Ex. 1507. Find the area of the base of a rectangular parallelopiped whose altitude is 6, if the solid is equivalent to a rectangular parallelopiped whose dimensions are 8, 12, and 15.

Ex. 1508. Find volume of a rectangular parallelopiped the dimensions of whose base are 12 and 20, and the area of whose entire surface is 800.

Ex. 1509. Find the volume of a rectangular parallelopiped the dimensions of whose base are 12 in. and a in., and the area of whose entire surface is (120 + 34 a) sq. in.

Ex. 1510. Find the edge of a cube having a surface equal to the sum of the surfaces of two cubes whose edges are 120 in. and 209 in.

Ex. 1511. The dimensions of a rectangular solid are proportional to 3, 4, and 5. Find the dimensions and the volume if the entire surface contains 2350 sq. in.

Ex. 1512.

Find the volume of a cube whose diagonal equals 18.

PROPOSITION X. THEOREM

596. The volume of a right parallelopiped is equal to the product of one of its lateral faces and the corresponding altitude.

. Given R, the volume of a right parallelopiped, B one of its lateral faces, and H the corresponding altitude.

Proof. Produce CD and the edges parallel to it. Take C'D' = CD, and pass planes C'E and D'F perpendicular to C'D'; thus forming parallelopiped c'F.

But the three angles at c' are right angles.

(581) (533)

597. The volume of any parallelopiped is equal to the product of its base and altitude.

6

D'

R

E

B

Given P, the volume of any parallelopiped, B, its base, and H, its altitude.

Proof. Produce edge CD and the edges parallel to it. Take C'D' CD, and through and D' pass the planes C'F' and D'G perpendicular to C'D'; thus forming the parallelopiped R.

=

Ex. 1514. Find the volume of a parallelopiped whose base is 20 and whose lateral edge is 10, if the inclination of the lateral edge to the base is 45°.

Ex. 1515. In the preceding figure, find the volume of the parallelopiped if AB = 6, AC = 5, ▲ BAC = 30°, E = 10, and the projection of E upon the base equals 8.

Ex. 1516. In the same figure find the volume of the parallelopiped if AB = 5, AC = 6, AD = 8, ▲ BAC = 60°, and the inclination of E to the base equals 60°.

Ex. 1517. Find the volume of a parallelopiped whose base is 10 square feet, if the projection of the lateral edge upon the base equals 2 feet, and the inclination of the lateral edge to the base is 60°.

Ex. 1518. A rectangular tank, 5 ft. long, 4 ft. wide, and 3 ft. deep, is to be lined with zinc, of an inch thick. How many cubic feet of zinc are required, if 3 sq. ft. are allowed for overlapping?

Ex. 1519. If one cubic inch of gold beaten into gold leaf will cover 20,000 sq. ft. of surface, what is the thickness of the gold leaf?

Ex. 1520. An open tank is made of ironin. thick. The outer dimensions are as follows: length equals 2 ft., width 1 ft. 6 in., height 1 ft. If one cubic foot of iron weighs 460 lb., find the weight of the tank.

Ex. 1521. In a rainfall of 14 in., how many tons of water fall upon an acre of ground, if 1 cu. ft. of water weighs 62.5 lb.?

PROPOSITION XII. THEOREM

599. The plane passed through two diagonally opposite edges of a parallelopiped divides it into two equiv alent triangular prisms.

Given BDD'B', a plane passing through edges DD' and BB' of parallelopiped AC'.

Ex. 1522.

If in the figure of Prop. XII, A ABD = 60 sq. in., and the altitude of the solid equals 10 in., find the volume of triangular prism A'-ABD.

Ex. 1523. In a right parallelopiped lettered as the figure above, AB 6, AD = 4, AA' = 8, and Z DAB = 30°. Find the volume of the triangular prism A'-ABD.

=

PROPOSITION XIII. THEOREM

600. The volume of a triangular prism is equal to the product of its base by its altitude.

Given v denoting the volume, B the base, and H the altitude of the triangular prism ABC-B'.

Proof. Upon the edges AB, BC, BB', construct the parallelopiped ABCD-D'.

ABC-B' = ABCD-B'.

The volume of ABCD-B' =base ABCD X H.

(599)

(597)

Ex. 1524. In the diagram for Prop. XIII, find the area of ▲ ABC, if the volume of ABC-B' is 200, and the altitude is 20.

Ex. 1525. In the same diagram find the volume of ABC-B' if BA=2, BC=6, 2 ABC 30°, and H = 7.

=

Ex. 1526. In the same diagram, find the volume of ABC-B', if AB4, BC= 6, BB' = 5, LABC = 60°, and the projection of BB' upon the base equals 4.