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EXTRACTION OF THE ROOTS OF ALL POWERS.
18. A ladder 50 feet in length being placed in the street, will reach a window 40 feet high on one side, and without moving the foot will reach a window 30 feet high on the opposite side-how wide is the street ? Ans. 70 feet.
Multiply one of the given numbers by the other, and extract the square root of the product, and the root will be the mean proportional required.
· NotE.—When the first number is as many times greater than the second, as the second is times greater than the third, the second number is called a mean proportional between the other two.
Ex. What is the mean proportional between 36 and 144? 36 x 144=5184, and 5184=72 Ans.
PROBLEM 2. To find the side of a square equal in area to any given superficies whatever.
Find the area, and the square root is the side of the square sought.
Ex. If the area of a triangle be 160, what is the side of a square equal in area thereto? ✓160=12.649+ Ans.
PROBLEM 3. If a pipe of 6 inches bore, will be 4 hours in running off a certain quantity of water, in what time will 3 pipes, each 4 inches bore, be in discharging double the quantity?
6 X 6=36 4x4=16, and 16 x 3=48 Then, as 48 : 36 :: 4h : 3h. and as lw. : 2w. :: 3h : 6h. ans.
Multiply the square root of the area by 1.12837, and the product will be the diameter. Or multiply the area by 1.2732, and take the square root of the product.
Ex. 1. Required the diameter of a circle whose area is 82 feet 84 inches.
Ans. 10 feet 3.13 inches. 2. I have driven a stake in my meadow, to which I wish to tie my horse by a rope of such a length as that he may graze exactly two acres-how long must the rope be ?
Ans. 55.5+ yards.
Multiply the square root of the area by 3.5449, and the product will be the circumference,
Ex. 1. Required the circumference of that circle whose area
USE OF THE CUBE ROOT. 2. When the area is 160 percheş, required the circumference.
Ans. 44.839 PROBLEM 6. Given the difference between two numbers, and the difference of their squares, to find the numbers.
Divide the difference between the squares by the difference of the numbers, and the quotient will be their sum ; then, to half the sum add half the difference, and the sum is the larger; and from half the sum take half the difference, and the remainder is the less.
Ex. 1. The difference between two numbers is 20, and the difference of their squares 2000—what are the numbers ?
Ans. 60 the greater, 40 the less. 2. Said Harry to Charles, My father gave me 12 apples more than he gave brother Jack, and the difference of the, squares of our separate parcels was 288—now tell me how many he gave us, and you
shall have half of mine.
SHarry's share 18
Jack's share 6 Further use of the Cube Root. I. To find two mean proportionals between two given numbers.
Divide the greater extreme by the less, and the cube root of the quotient multiplied by the less extreme, gives the less mean; multiply the said cube root by the less mean, and the product will be the greater mean proportional.
Ex. 1. What are the two mean proportionals between 7 and 189?
Ans. 21 and 63. 2. Required two mean proportionals between 4 and 256.
16 and 64. II. To find the side of a cube that shall be equal in solidity to any given solid, as globe, cylinder, prism, &c.
The cube root of the solid content of any solid body given, is the side of a cube of equal solidity.
Ex. 1. If the solid content of a globe be 10648, what is the side of a cube of equal solidity?
Ans. 22. 2. Required the side of a cubical vessel that shall contain 80 wine gallons, each 231 cubical inches ? Ans. 26.43+
III. The side of a cube being given, to find the side of a cube that shall be double, treble, &c. in quantity to the given cube.
Cube the given side, and multiply it by 2, 3, &c.; the cube root of the product is the side sought.
Ex. 1. There is a cubical vessel, whose side is 12 inches; it is required to find the side of another vessel that shall contain three times as much.
Ans. 17.306+ 2. Suppose the length of a ship's keel to be 125 feet, the breadth of the midship beam 25 feet, and depth of hold 15 feet-required the dimensions of another ship of the same form, that shall carry three times the hurthen.
Length of the keel 180.28+ft, Ans. Breadth of the beam 36.05+ft,
And depth of the hold 21.63+ft. IV. Having the dimensions and capacity of a solid, to find the dimensions of a similar solid of a different capacity.
Like solids are in triplicate proportion to their homologous sides; therefore it will be, as the cube of a dimension : is to the cube of any like dimension :: so is the given weight : to the weight required.
1. A brass bullet of 5 inches diameter weighs 20lbs.required the diameter of a like bullet that weighs 160lbs.
Ans. 10 inches. 2. If a ship of 300 tons burthen, be 75 feet long in the keel, I demand the burthen of another ship, whose keel is 100 feet long.
Ans. 711tons. APPLICATION. 3. What is the difference between half a solid foot, and la solid half foot?
Ans. 3 half feet.
EXCHANGE. 4. In a cubic foot, how many cubes of 6 inches, and how many of 4, of 3, of 2, and of 1, are contained therein ? Ans. 8 of 6in., 27 of 4in., 64 of 3in., 216 of 2in., and 1728 of lin.
5. Suppose a cubical cellar to contain 1728 solid feet: what will one of its cubic sides measure?
TABLE OF FOREIGN COINS, With their Federal value, as established by an act of Congress, for estimating
duties at the custom-houses of the United States. Sovereign, pound sterling of Great Britain,
$4.44 Livre of France,
.185 Franc of do.
.1875 Ruble of Russia,
.75 Florin or Guilder of the United Netherlands,
.40 Mark Banco of Hamburg,
.33 Real of Plate of Spain,
.10 Real of Vellon of do.
.05 Milrea of Portugal,
1.24 Tale of China,
1.48 Pagoda of India,
1.84 Rupee of Bengal,
We subjoin a few Examples for Practice. 1. Reduce 8764 livres to Federal money. Ans. $1621.34. 2. Reduce 10000 francs to dollars.
Ans. $187.5. 3. In $1000, how many francs?
Ans. 53335 4. In 300 milreas, how many
Ans. 372. 5. In 50 sovereigns, how many dollars ?
Ans. 222. 6. In $53.23, how many sovereigns?
Ans. 12. 7. In $106.56, how many tales of China ?
Ans. 72. 8. In $108, how many rubles of Russia ?
Anş. 144. 9. In $457.56, how many milreas of Portugal ? Ans. 369. 10. In $152.625, how many livres of France ? Ans. 825. 11. In 8540 reals of plate of Spain, how many dolls. ? Ans. 854.
12. In 9684 rupees of Bengal, how many dollars? Ans. 4842. NOTE.—The above examples more properly belong to the rule called Exchange.
EXCHANGE. EXCHANGE is the giving of the money, weight, or measure of one country, for the like value in money, bills, weight, or measure of another country.