(17) A reservoir is 48 ft. long and 30 ft. broad; how many cubic feet of water must be drawn off to make the surface sink 2 ft.? [The volume of the water drawn off will be that of a rectangular parallelopiped, 48 ft. long, 30 ft. broad, and 2 ft. deep.] (18) What is the length of the edge of a cubical cistern which contains as much water as a rectangular one whose edges are 154 ft. 11 in., 70 ft. 7 in., and 53 ft. 1 in. ? (19) Find the difference in cost of materials between paving a street 90 yds. long and 21 ft. broad with cubical wooden blocks each of whose edges measures 9 in., and laying down asphalte 4 in. deep; the wooden blocks costing 9d. per dozen, and the asphalte 4d. per cubic foot? 144. To find one dimension when the volume and the other two dimensions are given. RULE. Divide the volume by the product of the two given dimensions, and the quotient will be the dimension required. Since length x bread x depth = volume, volume breadth x depth volume length x depth volume length x breadth [The volume and the known dimensions must be expressed in corresponding denominations.] Ex. 1. What length must be cut off a straight plank 9 in. wide and 2 in. thick, in order that it may contain a cubic foot? Ex. 2. A rectangular parallelopiped contains 132 c. ft.; its length and thickness are 11 ft. and 3 ft. respectively; required its breadth. 132 Ex. 3. The value of a beam of timber, whose length is 20 ft. and breadth 3 ft., at 3s. 8d. per cubic foot, is £22 7s. 6d.; find its thickness. Find the third dimension of the following rectangular parallelopipeds, when the volume and the other two dimensions are respectively (1) Volume, 2160 c. yds.; breadth, 12 yds.; depth, 10 yds. (2) Volume, 121.975 c. yds.; breadth, 4.25 yds.; depth, 2.8 yds. (3) Volume, 207 c. yds.; length, 64 yds.; depth, 18 yds. (4) Volume, 46 c. yds. 8 c. ft. ; length, 6 yds. 2 ft.; breadth, 5 yds. (5) Volume, 77 c. yds. 4 c. ft. 576 c. in.; length, 18 ft. 9 in.; breadth, 13 ft. 4 in. (6) Volume, 13 c. yds. 15 c. ft. 1152 c. in.; length, 9 ft. 2 in.; depth, 2 yds. (7) What must be the length of a trench 5 ft. 6 in. deep and 10 ft. 8 in. wide, that it may contain 7040 c. ft.? (8) The cost of digging a cellar 25 ft. long and 15 ft. broad, at 2s. 3d. per cubic yard, was £14 1s. 3d.; find its depth. (9) The solid content of a rectangular parallelopiped is 5381 c. ft.; its length is 133 ft. and depth 5 ft.; find its breadth. (10) The cost of sinking a pit 4 yds. square, at 1s. 3d. per cubic yard, was £100; find its depth. (11) A rectangular cistern is 10 ft. long and 8 ft. broad; 560 c. ft. of water are drawn off; find how many feet the surface has sunk. [See Ex. LX. Quest. 17.] (12) The bottom of a tank contains 369 sq. ft. 48 sq. in.; what must be its depth that it may contain exactly 18432 gallons? [A gallon of water contains 277 c. in.] (13) A reservoir is 46 ft. 2 in. long and 21 ft. 6 in. broad; find through how many inches the surface will sink if 1032 gallons of water are drawn off. (14) A rectangular log of timber 1 ft. 8 in. broad and 18 in. thick cost five guineas, at 2s. 4d. per cubic foot. How long was it? (15) The cost of cutting a drain 5 ft. broad and 12 ft. deep, at 1s. 1d. per cubic yard, was £22 10s. Required its length. (16) A railway 600 chs. long is ballasted with gravel to a depth of 12 ft.; find the width of the roadway if 9240 truckloads are required, each truck holding 21 c. yds. of gravel. [1 chain=22 yds.] 145. The surface of a rectangular parallelopiped consists of six rectangles, whose areas may be found by Art. 50; the area of the whole surface is the sum of the areas of the rectangles. 