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8. How many cubic feet in 75.125 cords? How many cubic inches in the same? Ans. 9616 cu. ft. 9. How much gold may be obtained from a ton of quartz rock, if it yields .0016 of its weight in gold? Ans. 3.2 lb. 10. How much is the cost of 24 cwt. 87 lb. of sugar, at $6.50 per hundredweight? Ans. $161.65. 11. A man bought 4 hhd. 28 gal. 3 qt. of wine at $4.50 a gallon; what did it cost? Ans. $1263.375.

12. A grocer shipped 6120 eggs to Philadelphia in 6 barrels; how many did he pack in a barrel? Ans. 85 doz. 13. An apothecary bought 16 lb. 10 oz. of drugs, at $12.25 a pound; required the cost. Ans. $204.039. 14. Since 9 o'clock the sun has seemed to pass over 4° 23' 24"; what time is it? Ans. 9 h. 17 min. 333 sec. 15. A druggist purchased 20 lb. 83 oz. of opium at 483 cents an ounce; what did it cost? Ans. $160.265+.

16. What is the weight of $1,000,000 in gold dollars at 25.8 gr. each? What is the weight in silver half-dollars at 192.9 gr. each? Ans. 4479 lb. 2 oz.; 66979 lb. 2 oz. 17. If I start at St. Louis, latitude 38° 37′ 28′′ N., travel due north 1800 miles; what latitude do I reach? Ans. 64° 39' 3"+.

and

18. What cost 31 lb. 14 oz. of drugs, if 6 lb. 6 oz. cost $31.40? Ans. $157. 19. What cost 5 cwt. 65 lb. of sugar, if .96 of a cwt. cost $7.50? Ans $44.14.

20. What cost 10 cwt. 81 lb. of hay, if

£2 10 s. 6 d.?

21. If £124 16 s. 6 d. are worth

lars are £136 10 s. 6 d. worth?

5 cwt. 55 lb. cost Ans. £4 18 s. 4 d. +.

$599.16, how many dolAns. $655.32.

22. In what time will a man walk 120 mi. 160 rd. if he goes

12 mi. 16 rd. in 3 h. 20 min.

Ans. 83 h. 20 min.

23. If A travels 24 mi. 198 rd. 4 yd. in 6 h. 30 min., how far will he go in 9 h. 45 min.?

Ans. 36 mi. 298 rd. 1ft. 6 in.

SUPPLEMENTARY PROBLEMS.

To be omitted unless otherwise directed.

24. How many centals of wheat are equivalent to 1200 bushels? How many centals in 150 bar. ? Ans. 720; 294.

25. How many bushels of buckwheat in Kentucky are equivalent to 520 bu. in Illinois? To 650 bu. in Pennsylvania? (See Art. 286.) Ans. 400; 600.

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26. A merchant bought in Connecticut 32 bushels of oats at 24 a pound. and sold them in New York at 80g a bushel; what was his profit? (See Art. 286.) Ans. $4.48.

27. If.B digs 363 rd. 7 yd. of ditch in 35 wk. 5 da., how long will it take to dig 910 rd. 3 yd., working 12 11. a day, 6 da. a week, and 4 wk. a month? Ans. 22 mo. 1 wk. 31⁄2 da.

28. If a river current carries a raft of lumber at the rate of 4 mi. 265 rd. per hour, how long will it be in carrying it a distance of 309 miles? Ans. 2 da. 16 h.

29. Mr. Owen sold 15 bu. 3 pk. 4 qt. of apples at $2.75 a bushel, and took his pay in flour at 3 a pound, receiving only an exact number of barrels, and in sugar at 12 a pound for what remained; how many barrels of flour and how many pounds of sugar did he receive? Ans. 6 barrels; 2077 lb.

30. Two men start from different places on the equator, and travel toward each other till they meet; on comparing their watches with the time of the place of meeting, it is found that the first is 45 minutes slow and the second 1 h. 15 min. fast; how far apart were the points at which they started, and in what direction did each travel? Ans. 2074.8 mi.

31. The distance from a certain toll-gate east to a tavern is 3 mlles; from the toll-gate west to a school-house is 4510 rods; half way between the tavern and the school-house is a creek 100 yards wide; how far from the toll-gate to the middle of the creek; to the further bank of the creek?

Ans. 1 mi. 195 rd. 2 yd. 2 ft. 9 in.; 1 mi. 204 rd. 3 yd. 1 ft. 3 in. 32. A balloon started from Paris with despatches for Tours, and alighted near Bourges, 119 mi. 266.663 rods from Paris. Its actual route was 14 times this distance, which it made at the rate of 51 mi. 80 rd. an hour. Starting at 4 A. M., when did it alight?

