1. **Problem:** Find three different rectangular prisms each with surface area 72 square units. 2. **Formula:** Surface area of a rectangular prism with length $l$, width $w$, and height $h$ is $$SA = 2(lw + lh + wh)$$ 3. We want $2(lw + lh + wh) = 72$, so $$lw + lh + wh = 36$$ 4. Choose integer values for $l$, $w$, and solve for $h$: - Example 1: $l=3$, $w=3$ $$3\times3 + 3\times h + 3\times h = 36 \Rightarrow 9 + 3h + 3h = 36 \Rightarrow 6h = 27 \Rightarrow h = 4.5$$ So dimensions: $(3,3,4.5)$ - Example 2: $l=2$, $w=6$ $$2\times6 + 2\times h + 6\times h = 36 \Rightarrow 12 + 2h + 6h = 36 \Rightarrow 8h = 24 \Rightarrow h = 3$$ So dimensions: $(2,6,3)$ - Example 3: $l=1$, $w=8$ $$1\times8 + 1\times h + 8\times h = 36 \Rightarrow 8 + h + 8h = 36 \Rightarrow 9h = 28 \Rightarrow h = \frac{28}{9} \approx 3.11$$ So dimensions: $(1,8,\frac{28}{9})$ --- 1. **Problem:** Find the least surface area of a rectangular prism with volume 64 cubic inches. 2. **Formula:** Volume $V = lwh = 64$ 3. Surface area is $$SA = 2(lw + lh + wh)$$ 4. To minimize surface area for fixed volume, the prism should be a cube: $$l = w = h = \sqrt[3]{64} = 4$$ 5. Calculate surface area: $$SA = 2(4\times4 + 4\times4 + 4\times4) = 2(16 + 16 + 16) = 2 \times 48 = 96$$ **Final answers:** - Three prisms with surface area 72: $(3,3,4.5)$, $(2,6,3)$, $(1,8,\frac{28}{9})$ - Least surface area for volume 64: $96$ square inches