When the side length of the cut small square is closer to 10/3cm (that is, 1/6 times the side length of the original large square), the volume of the folded uncovered cuboid becomes larger and larger.
The range of the side length of a small square The volume refinement data of the side length cuboid of a small square is infinitely approximate.
You need to dig four small squares at four corners. Let the side length of the dug square be x.
It turns out that the side length of a square is a.
The volume of a cuboid is v = (a-2x) square * x.
V = 4 *X cube -4 a *X square +a square * x
The derivative is 12X square -8a * x+a square.
It increases from 0 to A/6 and decreases from A/6 to A/2, so A/6 is the maximum.
At this time, the cuboid volume is the cube of 1/27A.
1. Research content
Cut a square on each corner with 20cm×20cm paper to make a rectangular box without a lid. When is the side length of the square cut?
A cuboid box without a lid is the largest.
2. Research methods
Function, derivative
3. Research process
Let the cut side length of the square be L, and the volume converted into a cuboid is:
V = (20 - 2l) * (20 - 2l) * l
= 4L 3-80L 2+400L
Get the derivative v' =12L2-160l+400 = 4 (3l2-40l+100).
Let V' = 0 and the solution is l = 10/3 or l = 10.
Therefore, V(l) has local extreme values at 10/3cm and 10cm.
V( 10/3) = 16000/27 (about 529.6cm3).
V( 10) = 0
Therefore, when l= 10/3cm, the maximum volume is obtained, which is about 529.6cm3.
4. Research results
When l= 10/3cm, the maximum volume is about 529.6cm3.
5. Harvest and reflection
The derivative of the function can be used to find the maximum/minimum.
Conclusion: When the side length of the cut small square is closer to 10/3cm (that is, the side length of the original large square is 1/6 times), the volume of the folded uncovered cuboid becomes larger.
It is getting bigger and bigger.
The side length range of a small square The side length of a small square The volume of a cuboid.
Infinite approximation of accurate data
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.
It's huge.
(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 cuboids have the smallest volume when the side length of a small square is 0.5 cm, and cuboids have the smallest volume when the side length of a small square is 3.5 cm.
Maximum product. 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 an integer.
It's 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.