1. **Problem:** P varies directly as the square of Q and inversely as the cube of Z. Given P=5, Q=3, Z=1, find: (i) The relationship between P, Q, and Z. (ii) Z when P=3 and Q=5. **Step 1:** Write the variation formula: $$P = k \frac{Q^2}{Z^3}$$ where $k$ is the constant of proportionality. **Step 2:** Use given values to find $k$: $$5 = k \frac{3^2}{1^3} = k \times 9 \Rightarrow k = \frac{5}{9}$$ **Step 3:** Write the relationship: $$P = \frac{5}{9} \frac{Q^2}{Z^3}$$ **Step 4:** Find $Z$ when $P=3$ and $Q=5$: $$3 = \frac{5}{9} \frac{5^2}{Z^3} = \frac{5}{9} \frac{25}{Z^3} = \frac{125}{9 Z^3}$$ Multiply both sides by $9 Z^3$: $$27 Z^3 = 125$$ Divide both sides by 27: $$Z^3 = \frac{125}{27}$$ Take cube root: $$Z = \sqrt[3]{\frac{125}{27}} = \frac{5}{3}$$ **Final answers:** (i) $P = \frac{5}{9} \frac{Q^2}{Z^3}$ (ii) $Z = \frac{5}{3}$