First, the 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, tabulating, calculating and observing.

Third, the 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 unfolding diagram of a cube.

As shown in the figure: figure 1 figure 2.

As shown in Figure 2, you can cut a rectangular carton without a lid by cutting off the shadow.

Let the side length of this square be 20 cm.

If the length of the cut 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.

x = 1:V =(20- 1 * 2)2 * 1 = 324 cm2。

X=2: V=(20-2*2)2*2=5 12 cm2。

X=3: V=(20-3*2)2*3=588 cm2。

X=4: V=(20-4*2)2*4=576 cm2。

X=5: V=(20-5*2)2*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= 128 cm2。

X=9: V=(20-9*2)2*9=36 cm2。

Then I will make a statistical chart of the results:

As can be seen from the figure, when X=3, the cuboid carton has the largest volume, so is it the largest? Is the largest between 2 and 3 or between 3 and 4?

Let's first look at X=2.9cm and X=3. 1cm:

When X=2.9, v = (20-2.9 * 2) 2 * 2.9 = 584.756cm2.

When X=3. 1, v = (20-3.1* 2) 2 * 3.1= 590.364cm2.

It can be seen from the calculation results that the calculated volume when X=3. 1cm is larger than that when X=2.9cm.

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 =(20-3.4 * 2)2 * 3.4 = 592.4438+06 cm2。

x = 3.5:V =(20-3.5 * 2)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 = 587.4 12 cm2。

x = 3.8:V =(20-3.8 * 2)2 * 3.8 = 584.288 cm2。

x = 3.9:v =(20-3.9 * 2)2 * 3.9 = 580.476 cm2。

Let's make a statistical chart for everyone to see clearly.

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。

592.570764 square centimeters is greater than 592.548 square centimeters, so the maximum value of X must be greater than 3.3 centimeters. ..

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 = 592.472 cm2。

x = 3.33:v =(20-3.33 * 2)2 * 3.33 = 592.438+048 cm2。

x = 3.34:v =(20-3.34 * 2)2 * 3.34 = 592.438+06 cm2。

x = 3.35:V =(20-3.35 * 2)2 * 3.35 = 592.438+0500 cm2。

x = 3.36:V =(20-3.36 * 2)2 * 3.36 = 592.5224 cm2。

x = 3.37:v =(20-3.37 * 2)2 * 3.37 = 592.438+02 cm2。

x = 3.38:v =(20-3.38 * 2)2 * 3.38 = 592.5888 cm2。

x = 3.39:V =(20-3.39 * 2)2 * 3.39 = 592.4476 cm2。

Let's draw a statistical chart:

From this I know that X=3.33 is the maximum.

Research results:

Through repeated observation and experiments, I found that the maximum value of x is x = 3.33333 ... so I got that when the loop is infinite, the volume of the box is the largest.

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

Then 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 at the same time. 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.

2. Project learning

1. Do this.

( 1)

Cut off the volume of a cuboid with square sides.

1cm 324cm3

2cm 512cm3

3 cm 588 cubic cm

4 cm 576 cubic cm

5 cm 500 cubic cm

6 cm 384 cubic cm

7 cm 252 cubic cm

8 cm 128 cubic cm

9 cm 36 cubic cm

10cm 0 cm3

(2)

I found that the cuboid has the smallest volume when the side length of the small square is 10 cm, and the cuboid has the largest volume when the side length of the small square is 3 cm.

(3)

When the side length of the small square is 3 cm, the volume of the uncovered cuboid is the largest, and the volume of the uncovered cuboid is 588 cubic centimeters.

Do it

( 1)

Cut off the volume of a cuboid with square sides.

0.5cm 180.5cm

1.0 cm 324 cm 3

1.5cm 433.5cm。

2.0cm 512cm3

2.5 cm 562.5 cubic cm

3.0 cm 588 cubic cm

3.5 cm 59 1.5 cm.

4.0 cm 576 cubic cm

4.5 cm 544.5 cubic cm

5.0 cm 500 cubic cm

5.5 cm 445.5 cubic cm

6.0 cm 384 cubic cm

…… ……

(2)

I found that the cuboid has the smallest volume when the side length of a small square is 0.5 cm, and the cuboid has the largest volume when the side length of a small square is 3.5 cm. Moreover, when the side length of the cut square is an integer, the volume of the cuboid is also an integer, and when the side length of the cut square is a decimal, the volume of the cuboid is also a decimal.

(3)

When the side length of the small square is 3.5 cm, the volume of the uncovered cuboid is the largest, and the volume of the uncovered cuboid is 59 1.5 cubic cm.