434. Note. The theorem of § 433 may be expressed: Two rectangular parallelopipeds having a dimension of one equal to a dimension of the other, are to each other as the products of their other two dimensions. PROP. IX. THEOREM 435. Any two rectangular parallelopipeds are to each other as the products of their three dimensions. Given P and Q reet. parallelopipeds with the dimensions a, b, c, and a', b', c', respectively. (If R is a rect. parallelopiped with dimensions a', b', c, find Ex. 3. Find the lateral area of a right prism whose base is a regular hexagon, if one of the lateral faces has the base 10, and altitude 8. Ex. 4. The edges of two rectangular parallelopipeds are 9, 12, 15, and 15, 12, 18, respectively. Find the ratio of their volumes. 436. Def. The volume of a solid is its ratio to another solid, called the unit of volume, adopted arbitrarily as the unit of measure (§ 180). The usual unit of volume is a cube (§ 421) whose edge is some linear unit; for example, a cubic inch or a cubic foot, PROP. X. THEOREM 437. If the unit of volume is the cube whose edge is the linear unit, the volume of a rectangular parallelopiped is equal to the product of its three dimensions. Given a, b, and c the dimensions of rect. parallelopiped P, and Q the unit of volume; that is, a cube whose edge is 1. 438. Note. In all succeeding theorems relating to volumes, it is understood that the unit of volume is the cube whose edge is the linear unit, and the unit of surface the square whose side is the linear unit. The following are direct consequences of § 437: 439. The volume of a cube equals the cube of its edge. 440. The volume of a rectangular parallelopiped equals the product of its base and altitude. Ex. 5. The volume of a cube is 64 cubic inches. Using proportion, find the edge of a cube having eight times the volume. Ex. 6. A cubical tank holds 30 gallons of water; find an edge of the tank in feet. Ex. 7. Find the ratio of a diagonal of a cube to an edge. Is this ratio commensurable ? Ex. 8. The volume of a cube is 16 v3. Find the diagonal. 441. The volume of any parallelopiped is equal to the product Proof. 1. On AB prolonged take FG = AB; draw planes FK', GH' 1 FG, meeting A'B', D'C', DC prolonged, forming rt. parallelopiped FH' ≈ AC'. (§ 426) 2. On HG prolonged, take NM=HG; draw planes NP', NM, meeting H'G', K'F', KF prolonged, forming rt. parallelopiped LN' FH' (§ 426), and hence ML' AC'. (§ 390) (§ 387) 3. Since FG is 1 GH', planes LH, MH' are 1. 4. Since MM' is 1 MN, it is 1 plane LH. 5. Then LMM' is a rt. 4, and LM' a rectangle (§§ 350, 77); whence LN' is a rect. parallelopiped. 6. Then vol. LN' (or AC")= area LMNP × MM'. ($ 440) 7. Now rect. LN=rect. FH, having equal bases MN, GH, and same altitude (§ 111); also, rect. FH□ AC, having equal bases FG, AB, and same altitude (§ 283); then, LN AC. 8. Substitute in result of (6) area ABCD for area LMNP, and put MM' = AE (§ 374). Ex. 10. If the edge of a cube equals a, find the diagonal of a face, also the diagonal of the cube. Ex. 11. The diagonal of a cube is 12. Find its edge. |