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 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?
- in.

  3 in

Step-by-step explanation:

A graphing calculator is useful for finding the roots of a cubic equation. The one rational root is x=3. The other two solutions are irrational. As it happens, one of them is about 2.89, also rounding to 3 when rounded to the nearest inch.

The side length of the square cutout is 3 inches.

_____

The equation can be put into standard form as ...

  x^3 -18x^2 +80x -105 = 0

Possible rational roots are 1, 3, 5, 7 (divisors of 105). We can quickly rule out 1, since the sum of coefficients is not zero. Trial of 3 by synthetic division shows it to be a rational root, so the factorization is ...

  (x -3)(x^2 -15x +35) = 0

The solutions to the quadratic can be found by any of the usual methods. We can rewrite it to ...

  x^2 -15x +35 = (x^2 -15x +56.25) -21.25 = 0 = (x -7.5)^2 -21.25

  x = 7.5 ±√21.25 = {2.89, 12.11}

As we said above, the x value of 2.89 rounds to 3 anyway.

The side length of the cut square is 3 inches.

_____

Check

Volume = x(16 -2x)(20 -2x) = 3(10)(14) = 420 . . . cubic inches.

With a continuous thought process.

step-by-step explanation:

when the line is solid:

it's either < or >

when the line is dashed:

it's either < or >

this > inequality sign means:

greater thanmore thanaboveexceeds

this < inequality sign means:

less thanfewer thanbelow

this > inequality sign means:

greater than/equal tono less thanat leastminimum of

this < inequality sign means:

less than/equal tono more thanat mostmaximum of

about the problem:

since the line is solid, that means it's either < or >

srry i'm not done ! i'll