1. **Problem Statement:** Find the lateral area and surface area of each figure given. --- ### Graph 1: Cylinder with height $h=12$ km and diameter $d=22$ km - Radius $r=\frac{d}{2}=\frac{22}{2}=11$ km **Formulas:** - Lateral area $= 2\pi rh$ - Surface area $= 2\pi r^2 + 2\pi rh$ **Calculations:** - Lateral area $= 2\pi (11)(12) = 264\pi$ - Surface area $= 2\pi (11)^2 + 264\pi = 2\pi (121) + 264\pi = 242\pi + 264\pi = 506\pi$ **Numerical values:** - Lateral area $\approx 264 \times 3.1416 = 829.38$ km$^2$ - Surface area $\approx 506 \times 3.1416 = 1589.05$ km$^2$ --- ### Graph 2: Pyramid with base edges $12$ m and slant height $13.4$ m - Assuming a square base with side length $s=12$ m **Formulas:** - Lateral area $= \frac{1}{2} \times \text{perimeter} \times \text{slant height} = \frac{1}{2} \times 4s \times l$ - Surface area $= \text{base area} + \text{lateral area} = s^2 + \text{lateral area}$ **Calculations:** - Perimeter $= 4 \times 12 = 48$ m - Lateral area $= \frac{1}{2} \times 48 \times 13.4 = 24 \times 13.4 = 321.6$ m$^2$ - Surface area $= 12^2 + 321.6 = 144 + 321.6 = 465.6$ m$^2$ --- ### Graph 3: Pentagonal prism with side lengths $10$ ft, $5$ ft, and $6.9$ ft - Assuming the pentagon has sides $10$, $5$, $6.9$, and two other sides equal to $10$ ft and $5$ ft (typical pentagon with given sides) - Height $h$ is not given, so lateral area cannot be computed exactly without height. Since height is missing, we cannot compute lateral or surface area for this prism. --- ### Graph 4: Cone with height $h=20.1$ yd and base radius $r=9$ yd **Formulas:** - Slant height $l = \sqrt{r^2 + h^2}$ - Lateral area $= \pi r l$ - Surface area $= \pi r^2 + \pi r l$ **Calculations:** - $l = \sqrt{9^2 + 20.1^2} = \sqrt{81 + 404.01} = \sqrt{485.01} \approx 22.02$ yd - Lateral area $= \pi \times 9 \times 22.02 = 198.18\pi$ - Surface area $= \pi \times 9^2 + 198.18\pi = 81\pi + 198.18\pi = 279.18\pi$ **Numerical values:** - Lateral area $\approx 198.18 \times 3.1416 = 622.56$ yd$^2$ - Surface area $\approx 279.18 \times 3.1416 = 876.88$ yd$^2$