Ans. 6 h. 48 min. 21 sec. A. M.

SECTION VII.

PRACTICAL MEASUREMENTS.

332. The Applications of Measures to the farm, the household, the mechanic arts, etc., are so extensive that we now present a distinct treatment of the subject.

333. These Practical Measurements include Measures of Surface, Measures of Volume, Measures of Capacity, and Comparison of Weights and of Money.

MEASURES OF SURFACE.

334. A Surface is that which has length and breadth without thickness.

THE RECTANGLE.

335. A Rectangle is a plane surface having four sides and four right angles. A slate, a door, the sides of a room, etc., are examples of rectangles.

336. A Rectangle has two dimensions, length and breadth. A Square is a rectangle in which the sides are all equal.

337. The Area of a rectangle is the surface included within its sides. It is expressed by the number of times it contains a small square as a unit of measure.

Rule I. To find the area of a square or rectangle, multiply its length by its breadth.

For, in the rectangle above, the whole number of little squares is equal to the number in each row multiplied by the number of rows, which is equal to the number of linear units in the length multiplied by the number. in the breadth.

Rule II. To find either side of a square or rectangle, divide the area by the other side.

VOTES.-1. The sides multiplied must be of the same denomination, and the product will be square units of that denomination, which may be reduced, if necessary, to higher denominations.

2. In dividing, the linear unit of the side must be of the same name as the square unit of the area, and the quotient will be linear units of the came denomination.

WRITTEN EXERCISES.

1. How many square feet in a floor 32 ft. long by 21 ft. wide? how many square yards?

SOLUTION. To find the area, we multiply the length by the breadth, and we have 32×21=672 sq. ft.; reducing this to square yards, we have 74j sq. yd.

2. How many square yards in the surface of a blackboard 27 ft. long by 4 ft. wide? Ans. 12 sq. yd. 3. How many square yards in a garden 215 ft. long by 109 ft. wide? Ans. 2603 sq. yd. 4. What is the width of a room 25 feet long, whose floor contains 500 sq. ft.? Ans. 20 ft. 5. A rectangle contains 466 sq. ft., and one side is 16 ft 6 in. long; how long is the other side? Ans. 28 ft. 3 in. 6. How many square feet in the sides of a room 18 ft long, 14 ft. 6 in. wide, and 9 ft. 6 in. high?

7. A certain box is 3 ft. 6 in. long, 2 ft.

Ans. 617.

3 in. wide, and

1 ft. 4 in. high; how many square feet in its surface? Ans. 31 sq. ft. 12 sq. in.

8. What is the surface of a cubical box, each of whose dimensions is 1 ft. 6 in.?

Ans. 13 sq. ft.

THE TRIANGLE.

338. A Triangle is a plane surface having three sides and three angles; as, ABC.

339. The Base is the side upon which it seems to stand, as AB. The

B

D

Altitude is a line perpendicular to the base, drawn from the angle opposite; as, CD.

340. A triangle which has its three sides equal is called equilateral; when two sides are equal it is called isosceles; when its sides are unequal it is called scalene.

Rule I. To find the area of a triangle, multiply the base by one-half of the altitude.

Rule II. To find the base or altitude of a triangle, divide the area by one-half the other dimension.

WRITTEN EXERCISES

1. What is the area of a triangle whose base is 25 inches and altitude 18 inches?

SOLUTION.-To find the area, we multiply the base by one-half the altitude; 25x9=225; hence the area is 225 sq. in.

2. How many square feet in a triangle whose base is 18 ft. 6 in. and altitude 9 ft. 9 in.? Ans. 90 sq. ft. 27 sq. in. 3. What is the area of the gable end of a house 29 ft. wide the ridge being 12 ft. higher than the top of the wall?

Ans. 174 sq. ft.

4. The area of a triangular bed of flowers is 25 sq. ft., and its base 10 ft.; what is the altitude?

Ans. 5 ft.

5. The area of a triangular lot is 250 square yards, and its base is 250 ft.; what is its altitude?

Ans. 18 ft.

6. The area of the gable of a house is 378 sq. ft., the base being 14 yards; what is the height of the ridge?

Ans. 18 ft.

THE CIRCLE.

341. A Circle is a plane figure bounded by a curved line, every point of which is equally distant from a point within, called the centre.

A

342. The Circumference of a circle is the bounding line; any part of the circumference, as BC, is an Arc. An arc of one-fourth of the circumference is called a Quadrant.

343. The Diameter is a line passing through the centre and terminating in the circumference; as, AB. The Radius is a line drawn from the centre to the circumference; as, OD. Rule I. To find the circumference of a circle, multiply ine diameter by 3.1416.

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