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33. There is a conical glass, 6 inches high, 5 inches wide at the top, and which is part filled with water. What must be the diameter of a ball, let fall into the water, that shall be immersed by it?
Ans. 2.445+ inches. 34. A certain lady, the mother of three daughters, had a farm of 500 acres, in a circular form, with her dwelling-house in the center. Being desirous of having her daughters settled near her, she gave to them three equal parcels, as large as could be made in three equal circles included within the periphery of her farm, one to each, with a dwelling-house in the center of each; that is, there were to be three equal circles, as large as could be made within a circle that contained 500 acres.
How many acres did the farm of each daughter contain, how many acres did the mother retain, how far apart were the dwelling-houses of the daughters, and how far was the dwelling-house of each daughter from that of the mother? Ans. Each daughter's farm contained 107 acres 2 roods 31.22+ rods. The mother retained 176 acres 3 roods 26.34+ rods. The distance from one daughter's house to the other was 148.119817+ rods. The mother's dwelling-house was distant from her daugh
ters' 85.51+ rods. 35. James Page has a circular garden, 10 rods in diameter; how many trees can be set in it, so that no two shall be within ten feet of each other, and no tree within two and a half feet of the fence enclosing the garden ?
Ans. 241. 36. A and B engaged to reap a field for 90 shillings; and as A could reap it in 9 days, they promised to complete it in 5 days. They found, however, that they were obliged to call in C, an inferior workman, to assist them for the last two days, in consequence of which B received 3s. 9d. less than he otherwise would have done. In what time could B and C each reap the field ?
Ans. B could reap it in 15 days, and C in 18 days. 37. A merchant tailor bought 40 yards of broadcloth, 21 yards wide; but on sponging it, it shrunk in length upon every 4 yards half a quarter, and in width, one nail and a half upon every 14 yards. To line this cloth, he bought flannel 5 quarters wide, which, being wet, shrunk the whole width on every 20 yards in length, and in width it shrunk half a nail. Required the number of yards of flannel used in lining the cloth.
Ans. 71.75 yards. 38. I have a garden in the form of an equilateral triangle, whose sides are 200 feet. At each corner stands a tower; the height of the first is 30 feet, the second 40, and the third 50. At what distance from the base of each tower must a ladder be placed, that it may just reach the top of each ? And what is the length of the ladder, the garden being a horizontal plane ? Ans. The foot of the ladder from the base of the first tower 118.811+
feet; second tower, 115.827+ feet; third tower, 111.875+ feet. Length of the ladder, 122.535+ feet.
THE METRIC SYSTEM.
682. THE METRIC SYSTEM of Weights and Measures is based upon the decimal system of notation.
It was authorized to be used in the United States, and its use introduced into some departments of public service, in 1866, by an Act of Congress.
683. The Principal Units, or those from which the others are derived, are:
The METER, which was intended to be, and is nearly, one ten-millionth of the distance on the earth's surface from the equator to the pole.
The SQUARE METER, which is the square whose side is one meter; and the Are (pronounced air), which is the square whose side is ten meters.
The CUBIC METER, or STERE (pronounced stair), which is the cube whose edge is one meter; and the LITER (pronounced leeter), which is one thousandth of a cubic meter.
The GRAM, which is the weight, in vacuum, of a cubic centimeter of distilled water, at the greatest density.
684. The HighER DENOMINATIONS are expressed by prefixing to the name of the principal unit,
DEKA, 10; HECTO, 100; Kilo, 1000 ; MYRIA, 10,000 ; and the Lower DENOMINATIONS by prefixing
Deci, 10th ; CENTI, 100th ; Milli, 1000th.
685. MEASURES OF LENGTH. 10 millimeters (mm.) 1 cen'timeter,
.3937 inch. 10 centimeters,
1 dec'imeterdcm., 3.937 inches. 10 decimeters, 1 METER, me.,
39.37 inches. 10 meters,
1 dek'ameter, dkm., 393.7 inches. 10 dekameters,
1 hectometer, hm., 328 ft. 1 in. 10 hectometers,
1 kilometer, km., 3280 ft. 10 in. 10 kilometers, 1 myr'iameter, mym.,
NOTE 1. The meter is used as the unit of measure in measuring cloth, and common lengths and distances, and the kilometer in measuring long distances, as the length of roads and rivers, distances between cities, etc.
NOTE 2. A meter is very nearly 3 feet, 3 inches, and 3 eighths of an inch in length; and a kilometer is about 200 rods, or of a mile.
686. MEASURES OF SURFACE. 100 sq. millimeters (mm2)=1 sq. centimeter, cm'.,-.00155 sq. in. 100 sq. centimeters, 1 sq. decimeter, dem?., .1076 sq. ft. 100 sq. decimeters,
1 SQ. METER,
m2., 1.196 sq. yd.
Also, 100 cen'tiares (ca.), or sq. me.,
119.6 sq. yd.
1 hectare, ha., 2.471 acres. Note 1. The square meter is used in general in measuring surfaces. The are is the principal unit in measuring land; and the hectare is also taken as a unit of the same kind of measurement.
Note 2. A square meter is about 14 square yards, and an are nearly 4 square rods.
