1. **State the problem:** We are given an open box with dimensions involving $x$: - Height: $25$ cm - Width: $(x - 2)$ cm - Length: $(3x - 63)$ cm The volume of the box is $81,900$ cm$^3$. We need to find the value of $x$. 2. **Formula for volume of a rectangular box:** $$\text{Volume} = \text{length} \times \text{width} \times \text{height}$$ 3. **Set up the equation:** $$81,900 = (3x - 63)(x - 2)(25)$$ 4. **Simplify the equation:** Divide both sides by 25: $$\frac{81,900}{25} = (3x - 63)(x - 2)$$ $$3,276 = (3x - 63)(x - 2)$$ 5. **Expand the right side:** $$ (3x - 63)(x - 2) = 3x \cdot x - 3x \cdot 2 - 63 \cdot x + 63 \cdot 2 $$ $$= 3x^2 - 6x - 63x + 126$$ $$= 3x^2 - 69x + 126$$ 6. **Form the quadratic equation:** $$3x^2 - 69x + 126 = 3,276$$ Subtract 3,276 from both sides: $$3x^2 - 69x + 126 - 3,276 = 0$$ $$3x^2 - 69x - 3,150 = 0$$ 7. **Divide entire equation by 3 to simplify:** $$x^2 - 23x - 1,050 = 0$$ 8. **Solve quadratic equation using the quadratic formula:** $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-23$, $c=-1,050$. Calculate discriminant: $$\Delta = (-23)^2 - 4(1)(-1,050) = 529 + 4,200 = 4,729$$ Calculate square root: $$\sqrt{4,729} = 68.77 \text{ (approx)}$$ Calculate roots: $$x = \frac{23 \pm 68.77}{2}$$ Two possible values: - $$x = \frac{23 + 68.77}{2} = \frac{91.77}{2} = 45.885$$ - $$x = \frac{23 - 68.77}{2} = \frac{-45.77}{2} = -22.885$$ 9. **Check for valid solution:** Since dimensions must be positive, check $x=45.885$: - $x - 2 = 43.885 > 0$ - $3x - 63 = 3(45.885) - 63 = 137.655 - 63 = 74.655 > 0$ $x = -22.885$ would give negative dimensions, so discard. **Final answer:** $$\boxed{45.885}$$