A figure is a rectangular prism and AB = 5cm, AD = 6cm, and AE = 8cm.
Moreover, BF = 4 cm, AG = 2cm.
A rectangular prism is divided into two solids by the plane which passes along three point C, F, and G.
Find the ratio of the volume of a large solid to the volume of a small solid.
Answer
The volume of the rectangular prism is 5 × 6 × 8 = 240cm3.
(Volume of large solid) = (Volume of rectangular prism) - (Volume of a small solid).
△BFC and △AGH are homothetic and since AG = 2cm, AH = 6cm × 2/4 = 3cm.
Moreover, △IBF and △IAG are homothetic and a homothetic ratio is BF : AG = 4cm : 2cm = 2 : 1.
As IA = AB = 5cm, it turns out IB = 10cm.
Thus, the volume of triangular pyramid I-BFC is 4 × 6 / 2 × 10 × 1/3 = 40cm3.
The volume of triangular pyramid I-AGH is 2 × 3 / 2 × 5 × 1/3 = 5cm3.
Thus, the volume of a small solid is 40 - 5 = 35cm3.
The volume of a large solid is 240 - 35 = 205cm3.
Therefore, a volume ratio is 205 : 35 = 41 : 7.
Moreover, BF = 4 cm, AG = 2cm.
A rectangular prism is divided into two solids by the plane which passes along three point C, F, and G.
Find the ratio of the volume of a large solid to the volume of a small solid.
Answer
41 : 7
Solution
The cut surface becomes Fig.1 which is cut by the plane which passes along three point C, F, and G.
The volume of the rectangular prism is 5 × 6 × 8 = 240cm3.
(Volume of large solid) = (Volume of rectangular prism) - (Volume of a small solid).
As shown in Fig.2, AB, GF and HC are made to extend.
The volume of the small solid can be found by subtracting the volume of triangular pyramid I-AGH from the volume of triangular pyramid I-BFC. BF = 4 cm, BC = 6 cm.
△BFC and △AGH are homothetic and since AG = 2cm, AH = 6cm × 2/4 = 3cm.
Moreover, △IBF and △IAG are homothetic and a homothetic ratio is BF : AG = 4cm : 2cm = 2 : 1.
As IA = AB = 5cm, it turns out IB = 10cm.
Thus, the volume of triangular pyramid I-BFC is 4 × 6 / 2 × 10 × 1/3 = 40cm3.
The volume of triangular pyramid I-AGH is 2 × 3 / 2 × 5 × 1/3 = 5cm3.
Thus, the volume of a small solid is 40 - 5 = 35cm3.
The volume of a large solid is 240 - 35 = 205cm3.
Therefore, a volume ratio is 205 : 35 = 41 : 7.
