1. **Stating the problem:** We are given the formula for sag $S$ in terms of wire length $L$ and distance $d$: $$S = \sqrt{\frac{8d(4L - d)}{15}}$$ We need to change the subject of the formula to $L$, i.e., express $L$ in terms of $S$ and $d$. 2. **Start by squaring both sides** to eliminate the square root: $$S^2 = \frac{8d(4L - d)}{15}$$ 3. **Multiply both sides by 15** to clear the denominator: $$15S^2 = 8d(4L - d)$$ 4. **Expand the right side:** $$15S^2 = 32dL - 8d^2$$ 5. **Isolate the term with $L$:** $$32dL = 15S^2 + 8d^2$$ 6. **Divide both sides by $32d$ to solve for $L$:** $$L = \frac{15S^2 + 8d^2}{32d}$$ **Final formula:** $$\boxed{L = \frac{15S^2 + 8d^2}{32d}}$$ This expresses the wire length $L$ in terms of sag $S$ and distance $d$.