1. Research content: 1. How to cut a square cardboard into a cuboid carton without a lid? 2. How to cut the paper box to maximize it? Second, research methods: practice, drawing, tabulation, calculation and observation. Iii. Research process: 1. Through observation, I found that we can deduce how to cut a square cardboard into a cuboid carton without a cover through the expansion diagram of a cube. As shown in figure 1 and figure 2, you can cut a cuboid carton without a lid by cutting off the shadow as shown in figure 2. Let the side length of this square be 20 cm. If the side length of a square is X (x < 10), then the formula for calculating the volume of this box should be: v = (20-2x) 2x. I took out some pieces of paper and experimented with X = 1cm, 2cm, 3cm, 4cm, 5cm, 6cm, 7cm, 8cm and 9cm. When x =1:v = (20-1* 2) 2 *1= 324cm2 x = 2: v = (20-2 * 2) 2 * 2 = 512cm2 x = 3: v. 5 = 500 cm2 X = 6:V =(20-6 * 2)2 * 6 = 384 cm2 X = 7:V =(20-7 * 2)2 * 7 = 252 cm2 X = 8:V =(20-8 * 2)2 * 8 = 650。 Is the largest between 2 and 3 or between 3 and 4? Let's look at X=2.9cm, X=3. 1cm: X=2.9, v = (20-2.9 * 2) 2 * 2.9 = 584.756 cm2 x = 3.1,v = (20-3./kloc. When x = 3.2 cm, 3.3 cm, 3.4 cm, 3.5 cm, 3.6 cm, 3.7 cm, 3.8 cm, 3.9 cm? X = 3.2:V =(20-3.2 * 2)2 * 3.2 = 59 1.872 cm2 X = 3.3:V =(20-3.3 * 2)2 * 3.3 = 592.548 cm2 X = 3.4:V =(20-3.3 * 2)。 3.5 = 59 1.500 cm2 X = 3.6:V =(20-3.6 * 2)2 * 3.6 = 589.824 cm2 X = 3.7:V =(20-3.7 * 2)2 * 3.7 = 587.465438+。 We can see from the figure that when X=3. 3cm, the box has the largest volume. Let's consider whether it is the largest, between 3.2 and 3.3 or between 3.3 and 3.4. Let's first calculate when X=3. 29cm and X=3. 3 1cm。 V =(20-3.29 * 2)2 * 3.29 = 592.5 17 156 cm2 x = 3.3 1cm:V =(20-3.3 1 * 2)2 * 3。 So, X=3. 3 1cm maximum? Let's calculate the volume when X=3. 32~3.39cm。 x = 3.32:v =(20-3.32 * 2)2 * 3.32 = 592.585472 cm2 x = 3.33:v =(20-3.33 * 2)2 * 3.33 = 592.58472+00350606 3。 34 = 592.5908 16 cm2 X = 3.35:V =(20-3.35 * 2)2 * 3.35 = 592.5 1500 cm2 X = 3.36:V =(20-3.36。 3.37 = 592.5390 12 cm2 X = 3.38:V =(20-3.38 * 2)2 * 3.38 = 592.505888 cm2 X = 3.39:V =(20-3.39 * 2)2。 3.39 = 592.44876cm2 Let's draw a statistical chart. From this I know the biggest research result when X=3.33. Through repeated observation and experiments, I found that the maximum value of X is x = 3.3333 ... so I understand. The volume of the box is the largest in the infinite loop, that is to say, the volume of the box is the largest when x = 10/3. Generally speaking, if the side length of a square piece of paper is A, you can get X=A/6. Harvest and reflection: I benefited a lot from writing this research report, because it increased my knowledge of mathematics and computer. Writing research reports also cultivated my spirit of studying hard. But because it's the first time, I can't be perfect, and there must be some shortcomings, but I believe that through later study, I will write my second and third times better and better.