Answer:
the maximum volume of the open box is 3456 cm^3, and this occurs when the side length of the square cut from each corner is 12 cm.
Explanation:
To find the maximum volume of the open box, we need to determine the dimensions that would result in the largest possible volume.
Let's assume that the side length of the square cut from each corner is "x" cm. By cutting equal squares from all four corners and folding up the sides, the resulting box dimensions will be:
Length = (24 - 2x) cm
Width = (24 - 2x) cm
Height = x cm
The volume of a rectangular box is given by V = Length × Width × Height. Substituting the dimensions of our box, we have:
V = (24 - 2x) × (24 - 2x) × x
V = (24 - 2x)^2 × x
To find the maximum volume, we need to find the value of "x" that will maximize the expression (24 - 2x)^2 × x.
Let's expand this expression:
V = (24 - 2x) × (24 - 2x) × x
V = (576 - 48x - 48x + 4x^2) × x
V = (4x^2 - 96x + 576) × x
V = 4x^3 - 96x^2 + 576x
Now, we can differentiate V with respect to "x" to find the critical points:
dV/dx = 12x^2 - 192x + 576
Setting the derivative equal to zero to find the critical points:
12x^2 - 192x + 576 = 0
Dividing the equation by 12:
x^2 - 16x + 48 = 0
Factoring the quadratic equation:
(x - 4)(x - 12) = 0
Setting each factor equal to zero:
x - 4 = 0 or x - 12 = 0
Solving for "x":
x = 4 or x = 12
Since we are looking for the maximum volume, we need to determine which value of "x" will yield the larger volume. Let's substitute these values back into the expression for volume:
For x = 4:
V = 4(4)^3 - 96(4)^2 + 576(4)
V = 512 - 1536 + 2304
V = 1280 cm^3
For x = 12:
V = 4(12)^3 - 96(12)^2 + 576(12)
V = 4608 - 13824 + 6912
V = 3456 cm^3
Therefore, the maximum volume of the open box is 3456 cm^3, and this occurs when the side length of the square cut from each corner is 12 cm.