word problem: an open tray is to be manufactured from a 12-by-14-inch rectangular sheet of metal by cutting?

Question

squares from each corner and folding up the sides. If the volume of the tray is to be 160 cubic inches and x is to be an integer,what size squares should be cut from each corner?

Answer 1

Imagine the squares cut from the corners have a side length of x.

After cutting and folding squares from each corner of side x, you will have length 14 - 2x, width 12 - 2x, and height x.

(14 - 2x) (12 - 2x) (x) = 160 cubic inches

4x^3 - 52x^2 + 168x = 160

This cubic equation has three roots, but only one interger answer, which is 2.

Verification:

(14 - 2x2) (12 - 2x2) (2) = ?

(14 - 4) (12 - 4) (2) = ?

10 x 8 x 2 = 160

Answer 2

x(14 -- 2x)(12 -- 2x) = 160
=> x(x -- 7)(x -- 6) = 40
=> x^3 -- 13x^2 + 42x -- 40 = 0
=> (x -- 2)(x^2 -- 11x + 20) = 0
giving x = 2 so 2 inch squares from the corners are to be cut.

Answer 3

x(12 - 2x)(14 - 2x) = 160
- 2x(x - 6)(14 - 2x) = 160
4x(x - 6)(x - 7) = 160
4x(x^2 - 13x + 42) = 160
x^3 - 13x^2 + 42x - 40 = 0
x = 2 in.

8 - 52 + 84 - 40 = 0