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 (since not given explicitly): - Panel cost per $m^2$: $C_p = 100$ (units) - Strut cost per meter: $C_s = 50$ (units) 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 $\approx 6,996,555$ (within budget) - Coverage $\approx 61,915 m^2$ (satisfies coverage) **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.