An open rectangular box is to be made from a piece of cardboard 8 inches wide and 8 inches long by cutting a square from each corner and bending up the sides. Approximate the dimensions of the box with the largest volume.
2006-09-13 03:14:44 · 5 answers · asked by Jessie L 2 in Science & Mathematics ➔ Mathematics
This is a continuation of the last question, where we found that the volume of the box was given as,
V = x(8-2x)^2
The procedure for finding the maximum is to take the first derivative of the volume as a function of x. From the derivative expression, you will find two zeros, which indicate the max, min of the Volume function (what you are looking for). Compute the second derivative, and look for the sign. That will be your indicator of concavity, which will determine if the point is a max or a minimum.
The answer V = 37.925 in^3
Give me the 10 points and I will fill the details in between.
2006-09-13 03:26:21 · answer #1 · answered by alrivera_1 4 · 0⤊ 1⤋
Let the square that you cut out have a side of x
Therefore the volume of the box = (8-2x) * (8-2x) * x
vol = 64x - 32x^2 +4x^3
dv/dx = 64 - 64x +12x^2
now set equal to zero and factor
0 = 64 - 64x + 12x^2
6x^2 - 32x +32 = 0
(2x - 8) * (3x - 4) = 0
x = 4 or x = 4/3
x=4 is the trivial answer because each square you cut out would be equal to 1/4 of the entire sheet of cardboard and thus there would be nothing left to fold.
x = 4/3 is the correct answer which gives dimensions of
16/3 x 16/3 x 4/3 and a volume of 1024/27 = 37.93 in cubed
2006-09-13 03:30:55 · answer #2 · answered by Will 4 · 0⤊ 0⤋
A cube would have the largest volume (for calculus reasons). If you insist on a rectangle fudge some of the lines to make them a little closer to the edge (bigger bottom shorter sides) but the winner is always a cube.
If the cardboard is 8" wide you have to use the 8" to make one side, the bottom, and the other side. Divide the 8" by 3 to make them all equal in size (making a cube). That is 8/3 = 2.67". Draw a line parallel to one side 2.67" from the edge. Now draw another line 2.67" away from the first line and parallel to the first line. Turn the box a quarter turn and make two more lines, the first parallel to the edge and 2.67" away from the edge and a second line another 2.67 from the first line. If you look at all of the lines you will see that there is a little square in each corner. Cut out each square and discard them. Now fold up on each of those lines that are showing on the cardboard. You have a bottom sitting on the table and 4 sides sticking up. Tape together the edges of each pair of sides to make the final box. It is a 2.67" wide cube with an open top.
2006-09-13 03:28:19 · answer #3 · answered by Rich Z 7 · 0⤊ 1⤋
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2016-12-12 07:40:57 · answer #4 · answered by ? 4 · 0⤊ 0⤋
volumbe will be the same not matter how u cut it
2006-09-13 03:21:53 · answer #5 · answered by Trans Atlantic 2 · 0⤊ 2⤋