1. **Problem:** An open-topped rectangular box is made by cutting squares of side $x$ cm from each corner of a $30$ cm by $24$ cm cardboard and folding up the sides. The box volume is $648$ cm³. Find $x$. 2. **Formula:** Volume of the box $V = ext{length} \times \text{width} \times \text{height}$. 3. **Step 1:** After cutting squares of side $x$, the new length and width are $(30 - 2x)$ and $(24 - 2x)$ respectively, and height is $x$. 4. **Step 2:** Write volume equation: $$V = x(30 - 2x)(24 - 2x) = 648$$ 5. **Step 3:** Expand: $$x(720 - 60x - 48x + 4x^2) = 648$$ $$x(720 - 108x + 4x^2) = 648$$ 6. **Step 4:** Multiply out: $$720x - 108x^2 + 4x^3 = 648$$ 7. **Step 5:** Rearrange to standard polynomial form: $$4x^3 - 108x^2 + 720x - 648 = 0$$ 8. **Step 6:** Divide entire equation by 4 to simplify: $$x^3 - 27x^2 + 180x - 162 = 0$$ 9. **Step 7:** Solve cubic equation for $x$ (possible values must satisfy $0 < x < 12$ since $x$ must be less than half the smaller side). 10. **Step 8:** Test possible rational roots using Rational Root Theorem: try $x=3$: $$3^3 - 27(3)^2 + 180(3) - 162 = 27 - 243 + 540 - 162 = 162 eq 0$$ Try $x=1$: $$1 - 27 + 180 - 162 = -8 eq 0$$ Try $x=2$: $$8 - 108 + 360 - 162 = 98 eq 0$$ Try $x=6$: $$216 - 972 + 1080 - 162 = 162 eq 0$$ 11. **Step 9:** Use numerical methods or graphing to approximate root. Using a calculator or graphing, root near $x \approx 3$ cm. 12. **Answer:** The value of $x$ is approximately $3$ cm. This means squares of side 3 cm are cut from each corner to form the box with volume 648 cm³.