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10, and its perpendic. 81; sides of the smallest triangle 3.5 and the perpendicular of its triangle 3; and length
10x41=42.5 area largest end.
3:5X1:5=5.25 area smallest end. Areas 42-5X 5.25=223-1250=14:93 root added.
sum 62-68 sum 62.68x4=292.5 solidity, Ans. 3. What is the solidity of the frustum A B C D fig... ure third, case 5th ; sides of its fargest square 9, and, the smallest 3, and length 14?
Samne question by rule second, Sides of the two squares 9X3327 product. 9-3—6 dif. X6=3612 add.
39x143546 Ans. * What is the solidity of the frus tum ABCD figure fourth, case fifth; sides of the largest end 15 by 9; sides of the smallest end 5 by 3.2; and 14 long?
Sides of the largest end 15x9.5=142-5 area.
Sides of the smallest end 5X3.2 = 16•0 area. Areas. 142.5X165V2280.0=+7.77 root 47.7 add.
sum 206•2 206.2x142886-6=1=962-21 Anse 5. What is the solidity of the frustum of a conc that is 10 feet square at one end, and 4 at the other, and is 32 feet long?
Ans. 1664 ft. 6. What is the solidity of the frustum of a cone that is 16 in. square at one end, and 12 at the other, and is 16 ft. long?
Ans. 2134 7. What is the solidity of the frestum of a cone that is 24 in. square at one end and 21 at the oth er, and is 10 feet long?
To find the solidity of the segment of a cone, as the
parts above A B in the figures, case 5th. RULE.-This part forms a cone of itself, and must be measured by case fifth.
To find the solidity of a wedge, when the edge
and large end are of equal width. RULE.--Multiply the length and width together and that product by half the thickness, the last product is the solidity,
Examples 1. What is the solidity of a wedge that is 12 ft. long, 12 wide and 6 thick, at the large end?
12X12=144x6 =432 solidity, Ans. 2. What is the solidity of a wedge that is 10 ftlong, 6-inches wide, and 6 thick ? Ans. 1. ft.
To find the solidity of a wedge when the edge is nar
rower than the large end,
. long, and 18 in. wide at the largest end, and 12
in. wide at the edge ; and 6 in. thick at the large end ? Width of the edge
12 Width of large end 18 x 236
48 reserved sum. Length of wedge
144 inches. Thickness of large end 6 inches.
864 product. Pro. 864 X 4841472-66912 in. or 4 ft. Ans. 2. What is the solidity of a wedge that is 20 long, 15 wide at the large end, and 10 at the edge ; and 12 thick ?
Note: This rule will give the solidity of a cone when it is squa re, or in form of a parallelogram.
Example. What is the solidity of the cone, figure third, case fifth ; width of the large end 9, thickness 9, and length 21 ? Width of the edge
18 reserved sum. 21 X9 189 X18-3402-6567 Ans.
(S:e Ans. to Quust. 3, Case Fifth.)
CASE X. To find the solidity of a globe, or sphere. DEFINITION.-A globe or sphere is a round body bounded by a surface every point of which is equally distant from a point within called the centre ; a line passing from one side to the other through the centre is called the diameter, or axis.
RULE.-Cube the diameter, or axis, and multiply its cube by .5236 the last product is the solidity.
Eramples. 1. What is the solidity of a globe, or sphere whose diameter is 113?
#113x113X113 X.523637355500.8692 solidity.
2. What is the solidity of the globe which we inhabit, in solid miles; allowing its circumference to be 25000 miles ? As 22 is to 7 so is 25000 to 7954
diam. nearly. (Fractions omitted.) 263485304337 sol miles Ans.
3. What is the solidity of a cannon ball that is 9 inches on its diameter ? Ans. 381.7044 sol. in.
4. What is the solidity of a sphere whose diameter is 2 ft, 8 in.
Ans. 9.927 ft. solidity.
CASE XI. To find the solidity of a segment, or part of a globe, or sphere ; or part of a globe cut off parallel to the diameter, as part DGF.
See the figure.
B diam. A B.14
RULE.Square the radius of its base, (as D o, or FO) multiply its square by 3, and reserve the product ; square the depth o G, add the square and reserved product together ; and multiply the sum by the depth; and the last product by 5236 ;. the product is the solidity..
Examples. 1. What is the solidity of the segment D GF; radius D o, or F 0,7; and depth o G, 5?
Radius 7x7x3=147 reserved product.
172x depth 5. 860 X 5236.450.296 Ans.
:- 2. Required the solidity of a segment of a globe whose semidiameter, is 9 in. depth 9 in.
Ans. 1526.8176 in.
To find the solidity of the middle zone of a sphere, or globe ; or the part of a sphere after two segments have been cut off, parallel to the diameter or
ašis, as CD E F figure in case eleventh.
Rule.--Square the semidiameter of both ends and add the squares together; and reserve the sum ; square the height, or distance of the two ends as o M, and add of its square to the reserved sum ; multiply the sum by the height, or distance o M, and this product again by 1.5708, the last product is the solidity.
1. What is the solidity of the middle zone C D E F, (case eleventh); diameter CE, or DF, 14, and height 0 M, 3 ?
NOTE.-In this example the diameters are alike which is not always the case.
Diam. 14 semi.=7X7349
Diam. 14 semi.=7X7=49
2. What is the solidity of the middle zone of a sphere whose greatest diameter is 12, and least. 8, and height or length 10?