1. **Problem Statement:** We want to find the optimal side length $s$ of equilateral triangular panels to cover a tunnel dome with total surface area $A_{total} = 58058\,m^2$. 2. **Given Formulas:** - Area of one triangle: $A(s) = \frac{\sqrt{3}}{4}s^2$ - Ideal number of panels (no waste): $N_{ideal}(s) = \frac{A_{total}}{A(s)} = \frac{58058}{\frac{\sqrt{3}}{4}s^2} = \frac{58058 \times 4}{\sqrt{3} s^2}$ - Adjusted number of panels with 7% waste: $N_{procure}(s) = 1.07 \times N_{ideal}(s)$ - Total strut length: $L_{total}(s) = \frac{3}{2} N_{procure}(s) s$ - Total cost: $C_{total}(s) = N_{procure}(s) A(s) C_p + L_{total}(s) C_s = N_{procure}(s) \frac{\sqrt{3}}{4} s^2 C_p + \frac{3}{2} N_{procure}(s) s C_s$ 3. **Constraints:** - Coverage: $A_{covered}(s) = N_{procure}(s) A(s) \geq 58058$ (always true by construction) - Budget: $C_{total}(s) \leq 7000000$ - Panel size: $0.5 \leq s \leq 8$ meters 4. **Assumptions for costs:** Assuming example costs: - Panel cost per $m^2$: $C_p = 100$ - Strut cost per meter: $C_s = 50$ 5. **Express $N_{procure}(s)$ explicitly:** $N_{ideal}(s) = \frac{58058 \times 4}{\sqrt{3} s^2} \approx \frac{232232}{1.732 s^2} = \frac{134000}{s^2}$ $N_{procure}(s) = 1.07 \times \frac{134000}{s^2} = \frac{143380}{s^2}$ 6. **Calculate total cost function:** $C_{total}(s) = N_{procure}(s) \frac{\sqrt{3}}{4} s^2 C_p + \frac{3}{2} N_{procure}(s) s C_s$ Substitute $N_{procure}(s)$: $= \frac{143380}{s^2} \times \frac{\sqrt{3}}{4} s^2 C_p + \frac{3}{2} \times \frac{143380}{s^2} \times s C_s$ Simplify: $= 143380 \times \frac{\sqrt{3}}{4} C_p + \frac{3}{2} \times 143380 \times \frac{C_s}{s}$ Numerical values: $= 143380 \times 0.433 \times 100 + 215070 \times \frac{50}{s}$ $= 6200000 + \frac{10753500}{s}$ 7. **Budget constraint:** $C_{total}(s) \leq 7000000$ $6200000 + \frac{10753500}{s} \leq 7000000$ $\frac{10753500}{s} \leq 800000$ $s \geq \frac{10753500}{800000} = 13.44$ 8. **Check panel size bounds:** The required $s$ to meet budget is $13.44$ m, but max allowed $s$ is 8 m. 9. **Evaluate cost at $s=8$ m:** $C_{total}(8) = 6200000 + \frac{10753500}{8} = 6200000 + 1344187.5 = 7544187.5 > 7000000$ Budget exceeded. 10. **Evaluate cost at $s=0.5$ m:** $C_{total}(0.5) = 6200000 + \frac{10753500}{0.5} = 6200000 + 21507000 = 27707000$ Far exceeds budget. 11. **Conclusion:** The cost function decreases as $s$ increases, but even at max $s=8$ m, cost exceeds budget. 12. **Adjust cost assumptions or constraints:** To fit budget, either reduce $C_p$, $C_s$, or increase budget. 13. **If $C_s$ reduced to 10:** $C_{total}(s) = 143380 \times 0.433 \times 100 + 215070 \times \frac{10}{s} = 6200000 + \frac{2150700}{s}$ Budget constraint: $6200000 + \frac{2150700}{s} \leq 7000000$ $\frac{2150700}{s} \leq 800000$ $s \geq 2.69$ 14. **Check $s=2.7$ m:** - Number of panels: $N_{procure}(2.7) = \frac{143380}{2.7^2} = \frac{143380}{7.29} = 19655$ - Total coverage: $A_{covered} = N_{procure} \times A(s) = 19655 \times \frac{\sqrt{3}}{4} \times 2.7^2 = 19655 \times 3.15 = 61915\,m^2 \geq 58058$ - Total cost: $C_{total}(2.7) = 6200000 + \frac{2150700}{2.7} = 6200000 + 796555 = 6996555 \leq 7000000$ 15. **Final optimal values:** - Side length $s \approx 2.7$ m - Number of panels $N_{procure} \approx 19655$ - Total cost $\approx 6996555$ (within budget) - Coverage $\approx 61915\,m^2$ (satisfies coverage) **Verification against constraints:** - Coverage constraint: $61915 \geq 58058$ satisfied. - Budget constraint: $6996555 \leq 7000000$ satisfied. - Panel size constraint: $0.5 \leq 2.7 \leq 8$ satisfied. **Summary:** The optimal panel side length is about 2.7 meters, yielding about 19,655 triangular panels, covering the dome fully and fitting within the 7,000,000 budget for panels and struts under assumed costs. Adjusting material costs or budget will shift this value.