1. **Problem Statement:** We have a rectangular prism with surface area 80 square units. The prism shown is 5 units long, 4 units wide, and 2 units high. We want to find other rectangular prisms with the same surface area of 80 and determine their volumes. We also want to find the greatest volume possible for such prisms. 2. **Surface Area Formula:** The surface area $S$ of a rectangular prism with length $l$, width $w$, and height $h$ is given by: $$S = 2(lw + lh + wh)$$ 3. **Given:** $$2(lw + lh + wh) = 80$$ Divide both sides by 2: $$lw + lh + wh = 40$$ 4. **Known prism:** $l=5$, $w=4$, $h=2$ Calculate volume: $$V = lwh = 5 \times 4 \times 2 = 40$$ 5. **Find other prisms:** We want integer triples $(l,w,h)$ such that $$lw + lh + wh = 40$$ and $l,w,h > 0$. 6. **Method:** Fix $l$, solve for $w$ and $h$ satisfying $$lw + lh + wh = 40$$ Rewrite as $$w(l + h) + lh = 40$$ Try integer values for $l$ and $h$, then solve for $w$: $$w = \frac{40 - lh}{l + h}$$ 7. **Check integer solutions:** - For $l=5$, $h=2$: $$w = \frac{40 - 5 \times 2}{5 + 2} = \frac{40 - 10}{7} = \frac{30}{7} \notin \mathbb{Z}$$ But original prism is $w=4$, so check original equation: $$5 \times 4 + 5 \times 2 + 4 \times 2 = 20 + 10 + 8 = 38 \neq 40$$ This means the original prism surface area is actually: $$2 \times 38 = 76$$ which contradicts the problem statement. So we must trust the problem's surface area 80 and find other prisms. 8. **Try $l=4$, $h=4$:** $$w = \frac{40 - 4 \times 4}{4 + 4} = \frac{40 - 16}{8} = \frac{24}{8} = 3$$ Check surface area: $$4 \times 3 + 4 \times 4 + 3 \times 4 = 12 + 16 + 12 = 40$$ Volume: $$4 \times 3 \times 4 = 48$$ 9. **Try $l=8$, $h=1$:** $$w = \frac{40 - 8 \times 1}{8 + 1} = \frac{40 - 8}{9} = \frac{32}{9} \notin \mathbb{Z}$$ 10. **Try $l=10$, $h=1$:** $$w = \frac{40 - 10 \times 1}{10 + 1} = \frac{30}{11} \notin \mathbb{Z}$$ 11. **Try $l=2$, $h=6$:** $$w = \frac{40 - 2 \times 6}{2 + 6} = \frac{40 - 12}{8} = \frac{28}{8} = 3.5 \notin \mathbb{Z}$$ 12. **Try $l=1$, $h=7$:** $$w = \frac{40 - 1 \times 7}{1 + 7} = \frac{33}{8} \notin \mathbb{Z}$$ 13. **Try $l=2$, $h=5$:** $$w = \frac{40 - 2 \times 5}{2 + 5} = \frac{40 - 10}{7} = \frac{30}{7} \notin \mathbb{Z}$$ 14. **Try $l=3$, $h=3$:** $$w = \frac{40 - 3 \times 3}{3 + 3} = \frac{40 - 9}{6} = \frac{31}{6} \notin \mathbb{Z}$$ 15. **Try $l=5$, $h=3$:** $$w = \frac{40 - 5 \times 3}{5 + 3} = \frac{40 - 15}{8} = \frac{25}{8} \notin \mathbb{Z}$$ 16. **Try $l=4$, $h=5$:** $$w = \frac{40 - 4 \times 5}{4 + 5} = \frac{40 - 20}{9} = \frac{20}{9} \notin \mathbb{Z}$$ 17. **Try $l=1$, $h=9$:** $$w = \frac{40 - 1 \times 9}{1 + 9} = \frac{31}{10} \notin \mathbb{Z}$$ 18. **Try $l=2$, $h=8$:** $$w = \frac{40 - 2 \times 8}{2 + 8} = \frac{40 - 16}{10} = \frac{24}{10} = 2.4 \notin \mathbb{Z}$$ 19. **Try $l=1$, $h=10$:** $$w = \frac{40 - 1 \times 10}{1 + 10} = \frac{30}{11} \notin \mathbb{Z}$$ 20. **Try $l=2$, $h=7$:** $$w = \frac{40 - 2 \times 7}{2 + 7} = \frac{40 - 14}{9} = \frac{26}{9} \notin \mathbb{Z}$$ 21. **Try $l=4$, $h=3$:** $$w = \frac{40 - 4 \times 3}{4 + 3} = \frac{40 - 12}{7} = \frac{28}{7} = 4$$ Check surface area: $$4 \times 4 + 4 \times 3 + 4 \times 3 = 16 + 12 + 12 = 40$$ Volume: $$4 \times 4 \times 3 = 48$$ 22. **Summary of integer solutions found:** - $(l,w,h) = (4,3,4)$ volume 48 - $(l,w,h) = (4,4,3)$ volume 48 (same as above, just permuted) 23. **Greatest volume:** Among integer solutions with surface area 80, volume 48 is greater than the original prism's volume 40. **Final answer:** - Other rectangular prisms with surface area 80 include $4 \times 3 \times 4$ and $4 \times 4 \times 3$ with volume 48. - The greatest volume found is 48 cubic units.