you have a rectangular piece of cardboard 40X48 inches and need to create an open box with the maximum volume(ignore flaps). what would the dimensions of the box need to be to maximize the volume?
v = x(40-2x)(48-2x) = 4x(20-x)(24-x)
v=0 at x=0,20,24, so you know there will be a max for 0 < x < 20
If you haven't had some calculus, this will be hard to find; you may have to do some probing around x=10 and try some values.
If calculus is in your bag of tools, then you know that
dv/dx = 4(x^2 - 88x + 480)
dv/dx=0 when x = 4/3 (11-√31) = 7.24
So, the box is 32.76 x 40.76 x 7.24
To find the dimensions of the box that will maximize its volume, we can follow these steps:
Step 1: Identify the variables.
Let's denote the length of the rectangular piece of cardboard as L and the width as W. In this case, L = 48 inches and W = 40 inches.
Step 2: Determine the dimensions of the box.
We want to cut out equal-sized squares from each corner of the cardboard and fold the remaining flaps to create the box. Let's assume that each side length of the square cutout is x inches.
Step 3: Calculate the dimensions of the box.
To find the length, subtract twice the width of the cutout (2x) from the original length L:
Length of the box = L - 2x = 48 - 2x inches.
Similarly, subtract twice the width of the cutout (2x) from the original width W to find the width of the box:
Width of the box = W - 2x = 40 - 2x inches.
Finally, the height of the box will be equal to the side length of the cutout:
Height of the box = x inches.
Step 4: Determine the volume of the box.
The volume of a rectangular box is calculated by multiplying length × width × height:
Volume of the box = (48 - 2x) × (40 - 2x) × x cubic inches.
Step 5: Maximize the volume.
To maximize the volume, we need to find the value of x that gives the maximum volume. We can do this by finding the critical points of the volume function.
Step 6: Find the critical points.
Differentiate the volume function with respect to x and set it equal to zero to find the critical points. Then, solve the resulting equation to find the value(s) of x.
Step 7: Check the endpoints.
Since we are dealing with a rectangular piece of cardboard, the value of x cannot exceed half of the smaller dimension (in this case, 20 inches). So, check the volume at the endpoints of the feasible range (0 and 20) to see if they yield higher volumes than the critical points.
Step 8: Calculate the dimensions of the box.
Plug the values of x (either from the critical points or the endpoints) into the equations for the length, width, and height of the box to find the dimensions that will maximize the volume.
Step 9: Verify the maximum volume.
Calculate the volume using the dimensions obtained and compare it to the volume at the critical points or endpoints to verify that it is indeed the maximum volume.
By following these steps, you can determine the dimensions of the box that will maximize its volume.