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PROBLEM 3. The diameter of a globe or ball being given, to find the diameter of a globe or ball that shall be any number of times greater or less than the given one.
RULE.--Cabe the given diameter, and multiply or divide it by the given proportion as the question may require; and the cube root of the product, or quotient, will be the diameter of the globe required.
1. If the diameter of a globe be 12 inches. what will be the diam: eter of a globe one eighth part as large ?
Ans. 6 inches.
2. If the diameter of a globe be 6 inches, what will be the diameter of a globe 64 times as large ?
Ans. 24 inches.
À GENERAL RULE FOR EXTRACTING THE ROOTS
OF ALL POWERS,*
1. Prepare the given number for extraction, by pointing off from the units' place, as the required root directs.
2. Find the first figure of the root by trial, or by inspection into a table of powers, 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. 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 dividend, 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 periods in the root.
* The roots of most powers may be found by the square and cube root only; therefore when any eyen power is given, the easiest method will be (especially in a very high power,) to extract the square root of it, which reduces it to half the given power; then the square root of that power reduces it to half the same power; and so on till you come to a square or cube.
For example; suppose a twelfth power be given; extracting the square root of the twelfth power reduces it to à sixth power; and extracting the square root of the sixth power reduces it to a cube,
296. What is the general method of extracting roots of all powers ?
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 manner proceed till the whole be finished.
2. What is the biquadrate or fourth root of 19987173376 ?
ALLIGATION is the method of mixing two or more 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.
Is when the quantities and prices of several things are given, to find the
mean price of the mixture compounded of those things.
As the sum of the quantities, or the whole composition, is to the whole value ; so is any part of the composition to its mean price or value.
298. What is Alligation Medial? --299. What is
297. What is Alligation ? the rule in Alligation Medial?
1. A grocer mixed 4 cwt. of sugar at $10 a cwt. with 5 cwt. at $9.50 a cwt. and 3 cwt. at $8.75 a cut--what was the value of one cwt. of this mixture ? cwt. $ $ c.
cwt. $ C. cwt. 4 at 10 =40.00
12 : 113.75 : :1 5 9.50 =47.50
1 3 8.75 -26.25
12)113.75 cwt. 12 $113 75
2. If a bushel of Indian corn, at •75 cents a bushel, be mixed with 6 bushels of rye at •80 cents a bushel, and 15 bushels of oats at 30 cents a bushel ; what will be the value of a bushel of the mixture ?
Ans. '4421 cents.
3. A wine merchant mixes 12 gallons of wine at -75 cents a gallon, with 24 gallons at :90 cents, and 16 gallons at $1.10 ; what is a gallon of this composition worth ?
4. A goldsmith melted together 8oz. of gold of 22 carats fine, * 1 lb. Boz, of 21 carats fine, and 10oz. of 18 carats fine ; what is the fine. ness of the composition.
Ans. 2018 carats fine.
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.
* If an ounce or any other quantity of pure gold be divided into 24 equal parts, these parts are called carats, but gold is often mixed with some baser metal, which is called the alloy, and the mixture is said to be so many carats fine, according to the proportion of pure gold contained in it ; thus, if 22 carats of pure gold and 12 of alloy be mixed together, it is said to be 22 carats fine. -See page 43
300. What is Alligation Alternate ?
When the mean rate of the whole mixture, and the rates of all the
ingredients are given, and the quantity not limited.
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 reduced to the same denomination, at the left hand.
2. Connect with a continued line 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 mixture rate) and that of each of the simples, opposite to the rates with which they are connected.
4. Then, if only one difference stand against any rate, it will be the quantity belonging to that pate ; but if there be several, their sum will be the quantity.*
1. A merchant has spices, some at 1s. 6d. a lb. some at 2s. some at 4s. and some at 5s. a lb. : How much of each sort must he mix, that he may sell the mixture at 3s. 4d. a lb. ?
* By connecting the less rate to the greater, and placing the difference between them and the mean rate alternately, the quantities resulting are such, that there is precisely as much gatnod by one quantity as is lost by the other, and therefore the gain and loss upon the whole are equal, and are exactly the proposed rate.
In like manner, let the number of simples be what it may, and with how many soever each one is linked, since it is always a less with a greater than the mean price, there will be an equal balance of loss and gain between every two, and consequently an equal balance on the whole.
If any of the simples be of little or no value with respect to the rest, its rate is supposed to be nothing; as water mixed with wine, and alloy with gold or silver.
301. How do you find the quantity of each ingredient, when the mean rate of the whole mixture, and the rates of all the ingredients are given, and the quantity not limited ?
NOTE. Questions in this case admit of as many answers, as there are various ways of connecting the rates of the ingredients together.
2. A grocer would mix the following qualities of sugar; viz. at 10 cents, 13 cents, and 16 cents a pound; what quantity of cach sort must be taken to make a mixture worth 12 cents a pound?
5 lbs. at 10 cts. Aos. 2 lbs. at 13 cts.
2 lbs. at 16 cts.
3. It is required to mix ruw at 80 cents, and at 70 cents a gallon with water, that the mixture may be worth 75 cents a gallon ; what quantity of each must be taken ?
80 galls. at 80 cts. Ans. 5 galls. at 70 cts.
5 galls. of water.
4. A goldsmith would mis gold of 19 carats fine, with some of 16, 18, 23, and 24 carats fine, so that the compound may be 21 carats fine ; what quantity of each must be take?
5 oz. of 16 carats fine.
5 oz. of 18 carats fine. Ags. 5 oz. of 19 carats fine.
10 oz. of 23 carats tine. 10 oz. of 24 carats fine.