1. **State the problem:** We have a cuboid with dimensions $x$ cm, $x$ cm, and $(15 - 4x)$ cm. 2. **Formula for volume:** The volume $V$ of a cuboid is given by the product of its length, width, and height: $$V = \text{length} \times \text{width} \times \text{height}$$ 3. **Apply the formula:** Here, $$V = x \times x \times (15 - 4x) = x^2(15 - 4x)$$ 4. **Expand the expression:** $$V = 15x^2 - 4x^3$$ 5. **Find the maximum volume:** To find the maximum value of $V$, we take the derivative of $V$ with respect to $x$ and set it to zero: $$\frac{dV}{dx} = 30x - 12x^2$$ 6. **Set derivative to zero:** $$30x - 12x^2 = 0$$ 7. **Factor out $6x$:** $$6x(5 - 2x) = 0$$ 8. **Solve for $x$:** $$6x = 0 \Rightarrow x = 0$$ $$5 - 2x = 0 \Rightarrow x = \frac{5}{2} = 2.5$$ 9. **Check domain:** Since dimensions must be positive and $(15 - 4x) > 0$, $$15 - 4x > 0 \Rightarrow x < \frac{15}{4} = 3.75$$ So $x=2.5$ is valid. 10. **Second derivative test:** $$\frac{d^2V}{dx^2} = 30 - 24x$$ At $x=2.5$: $$30 - 24(2.5) = 30 - 60 = -30 < 0$$ This indicates a maximum. 11. **Calculate maximum volume:** $$V = 15(2.5)^2 - 4(2.5)^3 = 15 \times 6.25 - 4 \times 15.625 = 93.75 - 62.5 = 31.25$$ **Final answer:** The maximum volume of the cuboid is $31.25$ cm$^3$.