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The hypotenuse and base or perpendicular being given, to find the other leg of the triangle :
From the square of the hypotenuse subtract the square of the given leg, and extract the square root of the remainder.
1. If the diagonal of a rectangle be 50 rods, and its breadth 30, what is its length? Ans. 40 rods.
2. If the foot of a ladder 75 feet in length be 60 feet from the base of a wall, and the other end rests on the top of the wall, what is the height of the wall? Ans. 45 feet.
¶ 7. EXTRACTION OF THE CUBE ROOT.
To extract the cube root is to find out a number, which being raised to the third power, (or, which being multiplied into itself, and then into that product,) will produce the given number.
To extract the cube root :
1. Divide the given number into periods of three figures each, by placing a dot over the unit figure and every third figure from the place of units; then find the greatest cube in the left-hand period, and subtract it therefrom, placing the root in the quotient; then to the remainder annex the figures of the following period for a dividend.
2. Square the quotient, and triple the square for a divisor. Find how often it is contained in the dividend, (excepting units and tens,) and put the number in the quotient.
3. Square the last figure in the quotient, and annex its square to the divisor.*
* When the quotient figure is 1, or 2, or 3, its square being less than 10, a cipher must be put in the place of tens.
4. Triple the last figure in the quotient, which product multiply by the former figure or figures in the root, and putting it under the divisor, units under tens, and tens under hundreds, add them together, and multiply the sum by the last figure in the quotient, and, placing the product under the dividend, subtract it therefrom, and to the remainder annex the figures in the period following, and proceed as before; and so continue to do till all the periods are exhausted.
Square of 876×3=2302128.
Square of 5 annexed = 230212825
2. What is the cube root of 99252.847 ?
Square of 4×3=48, the divisor. 99252.847(46.3 Ans.
Square of 46x3=6348 divisor.
Square of 3 annexed to 6348-634809*
6. A cubic block of marble contains 389017 solid feet,-I demand the superficial contents of its 6 sides.
To find two mean proportionals between two given numbers:
Divide the greater number by the less, and the cube root of the quotient multiplied by the less extreme, or smaller number, will give the less mean proportional, which being multiplied by the said cube root will give the greater.
1. What are the two mean proportionals between 6 and 162? Ans. 18 and 54.
2. What are the two mean proportionals between 4 and 108? Ans. 12 and 36.
3. There are 4 weights which will weigh any number of pounds from 1 to 40, the least being 1 pound and the greatest 27, the other two are required. Ans. 3 and 9.
To find the side of a cube that shall be equal in solidity to any given solid, as a globe, cylinder, prism, cone, &c. :— Extract the cube root of the solid contents of the given body.
1. If the solid contents of a globe be 10648, what is the side of a cube of equal solidity? Ans. 22.
2. There is a cubical vessel, whose side is 12 inches; it is required to find the side of another vessel that will contain three times as much. Ans. 17.307 inches.
8. BIQUADRATE ROOT.
The biquadrate root (or fourth root) is that number which being involved four times into itself, will produce the given number.
Extract the square root of the given number, and then extract the square root of that root.
1. What is the biquadrate root of 5719140625?
2. If the principal and interest on one dollar for four years (the interest being compounded yearly) be 1.21550625, what is the rate per cent.? Ans. 5 per cent.
Extract the square root of the cube root of the given number ; and, to find the eighth root, extract the square root of the square root of the square root of the number; to find the ninth root of any number, extract the cube root of the cube root of the number; to find the fifth root of any number—if the number be less than 100, from half the sum of the biquadrate and sixth roots, subtract of the difference of said roots; but if the number be greater than 100 and less than 500, from the said half sum subtract of said difference, but when the number is more than 500, subtract of said difference, and the remainder will be the fifth root nearly.
1. If the amount of one dollar for six years at compound interest be $1.418519112256, what is the rate per cent.?
2. What is the eighth root of 5236 ?
6. What is the ninth root of 5236 ?
per cent. Ans. 2.91659.
Ans. 3.10377. Ans. 3.8919 nearly.
7. If the amount of one dollar for nine years is $1.55132, what is the rate per cent., compound interest?
Ans. 5 per cent.
T 9. SQUARE ROOT ON THE SLIDING RULE.
To square any number by the sliding rule; or, to extract the square root of any number :—
Place 1 on the line C over 1 or 10 on the line D, and over any number found on the line D, will be found its square on C; or, under any number found on the line C, will be found the square root on D.