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1. What is the convex surface of a cone, the circumfer. ence of whose base is 64 inches and slant height 40 inches ?

Ans. 1280 sq. in. 2. I bave a conical baystack whose slant heigbt is 8.25 ft., and the diameter of the base 6.5 ft. ; how many square yards of canvas will cover completely?

Ans. 9.35935 sq. yd. 774. Rule.-To find the contents of a cone, multiply the area of the base by one-third of the altitude.

1. Required the contents of a sugar-loaf, diameter of the base being 8 in. and height 18 in. Ans. 301.5936 cu. in.

2. How many cubic feet in a conical haystack 6 ft. higb and 20 ft. in circumference ?

Ans. 63.664 cu. ft.

THE FRUSTUM OF A PYRAMID AND CONE,

775. The Frustum of a Pyramid is the part of a pyramid which remains after cutting off the top by a plane parallel to the base.

776. The Frustum of a Cone is the part of a cone which remains after cutting off the top by & plane parallel to the base.

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777. Rule.- To find the convex surface of a frustum, take the sum of the perimeters or circumferences of the two bases, and multiply it by one-half the slant height.

1. Required the convex surface of the frustum of a square pyramid whose slant height is 24 feet, the side of the lower base 12 feet, and upper base 8 feet. Ans. 960 sq. ft.

2. Required the surface of a frustum of a cone wbose slant height is 20 feet, diameter of lower base 12 feet, and

Ans. 628.32 sq. ft. 778. Rule.--To find the contents of a frustum, take the sum of the two bases and the square root of their product, and multiply this sum by one-third of the altitude of the frustum.

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apper base 8 feet.

1. What are the contents of the frustum of a square pyramid the sides of whose bases are 2 and 3 feet, and whose altitude is 15 feet ?

Ans. 95 cu. ft. Sug.–28+32+20 x 32=4 +9+6= 19, and this multiplied by O equals 95 cu. ft.

2. What is the amount of timber in a log which measures 8C feet in length, the radius of one base being 6 feet and of the other 3 feet?

Ans. 5277.888 cu. ft.

THE SPHERE.

779. A Sphere is a volume bounded by & curved surface, every point of which is equally distant from a point within called the centre.

780. The Diameter of a sphere is a line passing through its centre and ending in the surface. The radius is half the diameter.

781. Rule.- To find the surface of a sphere, we multiply the circumference by the diameter, or square the radius and multiply it by 4 times 3.1416.

1. Required the surface of a sphere whose diameter is 24 inches.

Ans. 1809.5616 sq. in. 2. Required the surface of a sphere whose diameter is 96 inches.

Ans. 28952.9856 sq. in. 782. Rule.- To find the contents of a sphere, we multiply the cube of the diameter by of 3.1416.

1. Required the contents of a sphere whose diameter is 6 incbes.

Ans. 113.0976 cu. in. 2. If the diamet jr of the earth is 8000 miles, what are its surface and soli . contents ? Ans. Sur.; 201062400 sq. mi.

783. Rule.-To find the size of a cube which may be cut from a given sphere, we square the diometer, divide by 3, and extract the square root of the quotient.

1. What is the side of a cube which may be cau from a spbere 21 inches in diameter ?

Ans. 12.124 in.

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GAUGING. 784. Gauging is the process of finding the capac.ty of tasks and other vessels.

Barrels and casks differ from cylinders in bulging out in the middle. By ascertaining the approximate mean diaineter of the cask or barrel, the capacity can be obtained like that of a cylinder.

Rule I.-To find the mean diameter of a barrel or cask, add to the head diameter 3, or, if the staves are not much curved, $ of the difference between the head and bung diameters.

Rule II.-To find the capacity in gallons, multiply the square of the mean diameter by the length (both expressed in inches), and this product by .0034.

1. How many gal, in a cask whose head diameter is 28 in., bung diameter 36 in., and length 40 in.? Ans. 151. gal.

2. How many gallons in a barrel of slight curvature, 3 ft. long, the head diameter being 26 in., and the bung diameter 29 in.?

Ans. 94.59616 gal. SUPPLEMENTARY PROBLEMS IN MENSURATION.

1. Two towns, 42 mi. apart, are on a map located 10 in. apart; what is the scale on which the map is drawn? Ans. fin. to the mi.

2. How many feet of boards will cover the gable end of a house 34 ft. wide, the ridge being 18 feet high ? Ans. 306 sq. ft.

3. The rafters of a roof are 18 ft. long, and the distance between the eaves is 24 ft.; what is the height of the ridge? Ans. 13.41+ ft.

4. I have a triangular building lot whose sides measuru 25, 35, and 40 feet respectively; if I sell it at $5 per square foot, what do I receive?

Ans. $2165.05. 5. How many Belgian blocks, averaging 6 in. x 12 in. on the surface, will be required to lay a pavement on the roadway of a street 500 yd. long and 15 yd. wide?

Ans. 135,000. .6. How many bricks, 8 in. X4 in., will be required to lay a pave. ment on a sidewalk 7 feet wide, extending along 4 lots, each having 18 ft. 6 in. front?

Ans. 2331. 7. What is the expense of sodding a plot of ground 45 yd. long and 95 ft. wide, with sods 15 in. X 24 in., the sods when laid costing $1.50 per hundred ?

Ans. $76.95. 8. How much will it cost to fence a rectangular garden 20 rode

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long and 1.5 rods wide, with pickets 4 inches wide and 3 inches apart, at $9 ^ M.?

