1. **State the problem:** We have a square sheet with area 900 cm$^2$, so each side is $\sqrt{900} = 30$ cm. 2. **Form the box:** Squares of side $x$ cm are cut from each corner, and the edges are folded up to form an open box. 3. **Volume formula:** The resulting box has height $x$, and the base dimensions are reduced by $2x$ on each side, so the base is $(30 - 2x) \times (30 - 2x)$. Volume $V$ is: $$ V = x(30 - 2x)^2 $$ 4. **Expand the volume:** $$ V = x(900 - 120x + 4x^2) = 900x - 120x^2 + 4x^3 $$ 5. **Find critical points:** Take derivative with respect to $x$: $$ \frac{dV}{dx} = 900 - 240x + 12x^2 $$ Set derivative to zero to find maxima/minima: $$ 900 - 240x + 12x^2 = 0 $$ Divide entire equation by 12: $$ \cancel{12}75 - \cancel{12}20x + \cancel{12}x^2 = 0 \implies 75 - 20x + x^2 = 0 $$ Rewrite: $$ x^2 - 20x + 75 = 0 $$ 6. **Solve quadratic:** $$ x = \frac{20 \pm \sqrt{(-20)^2 - 4 \times 1 \times 75}}{2} = \frac{20 \pm \sqrt{400 - 300}}{2} = \frac{20 \pm \sqrt{100}}{2} $$ $$ x = \frac{20 \pm 10}{2} $$ Two solutions: - $x = \frac{20 + 10}{2} = 15$ - $x = \frac{20 - 10}{2} = 5$ 7. **Check domain and maximum:** Since $x$ must be less than half the side length (to form a box), $x < 15$ cm. At $x=15$, the base becomes zero, so volume is zero. Check volume at $x=5$: $$ V = 5(30 - 2 \times 5)^2 = 5(30 - 10)^2 = 5 \times 20^2 = 5 \times 400 = 2000 $$ 8. **Confirm maximum volume:** Second derivative: $$ \frac{d^2V}{dx^2} = -240 + 24x $$ At $x=5$: $$ -240 + 24 \times 5 = -240 + 120 = -120 < 0 $$ Negative second derivative means local maximum. **Final answers:** **a.** $x = 5$ cm **b.** Maximum volume = 2000 cm$^3$