Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

(6+8)+9, and regarding (6+8) as one number, we find, according to the foregoing statement, that the cube of (6+8)+9 is equal to the cube of (6+8) plus three times the product of the square of (6-8) into 9, plus three times the product of (6+8) into the square of 9, plus the cube of 9. But the cube of 6+8, has already been shown to be equal to the cube of 6, plus three times the product of the square of 6 into 8, plus three times the product of 6 into the square of 8, plus the cube of 8. Hence the cube of 6+8+9 is equal to the cube of 6, plus three times the square of 6 into 8, plus three times 6 into the square of 8, plus the cube of 8; plus three times the square of the sum of 6 and 8 into 9, plus three times the sum of 6 and 8 into the square of 9, plus the cube of 9. And in general, we have the cube of the sum of any number of numbers equal to the cube of the first number, plus three times the square of the first number into the second, plus three times the first into the square of the second, plus the cube of the second; plus three times the square of the sum of the first two into the third, plus three times the sum of the first two into the square of the third, plus the cube of the third; plus three times the square of the sum of the first three into the fourth, plus three times the sum of the first three into the square of the fourth, plus the cube of the fourth, and so on.

Thus:

(2+3)=23+3 x 22.3+3x2.32+33.

(5+7)=53+3x5.7+3x5.72+73.

(5+6+7)=5+3x52.6+3x5.6+63

+3x(5+6).7+3x(5+6).7+73.

(365)=(300+60+5)=3003+3x3002.60

+3×300.602+603+3× (300+60)2.5

+3×(300+60).52+53.

The cubing of a number may be illustrated geometrically as follows:

Let it be required to cube 45, the number before employed. To simplify the illustration, suppose we are required to find the number of cubic inches in a cube whose side is 45 inches. Separating 45 into

40+5, we will suppose the cube, (fig. 1,) to be 40 inches on a side; then 40 x 40×40 will give the solid contents of this cube, represented by 403.

Let fig. 2 represent the cube increased by three equal slabs; then 3 (the number of slabs) times 402 (the surface of one of the slabs,) multiplied by 5, the thickness of a slab, will give the solid contents of the slabs, represented by 3×402.5.

Let fig. 3 represent the solid, (as in fig. 2,) further increased by three equal corner pieces; then 3 (the number of corner pieces) times 40 (the length of one corner piece) multiplied into 52, the surface

of an end of a corner

Fig. 1.

[graphic][merged small][merged small][merged small]
[graphic]

402-40 × 40 =1600

× by 3

4800

× by

5

24000

[graphic]

3×40=120 × by 52= 25

600

240

3000

piece, will give the solid contents of the corner pieces, represented

by 3×40.5°

[merged small][merged small][merged small][merged small][merged small][graphic]

represented by 453-403+3 x 402.5+3×40.52+53

=64000+24000+3000+125=91125.

[blocks in formation]

135. We will now endeavor to deduce a rule for the extraction of the Cube Root.

Let it be required to find the cube root of 382657176.

For the sake of simplicity, we will suppose 382657176 to denote the number of cubic feet in a geometrical cube; we are required to find the number of linear feet in a side of this cube, that is, the length of one of its sides.

We will first inquire how many figures the root will have.

The smallest number, consisting of two figures, which is 10, becomes, when cubed, 1000, having more than three figures. Again, the largest number, 99, which consists of two figures, becomes, when cubed, 970299, which consists of six figures. Hence, when a number consists of more than three figures, and not of more than six, its cube root will consist of two figures. By a similar method it may be shown, that when a number consists of more than six, and of not more than nine figures, its cube root will consist of three figures. Therefore, if we separate a number into groups of three figures each, the number of groups will denote the number of figures in the cube root of that number.

In the present example, we know that there must be three figures in the root.

We know that the side of the cube sought must exceed 700 linear feet, since the cube of 700 is 343000000, which is less than 382657176, we also know that the side of this cube must be less than 800 linear feet, since the cube of 800 is 512000000, which is greater than 382657176. Hence the first figure of our root, or the figure in the

hundred's place is 7; whose cube, 343, is the greatest cube con

tained in 382, the first, or left-hand period. If we suppose each side of the cube, represented by figure 1, to be 700 linear feet, one of the equal faces, as the upper face DEFG, will be denoted by 700 × 700 490000 square feet. The solid contents of the cube will be represented by 700 x 700=490000 × 700 =343000000 cubic feet. Subtracting 343000000 cubic feet from 382657176

Fig. 1.
F

[graphic]

E

cubic feet, we find 39657176 cubic feet for a remainder.

Hence it is necessary to increase the cube, figure 1, by 39657176 cubic feet. We have seen (ART. 134) that such increase is effected by the addition of three equal slabs, three equal corner pieces, and an additional cube; and that the contents of the three slats will make by far the largest portion of the whole increase.

[merged small][merged small][graphic]

1,) which is 700 linear feet, we add BC, which is also 700 linear feet, we shall have AB+BC equal to 1400 linear feet, which, multiplied by DB, equal to 700 linear feet, will give 980000 square feet, for the area ABDG+BCED, which, added to DEFG, which is 490000 square feet,

*It will be noticed that the peculiar steps throughout this demonstration, have reference to the mode of extracting the Cube Root which follows. The object of these processes is, to make use of what has been obtained in one stage of the work for the stage next succeeding; to obtain a new quantity by adding to one already in hand, instead of multiplying an original quantity; thereby saving much time and labor.

will give 1470000 square feet, for the area of three faces of the cube, figure 1, which is the same as the area of the three slabs. Were we to multiply 1470000 by the thickness of the slabs, we should obtain the cubic feet in these slabs. And since the contents of the slabs make nearly the whole amount added, it follows that 1470000 multiplied by the thickness of slabs, will give nearly 39657176 cubic feet. Consequently, if we divide 39657176 by 1470000, the quotient will give the approximate thickness of the slabs. Using 1470000 as a trial divisor, we find it to be contained between 20 and 30 times in 39657176; hence the second or tens' figure of the root is 2.

We have already remarked that 1470000 multiplied by 20, the thickness of the slabs, will give their solid contents. But besides the slabs there must be added three corner picces, each of which is 700 feet long, and of the same thickness as the slabs, that is, 20 feet. Since each corner piece is the same length as a side of the cube, figure 1, it follows that adding 700 to 1400 or 700+700, the sum 2100 will represent the total length of the three corner pieces. Were we to multiply 2100 by 20, we should obtain the area of the three corner pieces, which might

be added to 1470000, the area of the three slabs. But, since there is also to be added a little cube, each of whose sides is 20 linear feet, we will add 20 to 2100, and thus obtain 2120 for the total length of the three corner pieces, and of a side of the little cube. Now, multiplying 2120 by 20, we obtain 42400 square feet for the surface of the three corner pieces

Fig. 3.

[graphic]

and a face of the little cube; which, added to 147000, the number of square feet in the faces of the three slabs, will give 1512400 square feet in all the additions. If we multiply 1512400 by 20, the thickness of these additions, we shall obtain 30248000 cubic feet for all the additions, which, subtracted from 39657176, leaves 9409176 cubic feet. The cube thus completed is 720 feet on a side, and is represented by figure 4,

« ΠροηγούμενηΣυνέχεια »