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12. A, of Boston, remits to B, of New York, a bill of exchange on London, the avails of which he wishes to be invested in goods on his account. B, having disposed of the bill at 7} per cent. advance, received $ 9675; and having reserved for himself per cent on the sale of the bill, and 2 per cent for commission, he invests the remainder. What is the amount invested, and for how much was the bill drawn? Ans. Investment, $ 9461.5834; the bill was £ 2025.
13. Bunker Hill Monument is 30ft. square at its base, and 15ft. square at its top; its height is 220 feet. From the bottom to the top, through its centre, is a conical avenue 15ft. in diameter at the bottom, and ilft. at the top. How many cubic feet are there in the monument?
Ans. 86,068.518+ ft. 14. A hired a house for one year for $ 300; at the end of 4 months he takes in M as a partner, and at the end of 8 months he takes in P. At the end of the year, what rent must each
Ans. A pays $ 183}; M pays $ 834; P pays $33. 15. A merchant receives on commission three kinds of flour; from A he receives 20 barrels, from B 25 barrels, and from C 40 barrels. He finds that A's flour is 10 per cent better than B's, and that B's is 20 per cent. better than C's. He sells the whole at $ 6 per barrel. What in justice should each man receive ?
Ans. A receives $ 139111; B, $ 158174; C, 211141. 16. Bought 100 barrels of flour, at $ 5 per barrel, and immediately sold it on a credit of six months. The note which I received for pay I got discounted at the Suffolk Bank, and, on examining my money, I found that I had gained 20 per cent. on my purchase. What did I receive per barrel for the flour?
Ans. $ 6.181935 17. Required the greatest possible number of hills of corn that can be planted on a square acre, the hills to occupy only a mathematical point, and no two hills to be nearer than three and a half feet. Ans. 4165.
18. Lent a friend $ 700, which he kept 20 months. Some years after I borrowed of him $ 300; how long should I keep it to balance the favor ?
Ans. 46 months. 19. John Lee gave 1 of his estate to his wife, } of the remainder to his oldest son, and of the residue to his oldest daughter, and 1 of what then remained, which was $ 1500, was to be equally distributed among his other children, who received $ 150 each; required the number of his children, and the value of his estate.
20. A and B set out to travel round a certain island, which is 80 miles in circumference. A travels 5 miles a day, and B 7 miles a day. How far must B travel to overtake A ? Ans. 280 miles.
21. If 24.4 cubic inches of lead weigh 16 pounds, required the number of feet of lead pipe that can be made from 80 pounds of lead, the caliber of the pipe to be 1 inch, and the thickness of it 1 of an inch.
Ans. 10.35+ feet. 22. How long a tube can be made from a cylinder of lead 8 inches long and 2 inches in diameter, and through the centre of which is a hole of an inch in diameter; the tube or pipe to be of an inch in caliber, and g of an inch in thickness ? Ans. 16.29+ in.
23. Four men, A, B, C, and D, bought a stack of hay, containing
8 tons, for $ 100. A is to have 12 per cent. more of the hay than B, B is to have 10 per cent. more than C, and C is to have 8 per cent. more than D. Each man is to pay in proportion to the quantity he receives. The stack is 20 feet high, and 12 feet square at its base, it being an exact pyramid; and it is agreed that A shall take his share first from the top of the stack, B is to take his share the next, and then C and D. How many feet of the perpendicular height of the stack shall each take, and what sum shall each pay? Ans. A takes 13.22+ft., and pays $ 28.9313431; B takes 3.14+ft.,
and pays $ 25.8311987; C takes 2.06+ft., and pays $ 23.481811; D takes 1.58+ft, and pays $ 21.7413 41. 24. A, B, and C bought a grindstone, for which they paid $ 10.60. B paid 20 per centmore than A, and 10 per cent. less than C. The diameter of the stone was 65 inches, and the diameter of the place for the shaft 3 inches. What sum did each pay, and how much must each grind off from the semidiameter to obtain his proper share of the stone ? Ans. A paid $3, B $ 3.60, and C $ 4. A grinds off 5 inches ; B
74 inches, and C 184 inches. 25. A servant draws off a gallon on each day, for 20 days, from a cask containing 10 gallons of wine, each time supplying the deficiency by the addition of a gallon of water; and then, to escape detection, he again draws off 20 gallons, supplying the deficiency each time by a gallon of wine. How much water still remains in the cask ?
Ans. 1.0679577 gallons, or more than a gallon and half a pint. 26. The dimensions of a bushel measure are 184 inches wide, and 8 inches deep; what should be the dimensions of a similar measure that would contain 8 bushels ?
Ans. 37in. wide, 16in. deep. 27. What is the weight of a hollow spherical iron shell 5 inches in diameter, the thickness of the metal being 1 inch, and a cubic inch of iron weighing 8 of a pound ?
Ans. 13.2387lb. 28. At a certain time between 2 and 3 o'clock, the minute-hand was between 3 and 4. Within an hour after, the hour-hand and minute-hand had exactly changed places with each other. What was the precise time when the hands were in the first position ?
Ans. 2h. 15m. 56.255. 29. Required the contents of the largest cube that can be inscribed in a sphere 20 inches in diameter.
Ans. 1539.58 +- cu. in. 30. If in a pair of scales a body weigh 90 pounds in one scale, and only 40 pounds in the other, required the true weight, and the proportion of the lengths of the two arms of the balance-beam on each side of the point of suspension.
Ans. Weight 60lb., and the proportions 3 to 2. 31. In turning a one-horse chaise within a ring of a certain diameter, it was observed that the outer wheel made two turns, while the inner wheel made but one; the wheels were each 4 feet high; and supposing them fixed at the distance of 5 feet asunder on the axletree, what was the circumference of the track described by the outer wheel ?
Ans. 62.83+ feet. 32. The ball on the top of St. Paul's Church is 6 feet in diameter. What did the gilding of it cost, at 34d. per square inch ?
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 her 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
the field ? Ans. B could reap it in 15 days, and C in 18 days. 37. A merchant tailor bought 40 yards of broadcloth, 24 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 11 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. 71,7 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.
682. TAE 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,
cm., = .3937 inch. 10 centimeters,
1 dec'imeter, dcm., 3.937 inches. 10 decimeters,
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, cm2.,.00155 sq. in. 100 sq. centimeters, 1 sq. decimeter, dcm., .1076 sq. ft. 100 sq. decimeters,
1 SQ. METER,
m., 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 11 square yards, and an are nearly 4 square rods.
687. MEASURES OF VOLUME. 1000 cu. millimeters, (mm.), 1 cu. centimeter, cmo., .061 cu. in. 1000 cu. centimeters, 1 cu. decimeter, dcm®., 61.022 cu. in. 1000 cu. decimeters, 1 cu. METER,
mo., 1.308 cu. yd.
Also, 10 dec'isteres (dcs.) =1 STERE, or cu. meter, st.,=1.308 cu. yd. 10 steres,
1 děk'astere, dks., 13.08 cu. yd.
.338 fluid oz.
1 dec'iliter, dcl., .845 liq. gill.
1.0567 liq. qt.
1 dek'aliter, dkl., 2.6417 liq. gal. 10 dekaliters, 1 hectoliter, hl., 2 bu. 3.35 pk. 10 hectoliters,
1 kiloliter, 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 11} liquid quarts, or 1 of a dry quart; 1000 liters are equal to 1 cubic meter.