1. Problem statement. A cuboid has length $12$, width $9$, and height $4$. A layer of uniform thickness $x$ is cut off from each face, producing a smaller cuboid whose volume is half the original. Find $x$. 2. Formula and important rules. Volume of a cuboid: $$V= lwh$$ When you cut off a thickness $x$ from both sides of a dimension, that dimension decreases by $2x$. Always keep units consistent and check that $00$, a root lies in $(0.5,1)$. Use bisection to narrow the root. Midpoint $x=0.75$ gives $$f(0.75)=2(0.75)^3-25(0.75)^2+96(0.75)-54\approx 4.78125.$$ So the root is in $(0.5,0.75)$. Midpoint $x=0.625$ gives $$f(0.625)=2(0.625)^3-25(0.625)^2+96(0.625)-54\approx -3.27734375.$$ So the root is in $(0.625,0.75)$. Midpoint $x=0.6875$ gives $$f(0.6875)\approx 0.83349609.$$ So the root is in $(0.625,0.6875)$. Midpoint $x=0.65625$ gives $$f(0.65625)\approx -1.2017827.$$ So the root is in $(0.65625,0.6875)$. Midpoint $x=0.671875$ gives $$f(0.671875)\approx -0.17864.$$ Midpoint $x=0.6796875$ gives $$f(0.6796875)\approx 0.3276.$$ Refine once more to get a good decimal approximation; the root lies near $x\approx 0.6745$. 9. (Optional check) Substitute $x\approx 0.6745$ into the volume expression to verify the half-volume condition. New dimensions approx: $12-2x\approx 10.651$, $9-2x\approx 7.651$, $4-2x\approx 2.651$. Their product $\approx 10.651\cdot 7.651\cdot 2.651\approx 216$ (within rounding error), confirming the solution. Final answer. The thickness cut off from each face is approximately $x\approx 0.6745$.