1. **Stating the problem:** We are given surface area, volume, and base area values for two shapes: a cylinder and a triangular prism. 2. **Cylinder (center-right):** Given: - Surface Area $SA = 421.5$ - Volume $V = 603.2$ - Base Area $B = 100.5$ We need to find: - Lateral Area $LA$ - Surface Area $SA$ (already given) - Volume $V$ (already given) - Perimeter $P$ of the base - Apothem $a$ - Base Area $B$ (already given) 3. **Formulas for cylinder:** - $LA = P \times h$ - $SA = LA + 2B$ - $V = B \times h$ Where $h$ is the height of the cylinder. 4. **Find height $h$ from volume:** $$V = B \times h \Rightarrow h = \frac{V}{B} = \frac{603.2}{100.5}$$ 5. **Calculate $h$:** $$h = \frac{603.2}{100.5} \approx 6.0$$ 6. **Find lateral area $LA$ from surface area:** $$SA = LA + 2B \Rightarrow LA = SA - 2B = 421.5 - 2 \times 100.5 = 421.5 - 201 = 220.5$$ 7. **Find perimeter $P$ using $LA = P \times h$:** $$P = \frac{LA}{h} = \frac{220.5}{6.0} = 36.75$$ 8. **Find apothem $a$ assuming base is a regular polygon:** Since $B = \frac{1}{2} P a$, rearranged: $$a = \frac{2B}{P} = \frac{2 \times 100.5}{36.75} \approx 5.47$$ --- 9. **Triangular prism (bottom-right):** Given: - Lateral Area $LA = 80.4$ - Surface Area $SA = 130.7$ - Volume $V = 83.8$ No further data or requests given for this shape, so we only report these values. --- **Final answers for the cylinder:** - Lateral Area $LA = 220.5$ - Surface Area $SA = 421.5$ - Volume $V = 603.2$ - Perimeter $P = 36.75$ - Apothem $a = 5.47$ - Base Area $B = 100.5$ **Triangular prism values given:** - Lateral Area $LA = 80.4$ - Surface Area $SA = 130.7$ - Volume $V = 83.8$