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