687. MEASURES OF VOLUME. 1000 cu. millimeters, (mm®.), 1 cu. centimeter, cm'., 1.061 cu. in. 1000 cu. centimeters, 1 cu. decimeter, demo., 61.022 cu. in. 1000 cu. decimeters, 1 cu. METER,
m., 1.308 cu. yd.
Also, 10 dec'isteres (des.) =1 STERE, or cu. meter, st.,=1.308 cu. yd. 10 steres,
1 děklastere, dks., 13.08 cu. yd.
1.0567 liq. qt. 10 liters,
1 dek'aliter, dkl., 2.6417 liq.gal. 10 dekaliters,
1 hectoliter, hl., 2 bu. 3.35 pk. 10 hectoliters, 1 kil'oliter, kl., 1.308 cu. yd.
NOTE 1. The cubic meter is used in general in measuring volume, and the stere in estimating wood.
Note 2. The liter is used in measuring liquids ; it is about 1} liquid quarts, or 7 of a dry quart; 1000 liters are equal to 1 cubic meter.
Note 3. The hectoliter, or for brevity hecto, is generally taken as the anit in measuring dry articles, such as grains, salt, etc., and is about 25 bushels, or 5 of a barrel.
688. WEIGHTS. 10 milligrams (mg.)=1 cen'tigram, cg.,
.1543 grains. 10 centigrams, 1 dec'igram, deg., 1.543 grains. 10 decigrams, 1 GRAM, gm.,
15.432 grains. 1 dek'agram, dkg.,
.3527 av. oz. 10 dekagrams, 1 hec'togram, hg., 3.5274 av. oz. 10 hectograms,
k., 2.2046 av. Ib. 10 kilograms, 1 myr'iagram, myg.,
22.046 av. lb. 10 myriagrams, 1 quin'tal, 9.,
220.46 av. Ib. 10 quintals, 1 tonneau, t.,
2204.6 av. Ib. Note 1. The gram is the unit in weighing gold, silver, and jewels, in mixing medicines, determining postage, etc.; the kilogram, or, for brevity, kilo, is the ordinary weight of commerce; and the tonnean, or metric ton, is used in weighing coal, hay, and all very heavy articles.
NOTE 2. A kilo is about 21 avoirdupois pounds, or 2.68 Troy pounds; and a metric ton a little more than 2200 avoirdupois pounds.
689. Equivalents of denominations of common weights and measures in metric denominations are exhibited in the following
COMPARATIVE TABLE. An inch 2.54 centimeters. A gallon 3.786 liters. A foot =30.48 centimeters. A bushel .3524 hectoliter. A mile =1.6094 kilometers. A cu. inch .01639 liter. inch=.0006452
sq. meter. A cu. yard = .7646 stere. A sq foot=.0929
meter. A cord 3.625 stere. A sq. yd. =.8362 sq. meter.
A grain A sq. rod=.2529 are.
A Troy lb. = .373 kilo. An acre =.4047 hectares. An av. Ib. - .4536 kilo. A
sq. mile=259 hectares. A com. ton .9071 tonneau.
The numbers in the tables are determined approximately, but are sufficiently exact for all ordinary purposes.
690. In expressing Metric Weights and Measures the decimal point, as in United States money, is placed between the unit, and its subdivisions decimally written.
One, two, or three orders of figures are required to express
each metric denomination, lower than the unit, according as the scale is 10, 100, or 1000. Thus,
We write according to the unit and scale, 6 kilometers, 3 dekameters, 5 meters, 7 decimeters, and 2 centimeters, as 6.03572 kilometers, or 6035.72 meters.
16 hectares, 35 ares, and 25 centiares, as 16.3525 hectares, or 1635.25 ares.
41 cubic meters, 5 cubic centimeters, and 428 cubic millimeters, as 41.000005428 cubic meters.
691. The metric system is made very simple of application by the practice of using in general but a few of its denominationsThus,
Of LENGTHS, the meter, or kilometer ; of strFACES, the square meter, are, or hectare; of voLUME, the cubic meter or stere, the liter, or hectoliter ; and of weights, the gram, kilo, or mctric ton.
692. In a metric expression, a decimal part, if any, may be either read as a decimal of the unit; or, which is better, as a number of the lowest metric denomination denoted. Thus,
15.75 meters may be read as fifteen meters and seventy-five centimeters.
25.0035 hectares, as twenty-five hectares and thirty-five centiares.
EXERCISES. 1. Copy and read 45.63 meters.
This mixed number may be read forty-five and sixty-three hundredths meters, or forty-five meters and sixty-three centimeters. Write and read, 2. 136.5 st.
6. 120.05 ha. 3. 170.301 km.
7. 31.55 hl. 4. 61.58 lt.
8. 561.038 k. 5. 3.6054 m.
9. 15.47894 m. 10. Write in figures eight meters, four centimeters.
Regarding a meter the unit, we write 8, which will denote the number of meters, as the integral part of a decimal expression, and place the decimal point. Then, as there are no decimeters, we write 0 in the place of decimeters, and 4, which will denote the number of centimeters, in the place of centimeters, and have as the required expression, 8.04 me.