An open box is formed from a piece of cardboard 12 inches square by cutting equal squares out of the corners and turning up the sides. Find the volume of the largest box that can be made.
Help!
let x be the edge length of the corner squares
the volume of the box is
... v = x (12 - 2x)^2
... v = 144 x - 48 x^2 + 4 x^3
the 1st derivative will show the maxima/minima
plug in the values to find the volume
To find the volume of the largest box that can be made from the given piece of cardboard, we need to maximize the volume by finding the optimal size of the cut-out squares.
Let's denote the side length of the squares cut out as 'x'.
To create the box, four squares would be cut out from the corners of the cardboard. The length of the sides of the resulting box would then be (12 - 2x) inches.
The height of the box would be 'x' inches since the sides are turned up from the cut-out squares.
Therefore, the volume of the box is given by multiplying the length, width, and height together:
Volume = length * width * height
= (12 - 2x) * (12 - 2x) * x
= x(144 - 48x + 4x^2)
To find the maximum volume, we need to find the value of 'x' that maximizes this volume. Let's take the derivative of the volume equation with respect to 'x' and set it equal to zero:
d/dx (x(144 - 48x + 4x^2)) = 0
Simplifying the expression, we get:
144 - 96x + 8x^2 = 0
Now, let's solve this quadratic equation to find the value of 'x' that maximizes the volume.
Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a), where a = 8, b = -96, and c = 144.
x = (-(-96) ± √((-96)^2 - 4 * 8 * 144)) / (2 * 8)
x = (96 ± √(9216 - 4608)) / 16
x = (96 ± √4608) / 16
x = (96 ± 67.82) / 16
Now, we need to determine which value of 'x' gives the maximum volume. Since the cardboard dimensions are 12 inches, the maximum value for 'x' should be less than or equal to half of that, i.e., x ≤ 6.
Let's substitute both values of 'x' into the volume equation and compare the results:
For x = (96 + 67.82) / 16 = 10.92:
Volume = 10.92(144 - 48 * 10.92 + 4 * 10.92^2) ≈ 640.08
For x = (96 - 67.82) / 16 = 2.32:
Volume = 2.32(144 - 48 * 2.32 + 4 * 2.32^2) ≈ 568.38
Hence, the larger volume is obtained when x ≈ 10.92 inches.
Therefore, the maximum volume of the box is approximately 640.08 cubic inches.
To find the volume of the largest box that can be made, we need to determine the dimensions of the box first. Let's break down the problem into steps:
Step 1: Visualize the problem
Imagine a 12-inch square piece of cardboard. We need to cut equal squares out of each corner and fold up the remaining sides to form an open box. The resulting box will have a length, width, and height.
Step 2: Determine the size of the squares to be cut out
Let's assume that the side length of the square to be cut out is "x" inches. Since the same size squares will be removed from each corner, there are four squares to be cut out in total.
Step 3: Calculate the dimensions of the box
When cutting out squares of side length "x," the length and width of the cardboard will each be reduced by 2x inches. Thus, the dimensions of the box can be expressed as:
Length = 12 - 2x
Width = 12 - 2x
Height = x
Step 4: Calculate the volume of the box
The volume of the box can be calculated by multiplying its length, width, and height. So, the volume (V) can be expressed as:
V = Length * Width * Height
Putting it all together:
V = (12 - 2x) * (12 - 2x) * x
Step 5: Determine the maximum volume
To find the maximum volume, we need to find the value of "x" that maximizes the volume function V. One way to do this is by differentiating V with respect to "x" and setting the derivative equal to zero. However, since this is likely to involve calculus, I will solve it differently.
To simplify the problem, let's expand the equation for V:
V = (144 - 24x + 4x^2) * x
V = 4x^3 - 24x^2 + 144x
Now, let's factor out an "x" from the equation:
V = x(4x^2 - 24x + 144)
We know that the volume of a box cannot be negative, so V > 0. Therefore, we can divide both sides of the equation by "x":
V/x = 4x^2 - 24x + 144
We can then solve for the value of "x" that maximizes the volume by finding the vertex of the quadratic function 4x^2 - 24x + 144. The formula to find the vertex of a quadratic function is x = -b/2a.
For 4x^2 - 24x + 144, "a" is 4 and "b" is -24:
x = -(-24)/(2*4)
x = 24/8
x = 3
By substituting x = 3 back into the equation for V, we can find the maximum volume.
V = (12 - 2*3) * (12 - 2*3) * 3
V = 6 * 6 * 3
V = 108 cubic inches
Therefore, the largest box that can be made has a volume of 108 cubic inches.