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volume of a sphere having its diameter equal to the inner diameter of the shell from the volume of a sphere having its diameter equal to the outer diameter of the shell. We thus obtain the following rule.

175. To find the volume of a spherical shell.

RULE. Subtract the cube of the inner diameter from the cube of the outer diameter, and multiply the remainder by 11.

(To find the outer diameter, twice the thickness of the shell must be added to the inner diameter.

To find the inner diameter, twice the thickness of the shell must be subtracted from the outer diameter. (Compare Art. 18).]

Ex. The outer diameter of a spherical shell is 10.5 in., and the thickness of the shell is 2.1 in. Find its solidity.

Here, the inner diameter will be 10.5 in.-4.2 in. = 6.3 in.

The cube of 10.5 is 1157.625; the cube of 6.3 is 250.047 ; 1157.625_250.047=907.578.

Hence, volume of shell= 31 x 907.578=475.398 c. in.

Ex. LXXVI.

Find the solid contents of spherical shells whose external and internal diameters are respectively:

(1) 10.5 in. and 8:4 in. (4) 8.4 in. and 6.3 in.
(2) 21 in. and 18.9 in. (5) 14:7 in. and 10.5 in.
(3) 6:3 in. and 2.1 in.

(6) 23.1 in. and 21 in. (7) How many c. in. of lead will be used in making a spherical shell 1 in. thick, whose external diameter is 7 in. ?

(8) What is the weight of a spherical shell 10.5 in. in diameter and 1.05 in. thick, composed of a substance of which 1 c. ft. weighs 216 lbs. ?

(9) Find the weight of a spherical shot of iron, 7 in. in diameter and 1 in, thick, supposing a cubic inch of iron to weigh 41 ozs.

(10) The exterior diameter of a hollow sphere is 21 in., and the thickness of the shell 4:2 in.; how many c. in. are there in it?

(11) Given that a cubic inch of iron weighs 4} oz8., find the weight of metal in 224 spherical shells 11 in. in diameter and in. thick.

(12) Find the volume of a spherical shell whose outer diam. eter is 11 in. and thickness 1 in.

Ex. LXXVII.

Miscellaneous Examples on the Mensuration of Solids

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(1) How many 3-inch cubes can be cut out of a 12-inch cube?

(2) A rectangular block of granite is 5 ft. 6 in. long, 4 st. 6 in. broad, and 2 ft. 8 in. thick; what is its cost, at 4s. 6d. per c. ft.?

(3) A piece of timber is 20 in. broad, and 16 in. thick ; at what distance from the end must it be cut, so that the part sawn off may measure exactly 5 c. ft. ?

(4) The diameter of a rolling-stone is 18:2 in., and its length 4 ft. 6 in.; required its solidity.

(5) What is the solidity of a sphere whose diameter is 14:7 ?

(6) The depth of a canal is 7 ft. 3 in., the width 20 ft. 4 in., and the length 10 miles; how many cubic feet of water will it contain?

(7) The length of a cellar is 27 ft., its breadth 16 ft., and its depth 8 ft. ; if the cost of digging be £2, what is the price per c. yd. ?

(8) Find the solid content of a square prism whose length is 48 ft., and its breadth and depth each 12 ft.

(9) What will be the expense of dressing a conical spire, at 8d. per sq. ft., the circumference of the base being 44 ft., and the slant height 45 ft. ?

(10) Find the surface of a globe 5.6 ft. in diameter.

(11) Find how many cubes whose edges are 4 in. may be cut out of a cube of which the edge is 8 in. ?

(12) A room 18 ft. long and 12 ft. broad is flooded with water to a depth of 6 in.; find the weight of the water.

(13) A pint contains 343 c. in. ; how many gallons of water will fill a cistern 4 ft. 4 in. long, 2 ft. 8 in. broad, and 1 ft. 1j in. deep?

(14) A reservoir is 26 ft. 8 in. long by 12 ft. 9 in. wide; how many cubic feet of water must be drawn off to make the surface sink 1 ft. ?

(15) What is the solidity of a spherical ball whose circumference measures 5 ft. 6 in. ?