146. To find the area of the whole surface of a rectangular parallelopiped. RULE. Add together the areas of the four side faces and of the two ends, and the sum will be the area of the whole surface. Ex. How many superficial feet of deal will be required to make a chest, closed at the top, 6 ft. long, 4 ft. broad, and 2 ft. deep? Here the surface consists of two rectangles, each measuring 6 ft. by 2 ft., two rectangles 6 ft. by 4 ft., and two 4 ft. by 2 ft., or, Surfaces of front and back=6×2×2=24 sq. ft. top and bottom=6×4x2=48 sq. ft. two ends Area of whole surface 24+48 +16=88 sq. ft. Ex. LXII. Find the area of the whole surfaces of rectangular parallelopipeds which have the following dimensions: (1) Length, 2 ft. 3 in.; breadth, 1 ft. 9 in.; depth, 9 in. (5) The length, breadth, and height of a rectangular parallelopiped are respectively 18, 12, and 10 in.; find the arca of its whole surface in square inches. (6) What will be the expense of lining a cistern 4 ft. long, 2 ft. 6 in. deep, and 2 ft. wide, with sheet lead of 10 lbs. to the square foot, estimating the lead at £1 17s. 4d. per cwt. ? (7) How much gilding will be required for the leaves of a book 18 in. long, 15 in. broad, and 2 in. thick? (8) The length, breadth, and depth of a rectangular vessel are 80, 75, and 36 in.; find, in square feet, the whole surface of a cube of the same capacity. (9) A room whose width is 10 ft. 4 in., and height 10 ft. 6 in., contains 1519 c. ft. of air; find the number of square feet in the floor. (10) A vessel with a square base is 12 ft. 6 in. deep, and holds 46 c. yds. 8 c. ft. of water; how many square feet of lead will be required to line it? (11) How many sheets of paper 8 in. by 6 in. will be required to cover the top and sides of a box 3 ft. 6 in. long, 2 ft. 4 in. wide, and 1 ft. 6 in. deep? (12) The value of a beam of timber whose length is 20 ft. and breadth 3 ft., at 2s. 6d. per cubic foot is £15; find the area of the whole surface. (13) A block of granite 16 ft. long, 8 ft. broad, and 4 ft. high, stands on one of its broadest faces; the other faces are polished, at a cost of £16; find the cost of polishing similarly another block 24 ft. long, 10 ft. broad, and 5 ft. high, similarly placed. THE RIGHT PRISM. 147. A Prism is a solid, whose two ends, or bases, are similar, equal, and parallel plane surfaces, and whose sides or faces are all parallelograms. The bases may be of any conceivable shape or figure, triangles, pentagons, hexagons, etc. 148. When the prism has a triangle for its end or base, it is called a triangular prism; when a pentagon, a penta- A gonal prism; when a hexagon, an hexagonal prism; and so on, according to the figure of its base. Thus ABCDEF is a triangular prism. 149. The axis of a prism is the straight line drawn through the centres of the two ends. When the axis is perpendicular to E the ends of a prism, it is said to be a right prism. E F D H Let there be two right prisms with the same height, one having the triangle ABC for base, and the other having the rectangle ABC D for base. Then it is evident that the prism on the triangular base is half the prism on the rectangular base. The solidity of the rectangular prism is found by Art. D 142 to be equal to the product of the area of the base and the height. Hence the solidity of the triangular G F E C prism will be found by taking half this product. If the base of the prism be not a triangle, but a pentagon, hexagon, or even a circle, it may be divided into triangles, as we have already seen (Art. 92), and the sum of the solid contents of the triangular prisms erected upon the triangular bases thus formed, and united to compose that prism, must be the solid content of the whole prism. The solid content of one of these is found by multiplying the area of its triangular base by the length, and therefore the solid content of the body formed from the union of all, is found by multiplying the sum of the triangular bases by the length; or, in other words, by multiplying the area of the base by the length or height. 150. To find the volume of a right prism. RULE. Multiply the area of the base by the height or length. Ex. 1. The base of a prism is a triangle, the sides of which are 13, 14, and 15 respectively, and the height or length is 25; find the volume. Area of base 84 (Art. 69, worked example). Ex. 2. The base of a prism is an equilateral triangle, each |