1. **State the problem:** We have two similar quadrilateral shapes, Q and R. - Quadrilateral Q has a diagonal of 95 cm (nearest 1 cm) and an area of 490 cm² (nearest 10 cm²). - Quadrilateral R has a diagonal of 3.1 m (nearest 0.1 m). We want to find the lower and upper bounds for the area of R, in cm², given these measurements. 2. **Convert measurements to consistent units:** Since Q’s measurements are in cm, convert R’s diagonal from meters to centimeters: $$3.1 \text{ m} = 310 \text{ cm}$$ The diagonal of R is 310 cm to the nearest 10 cm (because 0.1 m = 10 cm). 3. **Determine the bounds for each dimension:** - For Q’s diagonal (95 cm to nearest 1 cm): - Lower bound = $95 - 0.5 = 94.5$ cm - Upper bound = $95 + 0.5 = 95.5$ cm - For Q’s area (490 cm² to nearest 10 cm²): - Lower bound = $490 - 5 = 485$ cm² - Upper bound = $490 + 5 = 495$ cm² - For R’s diagonal (310 cm to nearest 10 cm): - Lower bound = $310 - 5 = 305$ cm - Upper bound = $310 + 5 = 315$ cm 4. **Calculate scale factor bounds:** The scale factor $k$ relates linear dimensions of R to Q: $$k = \frac{\text{diagonal of R}}{\text{diagonal of Q}}$$ Calculate lower and upper bounds of $k$: - Lower scale factor: $$k_{\text{low}} = \frac{305}{95.5} \approx 3.193$$ - Upper scale factor: $$k_{\text{high}} = \frac{315}{94.5} \approx 3.333$$ 5. **Calculate area bounds of R:** Area scales as the square of the scale factor: $$\text{Area}_{R} = k^2 \times \text{Area}_{Q}$$ Calculate lower bound of area: $$\text{Area}_{R, \text{low}} = (3.193)^2 \times 485 \approx 10.201 \times 485 = 4949.5 \text{ cm}^2$$ Calculate upper bound of area: $$\text{Area}_{R, \text{high}} = (3.333)^2 \times 495 \approx 11.111 \times 495 = 5499.9 \text{ cm}^2$$ 6. **Round answers to nearest 1 cm²:** - Lower bound area of R: $4950$ cm² - Upper bound area of R: $5500$ cm² **Final answer:** The area of quadrilateral R is between $4950$ cm² and $5500$ cm² to the nearest $1$ cm².