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3. Required the cube root of 729001101 ?
Ans. 900,000 QUESTIONS, Showing the use of the Cube Root. 1. The statute bushel contains 2150,425 cubic or solid inches. I demand the side of a cubic box, which shall contain that quantity ?
2150,425=12,907 inch. Ans. Note.—The solid contents of similar figures are in pro. portion to each other, as the cubes of their similar sides or diameters.
2. If a bullet 3 inches diameter weigh 4 lb, what will in bullet of the same metal weigh, whose diameter is 6 inches?
3x3x3=27.6x6x6=216. As 27 : 4 lb. : : 216: 32 lb. Ans.
3. If a solid globe of silver, of 3 inches diameter, be worth 150 dollars; what is the value of another globe of sjlver, whose diameter is six inches?
3x3x3=27 6X6X6=216, As 27 : 150.:: 216 ;.. $1200. Ans.
The side of a cube heing given, to find the side of that cube which shall be double, triple, &c. in quantity to the given cube.
Rule.-Cube your given side, and niuitiply by the given propose tion between the given and required cube, and the cube root of the product will be the side sought.
4. If a cube of silver, whose side is two inches, be word 20 dollars ; I demand the side of a cube of like silver whose : value shall be 8 times as mo? 2x2x2=8, and 8x8=64764=4 inches.
=4 . Ans. 5. There is a cubical vessel, whose side is 4 feet; I demand the side of another cubical vessel, which shall contain 4 times as much ? 4x4'4=64, and 644=256 2256=6,349 + ft. Ans. 6. A cooper having a cask 40 inches long, and 32 in.
ches at the bung diameter, is ordered to make another cask of the same shape, but to hold just twice as much; what will be the bung diameter and length of the new cask ? 40 x 40 x 40 x2=128000 then 128000=50,3+ length. 32 x 32 x 32x2=65536 and 65536–40,3+ bung diam.
4 General Rule for extracting the Roots of all Powers.
1. Prepare the given number for extraction, by pointing off from the unit's place, as the required root directs.
2. Find the first figure of the root by trial, and subtract its power from the left hand period of the given number.
3. To the remainder bring down the first figure in the next period, and call it the dividend.
4. Iuvolve 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 dividend, and the quotient will be another figure of the root. 6. I 54
the whole root to the given power, and subtra 2., 'If 20 bus from as many periods of the given number as y mixed with pl figures in the root.
7. Brittaa busliuthe first figure of the next period to the semainder -w dividend, to which find a new divisor
before, iacconiglike manner 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 hy the above general method.
135796744(51,4 the root.
132651=2d subtrahend. 7803) 31457=2d dividend.
5x5x3=75 first divisor.
514*514x514=135796744 3d subtrahend 2. Required the sursolid or 5th root of 6436343.
64363 13(23 root.
Note.--The roots of most powers may be found by the square and cube roots only; therefore, when any even power is given, the easiest method will be (especially in a very high power) to extract the square root om
given prop. 1 duces it to half the given power, then th cube root of the u that power
reduces it to half the same poi till you come to a square or a cube.
For example: suppose a 12th power beriches, ithe squilie root of that reduces it to a 6th power: anda silviquare rool of a 6th power to a cube.
3. What is the biquadrate, or 4th root of 19987173376 ?
Ans. 376. 4. Extract the square, cubed, or 6th root of 1223:1590 464.
Ans. 48. 5. Extract the square, biquadrate, or Sth root of 721:39 95789338336.
ALLIGATION, IS the method of mixing several simples of different qualities, 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 : : so is any part of the composition : to its mean price.
1. A farmer mixed 15 bushels of rye, at 64 cents a bushel, 18 bushels of Indian corn, at 55 cts. a bushel, and 21 bushels of oats, at 28 cts. a bushel; I emand what a bushel of this inixture is wortlı ? bu. cts. $cts. bu. $ cts.
bu. 15 at 64=9,60 As 54 : 25,38 : :1 18 5559,90
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 Is. 6d. per lb. 12 lb. at 2s. a pound, with 12 lb. at 1s. 10d. per ib. ; what is the price of a pound of this mixture ?
Ans. ls. 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. 20. per gallon, with 24 gallons at 6s. 8d. and 20 gallons at 6s. 3d. ; what is a gallon of this composition worth?
Ins. 5s. 10d. 241 975.
6. A grocer hath several sorts of sugar, viz. one sort at 8 dols. per cwt. another sort at 9 dols. per cwt. a third sort 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 3 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 put. 8 gr. fine. 8. Suppose 5 lb. of gold of 22 carats fine, 2 lb. of 21 carats tile', 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 Medial, and may be proved by it.
CASE I. When the mean rate of the whole mixture, and the rates of all the ingredients are given, without any limited quan tity.
RULE. 1. Place the several rates, or prices of the simples, being 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 ingredient, 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 number of the less.
3. Place the difference, between the mean price (or mix ture rate) and that of each of the simples, opposite to the rates with which they are connected.