterclockwise moments as positive and clockwise moments as negative. The distances from point A to the applied loads and the reaction at B are crucial.

From the image:
- Distance from A to the 8400 lb load: 36 ft
- Distance from A to the 10,000 lb load: 72 ft
- Distance from A to the reaction at B: 72 ft

The moment equation will look like this:

\[
-3160 \times 24 - 8400 \times 36 - 10000 \times 72 + R_B \times 72 = 0
\]

### Step 2: Solve for \( R_B \)
Now solve for \( R_B \):

\[
R_B = \frac{3160 \times 24 + 8400 \times 36 + 10000 \times 72}{72}
\]

Let me calculate this value.The reaction force at point **B** is approximately \( R_B = 15,253.33 \, \text{lb} \).

### Step 3: Sum of Vertical Forces
Next, we use the equation for the sum of vertical forces to solve for \( R_A \):

\[
\sum F_y = 0
\]

\[
R_A + R_B - 3160 - 8400 - 10000 = 0
\]

Substituting the value of \( R_B \):

\[
R_A + 15,253.33 - 3160 - 8400 - 10000 = 0
\]

Let me solve for \( R_A \).The reaction force at point **A** is approximately \( R_A = 6306.67 \, \text{lb} \).

### Summary of Reactions:
- \( R_A \approx 6306.67 \, \text{lb} \) (vertical reaction at A),
- \( R_B \approx 15,253.33 \, \text{lb} \) (vertical reaction at B).

Would you like more details on any step of the process, or do you have further questions?

### Related Questions:
1. How would the results change if an additional horizontal force acted at point A?
2. How does the location of forces affect the moment calculations?
3. Can this method be applied to other types of supports, such as fixed or free supports?
4. How would the reaction at B change if the truss were not symmetric?
5. What happens if a new force is applied halfway between the 8400 lb and 10000 lb loads?

**Tip:** Always verify that the sum of the moments and forces equals zero when analyzing static structures, ensuring equilibrium is maintained.

Find the reaction forces at points A and B for the given truss system.

Learn how to calculate the reaction forces at points A and B in a truss system. This problem involves the application of static equilibrium equations, including the sum of moments and forces, to determine the reaction forces based on given loads and distances.

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