1. **State the problem:** We are given the areas of the faces of a rectangular prism's net: two faces of 40 cm², two faces of 32 cm², and two faces of 20 cm². We need to find the dimensions of the prism (length, width, height). 2. **Recall the formula:** The areas correspond to the pairs of opposite faces of the prism. If the dimensions are $l$, $w$, and $h$, then the areas are: $$lw, lh, wh$$ Each area appears twice in the net. 3. **Assign the areas:** Let: $$lw = 20$$ $$lh = 32$$ $$wh = 40$$ 4. **Find the dimensions:** From $lw=20$, express $w=\frac{20}{l}$. 5. Substitute $w$ into $wh=40$: $$\left(\frac{20}{l}\right)h = 40$$ Simplify: $$\frac{20h}{l} = 40$$ Multiply both sides by $l$: $$20h = 40l$$ Divide both sides by 20: $$h = 2l$$ 6. Substitute $h=2l$ into $lh=32$: $$l(2l) = 32$$ $$2l^2 = 32$$ Divide both sides by 2: $$l^2 = 16$$ Take the square root: $$l = 4$$ 7. Find $h$: $$h = 2l = 2 \times 4 = 8$$ 8. Find $w$: $$w = \frac{20}{l} = \frac{20}{4} = 5$$ 9. **Answer:** The dimensions are $4$, $5$, and $8$. 10. **Check:** $$lw = 4 \times 5 = 20$$ $$lh = 4 \times 8 = 32$$ $$wh = 5 \times 8 = 40$$ All match the given areas. **Final answer:** 4, 5, 8