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ches at the bung diameter, is ordered to make another cask of the same shape, but to hold just twice as much; wha will be the bung diameter and length of the new cask ? 10 X 40 X 40x2=128000 then ✓ 128000=50,3+-length. 32 x 32 x 32 x2=65536 and 765536–40,3+ bung diam.
A General Rule for extracting the Roots of all Powers.
RULE. 1. Prepare the given number for extraction, by pointing våt from the unit's place, as the required root directs.
2. Find the first figure of the root by trial, and subtract Vis power from the left hand period of the given number.
3. To the remainder bring down the first figure in the #ext period, and call it the dividend.
4. Involve the root to the next inferior power to that which is given, and multiply it by the number denoting the given power, for a divisor. .
5. Find how many times the divisor may be had in the olividend, and the quotient will be another figure of the root.
6. Involve the whole root to the given power, and subtract it (always) from as many periods of the given number as you have found figures in the root.
7. Bring down the first figure of the next period to the remainder for a new dividend, to which find a new divisor as before, and in like marner proceed till the whole be finished.
Note.- When the number to be subtracted is greater than those periods from which it is to be taken, the last cuotient figure must be taken less, &c.
1. Required the cube root of 135796,744 by the abowe general method.
Note.—T'he roots of most powers may be found by the square and cube roots only; therefore, when any ever power is given, the easiest method will be (especially in e very high power) to extract the square root of it, which re duces it to half the given power, then the square rooi o that power reduces it to half the same power; and soul till you come to a square or a cube.
For example: suppose a 12th power be given; the square root of that reduces it to a 6th power: and the square rool of a 6th power to a cube.
EXAMPLES. 3. What is the biquadrate, or 4th root of 199871733761
Ans. 376. 4. Extract the square, cubed, or 6tb root of 12230590 464.
Ans. 48. 5. Extract the square quadrate, or Sth root of 72138 95789338336.
ALLIGATION, IS the method of mixing several simples of different quaijties, so that the composition may be of a mean or middle quality : It consists of two kinds, viz. Alligation Medial, and Alligation Alternate.
ALLIGATION MEDIAL, Is when the quantities and prices of several things are given, to find the mean price of the mixture composed of those materials.
RULE. As the whole composition : is to the whole value : : 80 Is any part of the composition : to its mean price.
EXAMPLES. 1. A farmer mixed 15 bushels of rye, at 64 cents a bushe, 18 bushels of ludian corn, at 55 cts. a bushel, and 21 bushels of oats, at 28 cts. a bushel; I demand what a bushel of this mixture is worth?
bu. cts. $cts. bu. $ cts. bu.
54)25,38(,47 Ans. 54 25,38 2. If 20 bushels of wheat at 1 dol. 35 cts. per bushel be mixed with 10 bushels of rye at 90 cents per bushel, what will a bushel of this mixture be worth ?
Ans. $1,20 cts. . 3. A tobacconist mixed 36 lb. of tobacco, at 1s. 6d. per lb. 12 lb. at 2s. a pound, with 12 lb. at Is. 10d. per Ib. ; what is the price of a pound of this mixture?
Ans. Is. 8d. 4. A grocer mixed 2 C. of sugar at 56s. per C. and i C. at 43s. per C. and 2 C. at 50s. per C. together; I demand the price of 3 cwt. of this mixture ? Ans. £7 13s.
5. A wine merchant mixes 15 gallons of wine at 4s. Bd. per gallon, with 24 gallons at 6s. 8d. and 20 gallons u Gs. 3d i what is a galon of this composition worth?
Ins. 5s. Ind. 24% grs.
6. A grocer hath several sorts of sugar, viz. one soft al 8 dols. per cwt. another sort at 9 dols. per cwt. a third sold at 10 dols. per cwt. and a fourth sort at 12 dols. per cwt and he would mix an equal quantity of each together; I. demand the price of 34 cwt. of this mixture ?
Ans. $34 12 cts. 5 m. 7. A goldsmith melted together 5 lb. of silver bullion, of 8 oz. fine, 10 lb. of 7 oz. fine, and 15 lb. of 6 oz. fine; pray what is the quality or fineness of this composition?
Ans. 6 oz. 13 pwt. 8 gr. fine. 8. Suppose 5 lb. of gold of 22 carats fine, 2 lb. of 21 carats fine, and 1 lb. of alloy be melted together; what is the quality or fineness of this mass?
Ans. 19 carats fine.
ALLIGATION ALTERNATE, IS the method of finding what quantity of each of the ingredients whose rates are given, will compose a mixture of a given rate; so that it is the reverse of Alligation Me. dial, and may be proved by it.
CASE I. When the mean rate of the whole mixture, and the ratos of all the ingredients are given, without any limited quar tity.
RULE. 1. Place the several rates, or prices of the simples, be. ing reduced to one denomination, in a column under each other, and the mean price in the like name, at the left hand
2. Connect, or link the price of each simple or ingredi. ent, which is less than that of the mean rate, with one or any number of those, which are greater than the mean rate, and each greater rate, or price, with one, or any num. ber of the less.
3. Place the difference, between the mean price (or mir. ture rate) and that of each of the simples, opposite to the rates with which they are connected.
4. Then, if only one difference stands against any rate, It will be the quantity belonging to that rate, but if there be several, their sum will be the quantity.
EXAMPLES. 1. A merchant has spices, some at 9d. per lb. some at Is. come at 2s. and some at 2s. 6d. per lb. how much of each sort must he mix, that he may sell the mixture at ls. 8d. per pound? d. 16. d.
d. 16. 9 , 10 at 9. 2. J 12) 4 12 ( Gives the d.) 12 ) 1 20 24 18 24 | Answer; or 20 24 ) 1 | 30 11 30
| 30— 8 2. A grocer would mix the following qualities of sugar; riz. at 10 cents, 13 cents, and 16 cents per lb. ; what quanity of each sort must be taken to make a mixture worth 2 cents per pound? ains. 5 lb. at 10 cts. 2 lb. at 13 cts, and 2 lb. at 16 cts. per lb.
3. A grocer has two sorts of tea, viz. at 9s. and at 158. cr lb. how must he mix them so as to afford the composiis for 12s. per Ib. ?
Ans. He must mir an equal quantity of each sort. 4. A goldsmith woull mix gold of 17 carats fine, with come of 19, 21, and 24 carats fine, so that the compound nay be 22 carats fine; whát quantity of each must he take? *. Ans. 2 of each of the first three sorts, and 9 of the last.
5. It is required to mix several sorts of rum, viz. at 5s. Ys. and 9s. per gallon, with water at 0 per gallon, togeTher, so that the mixture may be worth 6s. per gallon; how Touch of each sort must the mixture consist of ? Ans. I gal. of rum at 5s., I do. at 7s., 6 do, at 9s. and 3 gals. water. Or, 3 gals. rum at 5s., 6 do. at 7s., I do. at 9s, and
Igal, water. 6. A grocer hath several sorts of sugar, viz. one sort at 12 ets. per lb. another at 11 cts. a third at 9 cts..and a fourth at 8 cts. per lb.; I demand how much of each sort he must mix together, that the whole quantity may be afforded at 10 cents per pound ?