1. **Stating the problem:** Stéphanie wants to renovate her kitchen floor with ceramic tiles around a square island. We need to find a tile size that fits her budget of 1000 and meets the minimum tile count of 50. 2. **Given values:** - Kitchen floor dimensions: length = $3x + 4$, width = $\frac{14x^2 - 63x}{7x}$ (simplify this). - Island side length: $x + 6$. - $x = 9$. 3. **Simplify the floor width:** $$\frac{14x^2 - 63x}{7x} = \frac{7x(2x - 9)}{7x} = \cancel{\frac{7x}{7x}}(2x - 9) = 2x - 9$$ 4. **Calculate floor dimensions with $x=9$:** - Length: $3(9) + 4 = 27 + 4 = 31$ - Width: $2(9) - 9 = 18 - 9 = 9$ 5. **Calculate island side length:** $$x + 6 = 9 + 6 = 15$$ 6. **Calculate floor area and island area:** - Floor area: $31 \times 9 = 279$ - Island area: $15 \times 15 = 225$ 7. **Calculate tileable area:** $$279 - 225 = 54$$ 8. **Tile options:** - Tile A: Rectangle $x \times 2 = 9 \times 2 = 18$ area, price 10 each. - Tile B: Not available. - Tile C: Diamond with diagonals $x=9$ and $0.5$. Area of diamond = $\frac{d_1 \times d_2}{2} = \frac{9 \times 0.5}{2} = \frac{4.5}{2} = 2.25$ 9. **Minimum tiles needed:** 50. 10. **Check if Tile A fits budget and minimum tiles:** - Number of tiles needed: $\frac{54}{18} = 3$ tiles (less than 50, so fails minimum tile count). 11. **Check Tile C:** - Number of tiles needed: $\frac{54}{2.25} = 24$ tiles (less than 50, fails minimum tile count). 12. **Conclusion:** Neither tile A nor C meets the minimum tile count of 50 tiles for the given floor and island dimensions with $x=9$. **Final answer:** Stéphanie cannot meet the minimum tile count of 50 with the given tile sizes and floor dimensions at $x=9$. She needs to choose smaller tiles or adjust $x$ to meet constraints.