Make a large cube of 5cm one side by piling up 125 dices of 1cm one side without a gap.
This large cube is cut by the plane passing through the three vertex of the cube as shown in a figure.
As for the portion of the triangular pyramid below the cutting plane, find the following each number.
(1)The number of the dices which is not cut.
(2)The number of a solid with the volume larger than 0.5 cm3 in the solid made by cutting a dice.
(3)The number of a solid with the volume smaller than 0.5 cm3 in the solid made by cutting a dice.
This large cube is cut by the plane passing through the three vertex of the cube as shown in a figure.
As for the portion of the triangular pyramid below the cutting plane, find the following each number.
(1)The number of the dices which is not cut.
(2)The number of a solid with the volume larger than 0.5 cm3 in the solid made by cutting a dice.
(3)The number of a solid with the volume smaller than 0.5 cm3 in the solid made by cutting a dice.
Answer
The upper part of (A) is separated and the lower part (A) with smaller volume one remains.
The volume of (A) is smaller than 0.5 cm3.
The upper part of (B) is also separated and the lower part (B) with larger volume one remains.
The volume of (B) is larger than 0.5 cm3.
In Fig.1, the 1st step (Fig. 3) of big shadow and yellow right-angled isosceles triangle becomes a cutting plane of the 1st layer from the top of the large cube with 5th layer.
The 2nd layer of cutting plane is indicated in Fig. 4 in the 2nd step of a right-angled isosceles triangle.
Moreover, the 5th layer of cutting plane is indicated in Fig. 5 in the 5th step of a right-angled isosceles triangle.
(1) 10 pieces
(2) 10 pieces
(3) 15 pieces
(2) 10 pieces
(3) 15 pieces
Solution
The dices being cut are two kinds of shapes of Fig. 2.
The upper part of (A) is separated and the lower part (A) with smaller volume one remains.
The volume of (A) is smaller than 0.5 cm3.
The upper part of (B) is also separated and the lower part (B) with larger volume one remains.
The volume of (B) is larger than 0.5 cm3.
In Fig.1, the 1st step (Fig. 3) of big shadow and yellow right-angled isosceles triangle becomes a cutting plane of the 1st layer from the top of the large cube with 5th layer.
The 2nd layer of cutting plane is indicated in Fig. 4 in the 2nd step of a right-angled isosceles triangle.
Moreover, the 5th layer of cutting plane is indicated in Fig. 5 in the 5th step of a right-angled isosceles triangle.
As mentioned above, the number of small cubes, the number of the shadow triangles (A), the number of yellow triangles (B), and the number of dices which are not cut will become as it is shown in a lower table.
1st layer
|
2nd
|
3rd
|
4th
|
5th
|
Sum
| |
Number of (A)
|
1
|
2
|
3
|
4
|
5
|
15
|
Number of (B)
|
0
|
1
|
2
|
3
|
4
|
10
|
Number of dices
|
1
|
3
|
6
|
10
|
15
|
35
|
Number of dices
with no cut
|
0
|
0
|
1
|
3
|
6
|
10
|
