Question

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 in.$^{3}$.
the equation $4x^{3}-72x^{2}+320x = 420$ 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?
____ in.

Explanation:

Step1: Rearrange to standard cubic form

Subtract 420 from both sides.
$$4x^3 - 72x^2 + 320x - 420 = 0$$

Step2: Simplify the cubic equation

Divide all terms by 2.
$$2x^3 - 36x^2 + 160x - 210 = 0$$

Step3: Test integer values for x

Test x=2: $2(2)^3 - 36(2)^2 + 160(2) - 210 = 16 - 144 + 320 - 210 = -18$
Test x=3: $2(3)^3 - 36(3)^2 + 160(3) - 210 = 54 - 324 + 480 - 210 = 0$

Step4: Verify valid solution

x=3 is valid (since 20-2(3)=14>0, 16-2(3)=10>0). Volume: $3×14×10=420$, matches given volume.

Answer:

3 in.