An Open Box Of Maximum Volume Is To Be Made From A Square Piece Of Cardboard Twentyfour Inches On Each Side By Cutting Equal Suares Frm The Corners And Turning Up The Sides
as Reiny posted elsewhere,
let each side of the equal squares be x inches
length of box = 24-2x
width of box = 24-2x
height of box = x
a) Volume = x(24-2x)(24-2x)
now just find where dv/dx = 0, and you're done.
To find the maximum volume of an open box made from a square piece of cardboard, we need to determine the dimensions of the cut-out squares. Let's go step by step through the process:
1. Start with a square piece of cardboard measuring 24 inches on each side.
2. Let's assume that we cut x-inch squares from each corner of the cardboard.
3. After cutting out the squares, fold up the sides to form the open box.
4. The height of the box will be equal to the side length of the cut-out squares (x inches).
5. The length and width of the base of the box will be equal to the remaining sides of the cardboard after the squares are cut out. Since we cut out squares from all four corners, the dimensions of the base will be (24 - 2x) inches.
6. Therefore, the volume of the box can be calculated by multiplying the length, width, and height: Volume = (24 - 2x) * (24 - 2x) * x.
Now, we want to find the maximum possible volume by choosing the optimal value for x.
To do this, we can find the critical point by taking the derivative of the volume function with respect to x, setting it equal to zero, and solving for x:
(dVolume/dx) = 0.
Let's calculate the derivative:
(dVolume/dx) = 4x^2 - 96x + 1152.
Setting this equal to zero:
4x^2 - 96x + 1152 = 0.
Simplifying the equation:
x^2 - 24x + 288 = 0.
Now we can factor or use the quadratic formula to solve for x. Factoring the equation:
(x - 12)(x - 24) = 0.
The solutions are x = 12 and x = 24. We need to discard x = 24 since it would result in a non-existent box.
Therefore, the optimal value for x is 12 inches. By cutting out 12-inch squares from each corner, we can maximize the volume of the open box.
To find the maximum volume, substitute x = 12 back into the volume function:
Volume = (24 - 2*12)^2 * 12 = 144 cubic inches.