Question

If an open box is made from a tin sheet 6 in. square by cutting out identical squares from each corner and bending up the resulting flaps, determine the dimensions of the largest box that can be made. (Round your answers to two decimal places.)height in.length in.width in.

Answer 1

Answer:

height is 1 inch

length is 4 inches

width is 4 inches

Step-by-step explanation:

∵ An open box is made from a tin sheet 6 in²

∴ The tin sheet dimensions are 6 in by 6 in

∵ The box is made by cutting out identical squares from each

   corner and bending up the resulting flaps

- Assume that the side of the each cutting square is x

∴ The length and the width of the box are (6 - 2x) and (6 - 2x)

∴ The height of the box is x

Look to the attached figure for more understand

∵ The volume of the box = length × width × height

∴ The volume of the box = (6 - 2x)(6 - 2x)(x)

∵ (6 - 2x)(6 - 2x) = (6)(6) + (6)(-2x) + (-2x)(6) + (-2x)(-2x)

∴ (6 - 2x)(6 - 2x) = 36 - 12x - 12x + 4x²

∴ (6 - 2x)(6 - 2x) = 36 - 24x + 4x²

- Substitute it in the volume above

∴ The volume of the box = (36 - 24x + 4x²)(x)

∴ The volume of the box = 36x - 24x² + 4x³

To find the large box which means large volume differentiate the expression of the volume and equate the answer by zero to find the value of x which makes the volume of the box largest

∵ V = 36x - 24x² + 4x³

- By using differentiation

∴ V' = 36 - 24(2)x + 4(3)x²

∴ V' = 36 - 48x + 12x²

- Equate V' by 0

∴ 0 = 36 - 48 x + 12x²

- Switch the two sides and re-arrange the terms from the

   greatest power

∴ 12x² - 48x + 36 = 0

- Simplify the equation by dividing all terms by 12

∴ x² - 4x + 3 = 0

Factorize it into two factors

∵ x² = x × x

∵ 3 = -3 × -1

∵ x × -3 = -3x

∵ x × -1 = -x

∵ -3x + -x = -4x ⇒ middle term

∴ (x - 3)(x - 1) = 0

- Equate each factor by 0 to find x

∵ x - 3 = 0

- Add 3 to both sides

∴ x = 3

∵ x - 1 = 0

- Add 1 to both sides

∴ x = 1

Substitute the values of x in the expressions of length, width and height to find them

∵ Length = 6 - 2x

- If x = 3

∴ Length = 6 - 2(3) = 6 - 6 = 0 ⇒ it can not be

∴ x = 3 is rejected

∵ x = 1

∴ Length = 6 - 2(1) = 4 inches

∵ Width = 6 - 2x

∴ Width = 6 - 2(1) = 4 inches

∵ Height = x

∴ Height = 1 inch