1. **Problem 1: Find the length $x$ of the sector given the area is 46 cm$^2$.** The area $A$ of a sector of a circle is given by the formula: $$A = \frac{\theta}{360} \times \pi r^2$$ where $\theta$ is the central angle in degrees and $r$ is the radius (length $x$ here). Given: - $A = 46$ cm$^2$ - $\theta = 80^\circ$ - $\pi \approx 3.14$ Step 1: Substitute the known values: $$46 = \frac{80}{360} \times 3.14 \times x^2$$ Step 2: Simplify the fraction: $$\frac{80}{360} = 0.2222$$ Step 3: Rewrite the equation: $$46 = 0.2222 \times 3.14 \times x^2$$ Step 4: Multiply constants: $$46 = 0.6977 \times x^2$$ Step 5: Solve for $x^2$: $$x^2 = \frac{46}{0.6977} \approx 65.95$$ Step 6: Take the square root: $$x = \sqrt{65.95} \approx 8.1 \text{ cm}$$ 2. **Problem 2: Find the area of the card to make a cone-shaped hat with radius 14 cm and slope height 20 cm.** The surface area $S$ of the cone's lateral side (the card area) is: $$S = \pi r l$$ where $r$ is the radius and $l$ is the slant height. Given: - $r = 14$ cm - $l = 20$ cm - $\pi \approx 3.14$ Step 1: Substitute values: $$S = 3.14 \times 14 \times 20$$ Step 2: Calculate: $$S = 3.14 \times 280 = 879.2 \text{ cm}^2$$ **Final answers:** - Length $x$ of the sector: **8.1 cm** - Area of the card for the cone: **879.2 cm$^2$**