1. **State the problem:** We have the formula for sag $S$ of a wire stretched between two points: $$S = \sqrt{\frac{8d(4L - d)}{15}}$$ where $L$ is the length of the wire, $d$ is the distance between the points, and $S$ is the sag. We need to: (i) Change the subject of the formula to $L$. (ii) Find $L$ when $d = 25$ and $S = 0.18$. --- 2. **Change the subject to $L$:** Start with: $$S = \sqrt{\frac{8d(4L - d)}{15}}$$ Square both sides to remove the square root: $$S^2 = \frac{8d(4L - d)}{15}$$ Multiply both sides by 15: $$15S^2 = 8d(4L - d)$$ Divide both sides by $8d$: $$\frac{15S^2}{8d} = 4L - d$$ Add $d$ to both sides: $$4L = d + \frac{15S^2}{8d}$$ Divide both sides by 4: $$L = \frac{d}{4} + \frac{15S^2}{32d}$$ --- 3. **Find $L$ for $d=25$ and $S=0.18$:** Substitute values: $$L = \frac{25}{4} + \frac{15 \times (0.18)^2}{32 \times 25}$$ Calculate each term: $$\frac{25}{4} = 6.25$$ $$0.18^2 = 0.0324$$ $$15 \times 0.0324 = 0.486$$ $$32 \times 25 = 800$$ So: $$\frac{0.486}{800} = 0.0006075$$ Add terms: $$L = 6.25 + 0.0006075 = 6.2506075$$ **Final answer:** $$L \approx 6.25$$ meters (to 2 decimal places).