(16) A block of stone 4 ft. long, 24 ft. broad, and 14 ft. thick weighs 27 cwt. ; find the weight of 100 c. in. of the stone.

(17) Find the cost of re-gilding the surface of a globe whose radius is 1.75 ft., at jd. per sq. in.

(18) A cube contains 5832 c. yds.; find the length of a side. (19) A triangular prism has the three sides of its base 2 ft,

per sq. ft.

polishing the convex surfaces of four such columns at 1s. 6d.

(49) What is the weight in lbs. of 105 cast-iron cannon balls, each 4 in. in diameter, assuming that 1 c. in. of iron weighs 41 ozs. ?

(50) How many thousand bricks, each 8 in. long, 4 in. broad, and 3 in. thick, will be required to build a wall 200 ft. long, 10 ft. wide, and 1 ft. thiek ?

(51) A piece of timber is 20 in. broad, and 16 in. thick; at what distance from the end must it be cut that the part sawu off may contain exactly 5 cubic ft. ?

(52) The solid content of a square prism is 735 cubic ft., and each side of its base 7 ft. What will be its solidity when hewn into the largest possible cylinder ?

(53) A joiner has a block of wood 6 ft. long, 2 ft. wide, and 11 in. thick. How many laths, each 2 ft. long, 1} in. wide, and } in, thick, can be cut out of it, making no allowance for waste in sawing it up ?

(54) If the solid content of a piece of timber be 38.22 cubic ft., its breadth 2.6 ft., and its thickness 1.5 ft., what is its length ?

(55) What will be the expense of dressing a conical spire at 6d. per sq. ft., the circumference of the base being 30 ft. and the slant height 45 ft. ?

(56) The altitude of a triangular pyramid is 12 ft., and the three sides of the base 2 ft. 11 in., 3 ft. 8 in., and 6 ft. 3 in. respectively. What is its solidity ?

(57) The volume of a cube is 166.375 cubic ft. Find the side.

(58) A school-room is 32 ft. long, 18 ft. wide, and 101 ft. high. How many children will it accommodate allowing 3 sq. ft. of floor for each; and how many cubic feet of space .vill there be for each ?

(59) Find, in yards, the edge of a cube equal in volume to a rectangular parallelopiped whose length is 16 yds., breadth 12 yds., and height 8ft.

(60) The discharge-pipe of a cistern 8 ft. long, 5 ft. broad, and 4 ft. deep is capable of emptying it in 40 minutes; how long would the same pipe take to empty a cistern containing 128 c. ft. of water ?

(61) What is the height of a closet 87 ft. by 68 ft., which will exactly contain 12 boxes each 47 ft. long, 33 ft. wide, and 2} ft. deep?

(62) How many times may a cistern 7 ft. long, 6 ft. 8 in. broad, and 6 ft. deep be filled from another 26 ft. 8 in. long, 16 ft. broad, and 10 ft. 6 in, deep ?

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Butler & Tanner, The Selwood Printing Works, Frome, and London.

MENSURATION FOR BEGINNERS.

ANSWERS.

PART I.

Ex. I. (1) 1764 sq. yds.; 625 sq. ft. ; 47524 sq. in. ; 390625 sq. yds.; 93025 sq. ft. ; 1681 sq. in. ; 361 sq. ft.

(2) 915.0625 sq. yds.; 2:030625 sq. yds. ; 45.5625 sq. in. ; 600.25 sq. in. ; 207.36 sq. ft. ; 74.390625 sq. ft.

(3) 421 sq. yds.; 510 sq. yds.; 5047 sq. yds.; 34534 sq. ft.; 181657 sq. in. ; 237174 sq. ft. ; 40871 sq. in. (4) 841 sq. yds.

(10) 80 ac. 3425 sq. yds. (5) 1296 sq. yds.

(11) 7 ac. 3369 sq. yds. (6) 2232.5625 sq. yds. (12) 4096 sq. yds. (7) 66316 sq. ft.

(13) 21:16 sq. yds. (8) 289 sq. ft.

(14) 1936 sq. yds. (9) 210 sq. ft.

Ex. II. (1) 144 sq. yds.

