There is a solid which two rectangular prisms made of the clays of black and white have stuck fast as shown in the figure below.
Rectangular prism ABCD-EFGH is made of black clay.
Rectangular prism DCJI-HGKL is made of white clay.
This solid is cut by the plane which passes along three point A, F, and M.
Find the ratio of the volume of black clay to the volume of white clay by the ratio of the least integer in the solid including the point C.
Answer
37 : 19
Solution
When this solid is cut by the plane which passes along three point A, F, and M, it will become as it is shown in the figure below.
Triangular pyramid O-ABF, O-PCN, and O-QJM are homothetic.
BF = 8 cm, JM = 4 cm, thus CN = (8 + 4) / 2 = 6 cm.
The homothetic ratio of three trianbular pyramids is BF : CN : JM = 4 : 3 : 2.
The volume ratio is (4 × 4 × 4) : (3 × 3 × 3) : (2 × 2 × 2) = 64 : 27 : 8.
The volume ratio of the black clay and white clay in the solid including the point C is (64 - 27) : (27 - 8) = 37 : 19.
Rectangular prism ABCD-EFGH is made of black clay.
Rectangular prism DCJI-HGKL is made of white clay.
This solid is cut by the plane which passes along three point A, F, and M.
Find the ratio of the volume of black clay to the volume of white clay by the ratio of the least integer in the solid including the point C.
Answer
37 : 19
Solution
When this solid is cut by the plane which passes along three point A, F, and M, it will become as it is shown in the figure below.
Triangular pyramid O-ABF, O-PCN, and O-QJM are homothetic.
BF = 8 cm, JM = 4 cm, thus CN = (8 + 4) / 2 = 6 cm.
The homothetic ratio of three trianbular pyramids is BF : CN : JM = 4 : 3 : 2.
The volume ratio is (4 × 4 × 4) : (3 × 3 × 3) : (2 × 2 × 2) = 64 : 27 : 8.
The volume ratio of the black clay and white clay in the solid including the point C is (64 - 27) : (27 - 8) = 37 : 19.
