1. **State the problem:** Determine the reactions at the supports (R_A and R_B) of a bridge with given loads and distances. 2. **Given data:** - Optimus Prime mass = 7 tonnes = 7000 kg - Bumble Bee mass = 1 tonne = 1000 kg - Gravity = 10 m/s^2 - Loads on bridge: - 10 kN at 18 m from left end - 70 kN at 6 m from right end - 200 kN at center (30 m from left end, since total span is 30 m) - Bridge span = 30 m (two segments of 15 m each) 3. **Convert masses to forces:** - Force due to Optimus Prime = $7000 \times 10 = 70000$ N = 70 kN - Force due to Bumble Bee = $1000 \times 10 = 10000$ N = 10 kN 4. **Total loads on the bridge:** - Given loads: 10 kN, 70 kN, 200 kN - Additional loads from Optimus Prime and Bumble Bee: 70 kN and 10 kN - Total vertical loads = $10 + 70 + 200 + 70 + 10 = 360$ kN 5. **Assumptions:** - The loads act vertically downward. - The bridge is simply supported at two points: left (R_A) and right (R_B). 6. **Set up equilibrium equations:** - Sum of vertical forces = 0: $$R_A + R_B = 360$$ - Sum of moments about left support (taking counterclockwise as positive): $$\sum M_A = 0$$ 7. **Locate all loads from left end:** - 10 kN at 18 m - 70 kN at $30 - 6 = 24$ m - 200 kN at 15 m (center) - Optimus Prime and Bumble Bee positions are not given explicitly; assuming they coincide with the 10 kN and 70 kN loads respectively (since their masses correspond to those forces), so: - Optimus Prime (70 kN) at 18 m - Bumble Bee (10 kN) at 24 m 8. **Calculate moments about left support:** $$\sum M_A = 0 = R_B \times 30 - 10 \times 18 - 70 \times 24 - 200 \times 15 - 70 \times 18 - 10 \times 24$$ 9. **Calculate the sum of moments of loads:** $$10 \times 18 = 180$$ $$70 \times 24 = 1680$$ $$200 \times 15 = 3000$$ $$70 \times 18 = 1260$$ $$10 \times 24 = 240$$ $$\text{Total moment} = 180 + 1680 + 3000 + 1260 + 240 = 6360$$ 10. **Solve for $R_B$:** $$R_B \times 30 = 6360$$ $$R_B = \frac{6360}{30} = 212$$ 11. **Solve for $R_A$ using vertical forces sum:** $$R_A + 212 = 360$$ $$R_A = 360 - 212 = 148$$ 12. **Final answer:** - Reaction at left support: $R_A = 148$ kN - Reaction at right support: $R_B = 212$ kN These are the loads on each side of the bridge (reactions at the supports).