1. **Problem Statement:** Calculate the reaction force $R_A$ at point A for the garbage truck with the retrieval arm extended, given forces and distances. 2. **Given Data:** - $W_C = 90.0$ kN (downward force at C) - $W_D = 30.0$ kN (downward force at D) - $L_A = 1.0$ m (distance from A to C) - $L_B = 1.5$ m (distance from C to B) - $L_D = 2.4$ m (distance from B to D) 3. **Approach:** Use the sum of moments about point B to find $R_A$. The sum of moments about B must be zero for equilibrium: $$\sum M_B = 0$$ 4. **Moment Contributions:** - $R_A$ acts upward at distance $L_A + L_B = 1.0 + 1.5 = 2.5$ m from B (to the left) - $W_C$ acts downward at distance $L_B = 1.5$ m from B (to the left) - $W_D$ acts downward at point D, which is at $L_D = 2.4$ m to the right of B 5. **Set up moment equation about B (counterclockwise positive):** $$R_A \times 2.5 - W_C \times 1.5 - W_D \times 2.4 = 0$$ 6. **Substitute values:** $$R_A \times 2.5 - 90.0 \times 1.5 - 30.0 \times 2.4 = 0$$ 7. **Calculate:** $$R_A \times 2.5 - 135.0 - 72.0 = 0$$ $$R_A \times 2.5 - 207.0 = 0$$ 8. **Solve for $R_A$:** $$R_A \times 2.5 = 207.0$$ $$R_A = \frac{207.0}{2.5}$$ 9. **Simplify fraction with cancellation:** $$R_A = \frac{\cancel{207.0}}{\cancel{2.5}}$$ 10. **Final calculation:** $$R_A = 82.8$$ **Answer:** The reaction force $R_A$ is $82.8$ kN (upward).