1. **State the problem:** We need to determine if Olio will need more wire than Ellie to install around their painting and its border. 2. **Given information:** - Ellie's wire length is the perimeter of the painting. - Olio's wire length is around the painting plus a 2 cm border. - Use $\pi \approx 3.14$. 3. **Ellie's wire calculation:** The painting consists of a rectangle with a quarter circle cut out. - Rectangle perimeter without quarter circle: $2 \times (80 + 30) = 220$ cm. - Quarter circle perimeter (arc length): $\frac{1}{4} \times 2 \pi r = \frac{1}{4} \times 2 \times 3.14 \times 30 = 47.1$ cm. Since the quarter circle is cut out, the perimeter is: $$220 - 30 - 30 + 47.1 = 207.1 \text{ cm}$$ (We subtract the two straight edges of the quarter circle, each 30 cm, and add the arc length.) 4. **Olio's wire calculation:** The border is 2 cm outside the painting, so dimensions increase by 4 cm total (2 cm each side). - New rectangle dimensions: $80 + 4 = 84$ cm height, $30 + 4 = 34$ cm width. - New quarter circle radius: $30 + 2 = 32$ cm. Calculate the new perimeter: - Rectangle perimeter: $2 \times (84 + 34) = 236$ cm. - Quarter circle arc length: $\frac{1}{4} \times 2 \times 3.14 \times 32 = 50.24$ cm. Subtract straight edges replaced by arc: $$236 - 34 - 34 + 50.24 = 218.24 \text{ cm}$$ 5. **Compare wire lengths:** - Ellie's wire: $207.1$ cm - Olio's wire: $218.24$ cm 6. **Conclusion:** Olio needs more wire than Ellie because the border increases the perimeter. **Final answer:** Olio will need more wire than Ellie.