1. **Problem Statement:** We are given a volume function $$V(x) = (16 - 2x)(30 - 2x)x$$ representing the volume of a box formed by cutting squares of side length $$x$$ from each corner of a 16 by 30 rectangle and folding up the sides. 2. **Formula and Explanation:** The volume of the box is the product of its length, width, and height. - Length after cutting: $$30 - 2x$$ (since squares are cut from both ends) - Width after cutting: $$16 - 2x$$ - Height: $$x$$ (the size of the cut square) So, $$V(x) = (16 - 2x)(30 - 2x)x$$. 3. **Domain Explanation:** The domain is $$0 \leq x \leq 8$$ because: - $$x \geq 0$$ since the cut length cannot be negative. - $$x \leq 8$$ because if $$x$$ were larger than 8, the width $$16 - 2x$$ would become zero or negative, which is impossible for a physical box. 4. **Intermediate Work:** Expand the volume function: $$ V(x) = (16 - 2x)(30 - 2x)x = [480 - 32x - 60x + 4x^2]x = (480 - 92x + 4x^2)x $$ Multiply through by $$x$$: $$ V(x) = 480x - 92x^2 + 4x^3 $$ 5. **Summary:** - Volume function: $$V(x) = 480x - 92x^2 + 4x^3$$ - Domain: $$0 \leq x \leq 8$$ to ensure physical feasibility of the box dimensions. This explains why the domain must be less than or equal to 8.