1. **State the problem:** We are given the sag formula for a wire stretched between two points: $$S = \sqrt{\frac{8d(4L - d)}{15}}$$ We need to find the length of the wire $L$ when $d = 25$ and $S = 0.18$. 2. **Write down the formula and what is given:** $$S = \sqrt{\frac{8d(4L - d)}{15}}$$ Given: - $d = 25$ - $S = 0.18$ 3. **Square both sides to remove the square root:** $$S^2 = \frac{8d(4L - d)}{15}$$ Substitute values: $$0.18^2 = \frac{8 \times 25 (4L - 25)}{15}$$ 4. **Calculate $0.18^2$:** $$0.18^2 = 0.0324$$ So, $$0.0324 = \frac{200 (4L - 25)}{15}$$ 5. **Multiply both sides by 15 to clear denominator:** $$0.0324 \times 15 = 200 (4L - 25)$$ Calculate left side: $$0.486 = 200 (4L - 25)$$ 6. **Divide both sides by 200:** $$\frac{0.486}{200} = 4L - 25$$ Calculate: $$0.00243 = 4L - 25$$ 7. **Add 25 to both sides:** $$0.00243 + 25 = 4L$$ $$25.00243 = 4L$$ 8. **Divide both sides by 4 to solve for $L$:** $$L = \frac{25.00243}{4} = 6.2506075$$ 9. **Final answer:** The length of the wire is approximately $$L \approx 6.25 \text{ metres}$$