(5) 348 sq. ft. (2) 90000 sq. ft.

(6) 1296 tiles. (3) 301 sq. ft.

(7) 2944 stones. (4) Plantation, 8460 sq. yds.; (8) 1024 sq. in. ; 295 sq. in.

inner square, 15876 sq.
yds.

Ex. III.
(1) £70 68. 3d.

(5) 28. 8d. (2) £1953 28. 6d.

(6) 2 d. (3) £5 1s, 3d.

(7) £8 168. 4d. (4) £8 16s.

(8) £3 6s. 8d.

Ex. IV. (1) 13 sq. yds. 4 sq. ft. (5) 33 sq. ft. 9 sq. in. (2) 300 sq. yds. 4 sq. ft. (6) 16 sq. ft. 97 sq. in. (3) 93 sq. yds. 4 sq. ft. (7) 616 sq. ft. 100 sq. in.

264 sq. ft. 9 sq. in. (8) 10 sq. ft. 4 sq. in,

sq. in.

sq. in.

sq. in.

sq. in.

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(9) 193 sq. ft. 97 sq. in. (14) 3 sq. yds. 81 sq. in. (10) 14 sq. yds. 6 sq. ft. 36 sq. (15) 73 sq. yds. 6 sq. ft. 9 sq. in.

in. (11) 154 sq. yds. 1 sq. ft. 81 (16) 1 sq.yd. 5 sq.ft. 9 sq. in.

(17) 40 sq. yds. 7 sq. ft. 52 (12) 955 sq. yds. 7 sq. ft. 81

(18) 210 sq. ft. 36 sq. in.; (13) 273 sq. yds. 1 sq. ft. 73 £1 18s. 8d.

(19) £11 148. 1d.

Ex. V. (1) 2054 sq. yds.

(10) 1304294 sq. ft. 134 sq. yds.

(11) 878q. ft. (3) 561 sq. yds.

(12) 444 sq. ft. 562 sq. ft.

(13) 19138 sq. ft. 1076 sq. ft.

(14) 21294 sq. ft. (6) 6544 sq. ft.

(15) 10288 sq. ft. (7) 36738 sq. ft.

(16) £258 6s. 9d. (8) 1067 sq. ft.

(17) £38 10s. 5d. (9) 601 sq. ft.

Ex VI. (1) 342.25 sq. ft.

(5) 496.125 sq. ft. (2) 855.5625 sq. ft.

(6) 2016 tiles. (3) 1207.5625 sq. ft.

(7) £3 10s. 1d. (4) 37.515625 sq. ft.

(8) £2 6s.

Ex. VII.
(1) 72, 132, 96, 39, 112, 85, 82, 109, 62, 143 sq. in.
(2) 7', 2' 8", 11' 1", 11' 11", 3', 10' 9".

Ex. VIII.
(1) 21 sq. ft. 9'4", or 21 sq. ft. 112 sq. in.
(2) 152 sq. ft. 1° 4", or 152 sq. ft. 16 sq. in.
(3) 6 sq. ft. 3', or 6 sq. ft. 36 sq. in.
(4) 82 sq. ft. 6' 1", or 82 sq. ft. 73 sq. in.
(5) 132 sq. ft. 3', or 132 sq. ft. 36 sq. in.
(6) 458 sq. ft. 8' 1", or 458 sq. ft. 97 sq. in.

158 sq. ft. 4' 1", or 158 sq. ft. 49 sq. in.
(8) 117 sq. ft. 4' 4", or 117 sq. ft. 52 sq. in.

16 sq. ft. 8' 1", or 16 sq. ft. 97 sq. in.
(10) 604 sq. ft. 4' 1", or 604 sq. ft. 49 sq. in.
(11) 1100 sq. ft. Oʻ4", or 1100 sq. ft. 4 sq. in.
(12) 396 sq. ft. 8' 1", or 396 sq. ft. 97 sq. in.
(13; 600 sq. ft. 3, or 600 sq. ft. 36 sq. in.
(14) £46 178. 6d.
(15) £11 11s. 11d.
(16) £4 178. 13d.

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