A18. $17.82. 9. Required the length of a hand rail for a flight of stairs of 18 steps, each step being 7 in. high and 9f in. wide ? Ans. 171 ft.

10. What will be the cost of flooring at $33.25 per M., of a three story house, the inside measure being 58 ft. x 34 ft., deducting 15 ft. 6 in. by 8 ft. 3 in. for the stairs ?

Ans. $183.95. 11. What will be the cost of a thousand tiles in the shape of a rhombus 15 in, on a side, a line drawn from an obtuse angle perpendicular to the opposite side, meeting it 9 in. from the acute angle, at 75% a square foot ?

Ans. $937.50. 12. How much will it cost to roof a warehouse with slate 48 ft.X 60 ft., the height of the ridge being 10 ft, and the eaves projecting 6 inches, at $14.75 per square (100 sq. ft.)?

Ans. $469.05. 13. A yard 36 feet square has in the centre a fountain, the basin of which is 12 feet in diameter; there is a flower-bed, 4 feet wide, around 3 sides of the yard; what will be the expense of paving the remainder at $2.25 per sq. yard ?

Ans. $195.73. 14. The pressure of the atmosphere is 15 lb. to the square inch what is the pressure on a globe 4 ft. in dian,eter? Ans. 108573.696.

15. A horse is fastened by a rope 10 ft. long to the top of a post 6 ft. high; over how much space can he graze ? Ans. 201.0624 sq ft.

16. The circular course of 2 riding-school is 110 feet in.ils outer diameter, and 104 feet wide; what was the expense of its construction, at 104 per sq. foot.?

Ans. $328.22. 17. A room 27 ft. 6 in. long, and 16 it. 3 in. wide, has a sini-cir. cular bow, 22 feet in diameter, thrown out on one side; find the area of flooring in the whole room.

Ans. 636.9418 sq. ft. 18. I have a fish pond in the form of an ellipse, 20 ft. long, 15 ft. wide; how many hogsheads of water are required to fill it to the depth of 4 ft.?

Ans. 111.909 hhd. 19. In a circular grass-plot whose diameter is 50 yd., there is a gravel walk 1 yd. wide, running round it 1 yd. within the edge; what will be the cost of sodding the plot at 124 per sq. yd. ? Ans. $217.90.

20. The steeple of a church in the form of a cone is 30 feet in di ameter at the base, the slant height being 90 ft.; what will it cost t paint it at 25% per square yard ?

Ans. $117.81. 21. In a tin funnel, one part is conical, the slant height of the conical part is 4 in., the circumference at one end 10 in., and at the nther end 1 in.; the other part is cylindrical, the length being 5 in.; required the number of sq. in. of tin in it.

Ans. 27 sq.in

22. A hip has the bottom 2 ft. 6 in. square, the top 3 ft. 6 in. square, the height 2 ft. 6 in.; what is the cost of lining it with zinc at 209 per sq. ft.? How many bu. will it hold? Ans. $7.37; 18.24 bu.

23. How many cu. ft. in a telegraph pole 40 ft. long, 18 in. in diam. eter at the base, and 8 in. in diameter at top? Ans. 38.688cu, ft.

24. How much map-surface on a school globe 12 inches in diame ter; and how many cubic inches of material are there in it, if it is a hollow sphere 1 in. thick ? Ans. 452.3904 sq. in.; 381. 1808 cu. in.

25. A room 24 ft. 6 in. long, 13 ft. 4 in. wide, is flooded with water 8 in. deep; what is the weight of the water ? Ans. 51047 lb.

26. A circular room, of which the diameter is 25 ft., and height of wall 14 ft., is covered with a hemispherical dome, what is the cost of plastering the wall and dome at 254 per sq. ft. ? Ans. $520.324

27. How many dozen boxes of perfumery, 2 in. on each side, can be packed in a rectangular box whose dimensions are respectively 1 ft. 6 in., 1 st., and 8 in.?

Ans. 18 doz. 28. A cubic inch of gold is hammered out to cover a square (10C sq. ft.); what is its thickness ?

Ans. .000069 inch. 29. An ice house is 40 ft. long, 30 ft. wide, and 20 ft. deep; what area of ice 6 in. thick, will be required from a pond to fill it? What would it weigh at 56 lb. the cu. ft. ? Ans. 48000 sq. ft.; 1344000 lh.

30. A company wish to excavate a canal 25 miles long, with an average width of 9 yards, and an average depth of 5 ft. 6 in.; how long will it take 500 men to do the work, if each man averages 12 cubic yards a day?

Ans, 121 days. 31. If the men are paid $1.50 each per day, and the contractor estimates 5% profit, what proposal will he make for digging the above canal ?

Ans. $95287.50. 32. If 50 cubic feet of air are required per person in a well-venti. lated room, how many persons can safely remain in a room 60 ft. long, 40 ft. wide, and 12 ft. high?

Ans. 576 persons. 33. The cost of a cube of metal, at $12 per cubic inch, is $4116; find the cost of gilding it over at 29 per square inch. Ans. $5.88.

31. A cast-iron garden roller is 40 in. long, 22} in, in diameter, and the iron is į in. thick; required its weight, if a cubic inch of iron weighs 4} ounces.

Ans. 576.5321} lb. 35. If a cannon ball, 6 in. in diameter, is melted and cast in a cuni. cal mould 6 in. in diameter at the base, what is the length of the cone ?

Ans. 12 in. 36. A railroad embankment has an average depth, for 5 miles, of 12 feet, width at top 8 yards, and at bottom 12 yards; what was the cost of carting at 15% per load (cu. yd.) ?

Ans. $52800

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