1. **Problem statement:** Given that $y$ varies directly with $(x-1)^2$, and when $x=4$, $y=18$, find: i. The relationship between $x$ and $y$ ii. The value of $y$ when $x=5$ iii. The value of $x$ when $y=8$ 2. **Formula and explanation:** Direct variation means $y = k \cdot (x-1)^2$ where $k$ is the constant of proportionality. 3. **Find $k$ using given values:** Substitute $x=4$ and $y=18$: $$18 = k \cdot (4-1)^2 = k \cdot 3^2 = 9k$$ Solve for $k$: $$k = \frac{18}{9} = 2$$ 4. **Relationship between $x$ and $y$:** $$y = 2 \cdot (x-1)^2$$ 5. **Find $y$ when $x=5$:** Substitute $x=5$: $$y = 2 \cdot (5-1)^2 = 2 \cdot 4^2 = 2 \cdot 16 = 32$$ 6. **Find $x$ when $y=8$:** Substitute $y=8$: $$8 = 2 \cdot (x-1)^2$$ Divide both sides by 2: $$(x-1)^2 = 4$$ Take square root: $$x-1 = \pm 2$$ So, $$x = 1 + 2 = 3 \quad \text{or} \quad x = 1 - 2 = -1$$ **Final answers:** - Relationship: $y = 2(x-1)^2$ - When $x=5$, $y=32$ - When $y=8$, $x=3$ or $x=-1$