You cut square corners from a piece of cardboard that has dimensions 32 cm by 40 cm. You then fold the cardboard to create a box with no lid. To the nearest centimeter, what are the dimensions of the box that will have the greatest volume?
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To find the dimensions of the box that will have the greatest volume, we need to consider the dimensions of the cut-out corners. Let's assume the side length of each cut-out corner is "x" centimeters.
When the corners are cut out, the dimensions of the cardboard will be reduced by 2x on each side. This means the new dimensions of the cardboard will be (32 - 2x) cm by (40 - 2x) cm.
To create a box with no lid, we need to fold the cardboard along the remaining flaps. The height of the box will be equal to the length of the cut-out corners, which is "x" cm.
So, the dimensions of the box will be (32 - 2x) cm by (40 - 2x) cm by "x" cm.
Now, let's find the volume of the box. The volume of a rectangular box is given by the formula:
Volume = Length × Width × Height.
In this case, the volume of the box will be:
V = (32 - 2x)(40 - 2x)(x).
To find the dimensions of the box that will give the greatest volume, we need to maximize the volume function V.
To do this, we can differentiate V with respect to x, set it equal to zero, and solve for x:
dV/dx = 0.
Let's differentiate V with respect to x:
dV/dx = (40 - 2x)(x) + (32 - 2x)(x) + (32 - 2x)(40 - 2x).
Simplifying the equation:
0 = (40x - 2x^2) + (32x - 2x^2) + (1280 - 104x - 4x^2).
Now, let's combine like terms:
0 = 40x - 2x^2 + 32x - 2x^2 + 1280 - 104x - 4x^2.
Combining further:
0 = 72x - 8x^2 + 1280 - 104x - 4x^2.
Rearranging the terms:
0 = 72x - 8x^2 - 104x - 4x^2 + 1280.
Combining like terms once again:
0 = -12x^2 - 32x + 1280.
Now, we can solve this quadratic equation for x. We can either factor it or use the quadratic formula to find the solutions.
Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a), where a = -12, b = -32, and c = 1280.
Calculating the value under the square root:
√(b^2 - 4ac) = √((-32)^2 - 4(-12)(1280)) = √(1024 - (-61440)) = √(62464) ≈ 249.93.
Let's find the values of x using the quadratic formula:
x = (-(-32) ± 249.93) / (2(-12)).
For the positive value:
x = (32 + 249.93) / (-24) = 9.99.
For the negative value:
x = (32 - 249.93) / (-24) ≈ -9.29.
Since the dimensions cannot be negative, we discard the negative value.
So, the side length of each cut-out corner, which is also the height of the box, is approximately 10 cm.
Plugging this value back into the expressions for the length and width of the box:
Length = 32 - 2x = 32 - 2(10) = 32 - 20 = 12 cm.
Width = 40 - 2x = 40 - 2(10) = 40 - 20 = 20 cm.
Therefore, the dimensions of the box that will have the greatest volume to the nearest centimeter are approximately 12 cm by 20 cm by 10 cm.