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95. Find the side of a square equal in area to a circle of 20 rods in diameter.
96. Find the diameter of a pond, that shall contain as much surface, as a pond of 6.986 miles circumference.
97. Find the length and breadth of a right-angled parallelogram, which shall be 4 times as long as it is wide, and equal in area to a circle of 43.9822971 rods circumference.
99. Find the circumference of a pond, which shall contain as much surface, as 9 ponds of of a mile diameter each.
EXTRACTION OF THE CUBE ROOT A cube is represented by a solid block-like either of those annexed—with six plane surfaces; having its length, breadth, and height all equal. Consequently, the solid contents of a cube are found by multiplying one of its sides twice into itself. For this reason, the third power of any number is called a cube.
Therefore, if we multiply the square of a number by its root, we
obtain a product, which is called a cube, or a cubic number. For instance, 4 multiplied by 4 produces 16, which is the square of 4, as shown on one of the sides of this larger block; and 16 multiplied by 4 produces 64, which is the cube of 4, as shown by the whole of the larger block.
Thus the cube of any quantity is produced by multiply ing the quantity by itself, and again multiplying the product by the original quantity. When the quantity to be
cubed is a mixed number, it may be reduced to an improper fraction, and the fraction cubed, and then reduced back to a mixed number.
As we can, in the manner explained, find the cube of a given number, so also, when a number is proposed, we may reciprocally find a number, which being cubed will produce the given number. In this case, the number sought is called, in relation to the given number, the
Therefore, the cube root of a given number is the number, whose cube is equal to the given number. For instance, the cube root of 125 is 5; the cube root of 216 is 6; the cube root of f is ž; the cube root of 33 is 13.
A cube cannot have more places of figures than triple the places of the root, and, at least, but two less than triple the places of the root. Take, for instance, a number consisting of any number of places, that shall be the greatest possible in those places, as 99, the cube of which is 970299; here the places are triple. Again, take a number, that shall be the least possible in those places, as 10, the cube of which is 1000; here the places are two less than triple.
It is manifest from what has been said, that a cubic number is a product resulting from three equal factors. For example, 3375 is a cubic number arising from 15X 15X15. To investigate the constituent parts of this cubic number, we will separate the root, from which it was produced, into two parts, and instead of 15, write 10+5, and raise it to the third power in this form.
10 + 5
10 + 5 Product of 10+5 by 5,
50 + 25 Product of 10+5 by 10,
10 + 5 Prod. of 100+100+25 by 5,
500+500 +125 Prod. of 100+100+25 by 10, 1000+1000+250
1000+1500+750+ 125 This product contains the cube of the first term, three
The third power,
times the square of the first term multiplied by the second term, three times the first term multiplied by the square of the second term, and the cube of the second term: thus, 10x10x10=1000; 10X10X3X5=150010X3X25=750; 5X5X5=125.
Now, if the cube be given, viz. 1000+-1500+750+ 125, and we are required to find its root, we readily perceive by the first term 1000, what must be the first term of the root, since the cube root of 1000 is 10; if, therefore, we subtract the cube of 10, which is 1000, from the given cube, we shall have for a remainder, 10X10X3X5=1500, 10X3X25=750, and 5X5X5 =125; and from this remainder we must obtain the second term in the root. As we already know that the second term is 5, we have only to discover how it
be derived from the above remainder. Now that remainder may be expressed by two factors; thus, (10X10X3+ 10X3X5+5X5) 5: therefore, if we divide by three times the
square of the first term of the root, plus three times the first term multiplied by the second term, plus the square of the second term, the quotient will be the second term of the root, which is 5.
But, as the second term of the root is supposed to be unknown, the divisor also is unknown; nevertheless we have the first term of the divisor, viz. three times the square of the root already found; and by means of this, we can find the next term of the root, and then complete the divisor, before we perform the division. After finding the second term of the root, it will be necessary, in order to complete the divisor, to add thrice the product of the two terms of the root, and the square of the second term, to three times the square of the first term previously found.
The preceding analysis explains the following rule for the extraction of the cube root.
RULE. First-Point off the given number into periods of three figures each, beginning at the unit's place, and pointing to the left in integers, and to the right in deci. mals; making full periods of decimals by supplying the deficiency, when any exists.
2dly--Find the root of the left hand period, place it in the quotient, and subtract its cube from the given number. The remainder is a new dividend.
3dly— Square the root already found and multiply its quare by 3, for a divisor.
4thly--Find how many times the divisor is contained in the dividend, and place the result in the quotieni.
5thly-In order to complete the divisor, multiply the root previously found, by the number last put in the root, triple the product, and add the result to the divisor; also square the number last put in the root, and add its square to the divisor.
Lastly-Multiply the divisor thus completed, by the number last put in the root, and subtract the product from the dividend. The remainder will be a new dividend
Thus proceed, till the whole root is extracted.
We will extract the cube root of 34965783, denoting each step of the operation, from first to last, by a reference to that part of the rule, under which it falls First,
34965783 2dly. Cube of 300, subtracted, - 27000000(300 New dividend,
1.7965783 3dly. 300X300X3[a divisor] 270000 4thly. Divisor in new dividend,
(20 5thly. Triple prod.of 300X20, 18000
Square of last number, 400
Divisor completed, 288400 Lastly. 283400 X 20, subt’ed,
5768000 New dividend,
2197783 3dly. 320X320X3 [a divisor] 307200 4thly. Divisor in new dividend,
17 5thly. Triple prod. of 320X7, 6720
Square of last number, 49
Divisor completed, 313969 Lastly 313969X7, subtracted,
In completing every divisor, we have three products to add together; viz. three times the square of the root already found; three times the product resulting from the multiplication of the root already found, by the number last put in the root; and the square of the last number.
If the ciphers be removed from the right hand of each of these products, the remaining figures in each succeeding product will stand one place to the right of each preceding product; therefore, the work will be considerably abridged by adopting the following
RULE. First-Point off the given number into periods of three figures each, as before directed.
2dly— Find the root of the left hand period, place it in the quotient without regard to local value, and subtract its cube from that period; and to the remainder bring down the next period for a dividend.
3dly— Square the root already found, without gard to its local value, and multiply its square by 3, for a divisor.
4thly—Find how many times the divisor is contained in the dividend, amitting the two right hand figures, and place the result in the quotient.
5thy-To complete the divisor, multiply the root pre. viously found, by the figure last placed in the quotient, without regarding local value, triple the product, and write it under the divisor, one place to the right; square the figure last put in the quotient, and write its square under the preceding product, one place to the right. Add these three together, and their sum is the divisor completed.
Lastly—Multiply the divisor thus completed, by the figure last placed in the root, and subtract the product from the dividend; and to the remainder bring down the next periud for a new dividend.
Thus proceed, till the whole root is extracted
Observe, that, when the divisor is not contained in the dividend, as sought in the fourth part of the rule, a cipher must be put in the root, and the next period brought down for a new dividend.