1. **State 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 with the given formula:** $$S = \sqrt{\frac{8d(4L - d)}{15}}$$ 3. **Square both sides to eliminate the square root:** $$S^2 = \frac{8d(4L - d)}{15}$$ 4. **Multiply both sides by 15 to clear the denominator:** $$15S^2 = 8d(4L - d)$$ 5. **Expand the right side:** $$15S^2 = 32dL - 8d^2$$ 6. **Add $8d^2$ to both sides:** $$15S^2 + 8d^2 = 32dL$$ 7. **Divide both sides by $32d$ to isolate $L$:** $$L = \frac{15S^2 + 8d^2}{32d}$$ **Final formula:** $$\boxed{L = \frac{15S^2 + 8d^2}{32d}}$$ This formula expresses the wire length $L$ in terms of the sag $S$ and the distance $d$ between the two points. **Explanation:** - We started by squaring both sides to remove the square root. - Then we cleared the fraction by multiplying both sides by 15. - We expanded and rearranged terms to isolate $L$. - Finally, we divided to solve for $L$. This method is a standard approach to changing the subject of a formula involving square roots and fractions.