1. **State the problem:** Reese's old bike mirror is a rectangle with length 10 cm and width 5 cm. She wants a new circular mirror with approximately the same area. 2. **Formula for area of rectangle:** $$\text{Area}_{rectangle} = \text{length} \times \text{width}$$ 3. **Calculate area of old mirror:** $$10 \times 5 = 50 \text{ cm}^2$$ 4. **Formula for area of circle:** $$\text{Area}_{circle} = \pi r^2$$ 5. **Calculate areas of circular mirrors:** - For radius 4 cm: $$\pi \times 4^2 = 16\pi \approx 50.27 \text{ cm}^2$$ - For radius 7 cm: $$\pi \times 7^2 = 49\pi \approx 153.94 \text{ cm}^2$$ - For radius 8 cm: $$\pi \times 8^2 = 64\pi \approx 201.06 \text{ cm}^2$$ 6. **Compare areas:** The circular mirror with radius 4 cm has an area closest to 50 cm², which matches the old mirror's area. **Final answer:** Reese should buy the circular mirror with radius 4 cm.