Maximization of a lidless box

We are given a rectangular sheet of cardboard 15 in. by 25 in. If a small square of same size is cut from each corner and each side folded up along the cuts to form a lidless box. We know that the volume of the box will be v=(base)(width)(height).
What will vary for this box will be the height so we will let x equal the height. Therefore the equation of our box made form a sheet of cardboard 15 in. by 25 in. will be
v=(15-2x)(25-2x)(x)
If we graph this on Algebra Xpressor we will be able to tell the maximum volume.

We can tell from this graph that the maximum volume, of about 310, will be achieved when x is approximately 3.

Now lets consider what this actually looks like with GSP
click here for GSP sketch

With GSP, we are able to get a more visual idea of what the cut out piece of the card board would look like. We get the same answer of the maximum volume being about 510 cubic inches.
Now lets get a little more exact. We will create a spreadsheet using Excel and see how some decimal places.
height base width height volume
0.1 24.8 14.8 0.1 36.704
0.2 24.6 14.6 0.2 71.832
0.3 24.4 14.4 0.3 105.408
0.4 24.2 14.2 0.4 137.456
0.5 24 14 0.5 168
0.6 23.8 13.8 0.6 197.064
0.7 23.6 13.6 0.7 224.672
0.8 23.4 13.4 0.8 250.848
0.9 23.2 13.2 0.9 275.616
1 23 13 1 299
1.1 22.8 12.8 1.1 321.024
1.2 22.6 12.6 1.2 341.712
1.3 22.4 12.4 1.3 361.088
1.4 22.2 12.2 1.4 379.176
1.5 22 12 1.5 396
1.6 21.8 11.8 1.6 411.584
1.7 21.6 11.6 1.7 425.952
1.8 21.4 11.4 1.8 439.128
1.9 21.2 11.2 1.9 451.136
2 21 11 2 462
2.1 20.8 10.8 2.1 471.744
2.2 20.6 10.6 2.2 480.392
2.3 20.4 10.4 2.3 487.968
2.4 20.2 10.2 2.4 494.496
2.5 20 10 2.5 500
2.6 19.8 9.8 2.6 504.504
2.7 19.6 9.6 2.7 508.032
2.8 19.4 9.4 2.8 510.608
2.9 19.2 9.2 2.9 512.256
3 19 9 3 513
3.1 18.8 8.8 3.1 512.864
3.2 18.6 8.6 3.2 511.872
3.3 18.4 8.4 3.3 510.048
3.4 18.2 8.2 3.4 507.416
3.5 18 8 3.5 504
3.6 17.8 7.8 3.6 499.824
3.7 17.6 7.6 3.7 494.912
3.8 17.4 7.4 3.8 489.288
3.9 17.2 7.2 3.9 482.976
4 17 7 4 476
4.1 16.8 6.8 4.1 468.384
4.2 16.6 6.6 4.2 460.152
4.3 16.4 6.4 4.3 451.328
4.4 16.2 6.2 4.4 441.936
4.5 16 6 4.5 432
4.6 15.8 5.8 4.6 421.544
4.7 15.6 5.6 4.7 410.592
4.8 15.4 5.4 4.8 399.168
4.9 15.2 5.2 4.9 387.296

We can see from the spreadsheet that the maximum volume, of 512.864 cubic inches will be achieved when the height is 3.1 inches. The reason that I copied the height over by the volume was to create the following graph.

Using the different programs makes this problem fun.