Question

solving a real - world problem involving a polynomial equation
a rectangular sheet of metal has identical squares cut from each corner. the sheet is then bent along the dotted lines to form an open box. the volume of the box is 420 in³.
the equation $4x³ - 72x² + 320x = 420$ can be used to find $x$, the side length of the square cut from each corner.
what is the side length of the square that is cut from each corner, to the nearest inch?
\boxed{} in.
(there is also an image of a rectangular metal sheet with length 20 in, width 16 in, and squares of side length $x$ cut from each corner.)

Explanation:

Step1: Simplify the equation

Subtract 420 from both sides: $4x^3 - 72x^2 + 320x - 420 = 0$. Divide by 4: $x^3 - 18x^2 + 80x - 105 = 0$.

Step2: Test integer roots

Possible roots: ±1,±3,±5,±7,±15,±21,±35,±105. Test x=3: $3^3 -18(3)^2 +80(3)-105=27-162+240-105=0$. So x=3 is a root.

Step3: Verify validity

Check if x=3 makes sense: length=20-23=14, width=16-23=10, height=3. Volume=14103=420, which matches.

Step4: Round to nearest inch

x=3 is already an integer, so no rounding needed.

Answer:

3 in