Fig. 7.

74088

MODEL OPERATION.

74,088(40+2=42, root.

64000 (a.) 40X40X40=64000

10088 40X40X3=4800)10088

9600 (b.) 4800X2 9600

488

480 (c.) 40X2X2X8

=480

8

nof 8

(d.) 2X2X2= 8

0

ANALYSIS. 1. Since the product of the length, breadth, and thickness gives the solid contents of a cube, therefore the cube root of 74088 cu. in. will give the length of each side of the cubical block.

2. Separate the number into periods of three places, each counting from the right (329., a.); thus, 74'088, which shows that the root will contain two places.

3. By trial the length of the side of the largest cube is 40 inches, and the solid contents of this cube are 64000 cu. in., (40x40x40=64000; Fig. 1. a.) which subtracted from 74088 cu. in. leaves a remainder of 10088 cu. in. (Fig. 7. a.)

In order to preserve the form of a cube, the remainder must be added to three of the sides of the cube. (Fig. 3.)

4. Since the addition to each side of the cube must be 40 in. long and 40 in. wide, the addition to each side at 1 in. thick will contain 40 times 40 cu. in., which are 1600 cu. in., and the 3 sides will contain 3 times 1600 cu. in., which are 4800 cu. in. (Fig. 2., b.)

5. If it require 4800 cu. in. to make an addition 1 in. thick to the three sides of the cube, 1008 cu. in. will make the addition as many inches thick as 4800 cu. in. are contained times in 1008 cu. in., which are 2. If the addition at 1 in. thick contain 4800 cu. in., at 2 in. thick it will contain 2 times 4800 cu. in., which are 9600 cu. in. (Fig. 2., b.)

6. More nearly to complete the cube, 3 blocks are required,

each 40 in. long, 2 in. wide, and 2 in. thick (Fig. 4), which contain 480 cu. in. (40 × 2 × 2 ×3=480) (Fig. 4, c), which subtracted from 488 cu. in. leave 8 cu. in.

7. To complete the cube a block is required (Fig. 6) 2 in. long, 2 in. wide, and 2 in. thick, which contains 8 cu. in. (2×2 ×2=8) (Fig. 6., d.) which subtracted from 8 cu. in. leaves no remainder. (Fig. 7., a.)

Therefore, the block is 42 in. long, 42 in. wide, and 42 in. thick.

QUESTIONS.-What is revenue? (228.) When are duties said to be ad valorem? (229.) When are duties said to be specific? (230.) What is an invoice? (231.) What is tare? (232.) What is leakage? (233.) What is breakage? (234.) What is gross weight? (235.) What is net weight? (236.) What is interest? (238.) What is the principal? (239.) What is rate per cent.? (240.)

Complete divisor, 475 2375, product of the complete div., by 5 Trial divisor, (15)2×300=67500. 499.230, new dividend.

15X7X30= 3150.

7x7= 49.

[divisor by 7.

Complete divisor, 70699 494.893, product of the complete Trial divisor, (157)2×300=7394700. 4.337000000, new dividend.

Trial div. (1570)2×300=739470000.

1570X5X30=235500.

5X5= 25.

[plete div. by 5.

Complete divisor, 739705525 3.698527625, prod't of the com

.638472375, remainder.

For convenience in practice, the following rule is adapted to the above contracted operation:

RULE.-I. Separate the numbers into periods of three places each, counting toward the left from units' place.

II. Find the root of the left hand period, subtract its cube from the period, and to the remainder annex the next period for a dividend.

III. Multiply the square of the root found by 300 for a trial divisor, and write the quotient by the divisor as the second figure in the root.

IV. Multiply the product of the root found, and the last figure by 30, and the result, plus the square of the last figure, added to the trial divisor, will form the complete divisor.

V. Multiply and subtract, as in simple division, and, to the remainder, annex the next period for a new dividend.

VI. Proceed in the same manner as before, until all the periods are brought down..

NOTES.-1. If the dividend does not contain the complete divisor, write a cipher in the root, and annex the next period for a new dividend, with which proceed as before.

2. If the given number contains a decimal whose number of decimal places is not divisible by three, make it divisible by annexing ciphers, before separating into periods; thus, 4,473.2213=4,478.221,300.

QUESTIONS.-What is the amount? (241.) What is simple interest? (242.) What is compound interest? (243.) What is legal interest? (244.) What are partial payments? (248.) What is an indorsement? (249.) What is the rule of the Supreme Court of the United States? (250.)

LESSON VI.

EXAMPLES FOR PRACTICE.

Find the cube root of each of the following numbers;

QUESTIONS.-Write a rule for problem first in interest. (252.) For problems second, third, fourth, and fifth. (253.) (254.) (255.) (256.) What is discount? (259.) What is present worth? (260.) What is a bank? (261.) What is bank discount? (262.)

LESSON VII.

3

8.144865728=what? | 43. 81, what?

3

4176.3841 what? 44. √72=what?

Fig. 3. A

point of meeting is called the vertex of the angle; and when the angle is named the letter at the vertex is placed second; as, A B C.

B

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