2 60. From ✓152399025 take (√4120900++.00060025.) 025 A. 10314.7255 61. What is the square root of 15241578750190521? A. The 9 digits. 62. Find the sum of the roots or numbers involved in all the per ect squares between 1 and 100. A. 44. 63. Find the sum of the squares whose roots are surds, between 1 and 20? A. 160. 64. Suppose that a commandant of an army has 180625 effective men, and would form them into a solid square, how many would there be in each rank and file? A. 180625=425. 65. Suppose a town proposes to levy a poll tax of $216.09 so that each man shall pay as many cents as there are men to be taxed; what is each man's tax on his head? A. $1.47. 66. Suppose there are two portions of land each in the form of a square, and that one is 30 miles square, and the other contains 30 square miles; what is the sum of the distances round both squares? A. 143 miles. 67. If the surface of the earth, which is computed to contain 196.000,000 square miles were in the form of a square, what would be the distance round it? A. 56,000 miles. 68. If a tract of land 61 miles long, and 4 miles wide, which cost $14 per acre be exchanged for the same quantity in the form of a square, and subsequently be divided into one hundred equal and square farms, 3 of which should bring at auction $113 per acre; of them $12 per acre, and the rest $10 per acre; what would be the profit in the transaction, and what the sum of the distances round all the squares? A. 200 miles; $164020 profit. PROPORTIONS INVOLVING ROOTS AND POWERS. 69. The product of the square roots of any two numbers, is equal to the square root of their product. 70. Prove 81× √225=√81 × 225. 71. To find a mean proportional between any two numbers :--Extract the square root of their product. 72. For in the proportion 2: 10 :: 10: 50; of which the 10 is a mean proportional between 2 and 50; we have on geometrical principles, 2×50=102. 73. What is the mean proportional between 3 and 12? 4 and 36? 24 and 96? 16 and 64? A. 6; 12; 48; 32. 74. What is the mean proportional between 25 and 289? 25 and 156.25? A. 85; 621. 75. What is the mean proportional between 7 and 13? 10 and 41} ? A. 3: 20.6. 3 19 Q. To what is the product of the square root of any two numbers equal? 69. How is a mean proportional between any two numbers found? 71 What is the mean proportional between 4 and 9? between 2 and 18? 76. The mean proportional between any two numbers, has the same ratio to those numbers, that the square roots of those numbers have to each other. 77. Find the mean proportional between 25 and 36, and the ratio between it and those numbers, and see if it is the same as the ratio between 25: √36. 78. * To find any two numbers from having their sum and product given :-From the square of half their sum, subtract their product: extract the square root of the remainder, and add it to half their sum, for the larger number; or subtract it therefrom for the smaller number. 79. A certain field contains an area of 30 acres 2 roods and 20 rods: required its length and breadth, the sum of these being 148 rods. A. 98rd.: 50rd. 80. A gentleman having purchased a certain quantity of flour, for $1935, found that if he added the number of dollars it cost per barrel to the number of barrels, the sum would be 224. How many barrels must he have bought? A. 215 barrels. 81. To find any two numbers from having their sum and the sum of their squares given :-Find the difference between the square of their sum, and the sum of their squares: half this difference subtract from the square of half their sum, and add the square root of the remainder to their half sum for the greater number, or subtract it therefrom for the smaller number. 82. Suppose that two square fields contain together 9A. 2R. 5rd and that the sum of either their length or breadth is 55 rods; pray what is the length of each lot? A. 25rd.: 30rd. EXTRACTION OF THE CUBE ROOT. C. 1. The CUBE of any given number is the product of that number multiplied by its square. [xcvii. 3.] 2. The CUBE Roor of any given number, is such a number as will, on being multiplied by its square, produce the given number. [xcviii. 2.] 3. A body in the form of a cube is a solid of six equal sides, each containing an exact square. [See the block accompanying this work.] 4 A CUBE then has three dimensions, viz., length, breadth, and thickness or depth; the product of which multiplied into each other is called its solid content. [VII. 60.] 5. The length, breadth, and thickness of a cube being equal, the cube of either of its sides must be equal to its solid contents; of course the cube root of its solid contents must be equal to the length of either of its sides. Q. To what is the ratio of any mean proportional equal? 76. How are any two numbers found from having their sum and product given? 78.-from hav ing their sum and the sum of their squares given? 81. C. Q. 1. What is the cube of any number? 1. What is the cube root? 2. What is a cubical body? 3. What its dimensions? 4. How are its solid conents found? 5. How either of its dimensions? 5. This and the following proportion are deduced from Algebrais prososSOS 6* The blocks which accompany this work for the purpose of illustrating the operation of the following example are eight in all, and when put together, they should form a perfect cube of 24,389 sd. feet. 7. These blocks are marked by the letters A, B, C, and D, whose proportional dimensions are supposed to be as follows: A is a cube, 20 feet long, 20 feet wide, and 20 feet thick. 8. If a cubical block which is formed by the 8 small ones above, contains 24,389 solid ft. ; what must be the length of each of its sides? In this example, we know that one side cannot be 30ft., for 303 27000 solid feet, are more than 24389, the given sum -therefore, we will take 20 for the length of one side of the cube. Then 20 x 20 × 20 -8000 solid feet, which we must, of course, deduct from 24389 leaving 16389 These 8000 solid feet the pupil will perceive, are the solid contents of the cubical block marked A. This corresponds with the operation; for we write 20 feet, the length of the cube A, at the right of 24389, in the form of a quo tient; and its square 8000, under 24389; from which subtracting 8000, leaves 16389 as before. As we have 16389 cubic feet remaining, we find the sides of the cube A are not so long as they ought to be; consequently we must enlarge A; but in doing this we must enlarge three sides of A, in order that we may preserve the cubical form of the block. We will now place the three blocks each of which is marked B, on these three sides of A. Each of these blocks, in order to fit, must be as long and as wide as A; and, by examining them, you will see that this is the case; that is, they are 20 feet long and 20 feet wide; then 20×20=400, the square contents in one B; and 3 × 400=1200, square contents in threc Bs; then it is plain, that 16389 solid contents, divided by 1200, the sq. contents will give the thickness of each block. But an easier method is to square the 2, (tens,) in the root 20, making 4, and multiply the product 4, by 300, making 1200, a divisor, the same as before. We do the same in the operation (which see); we multiply the square of the *This rule is best illustrated by means of blocks which may be supposed to contain a certain proportional number of feet, inches, &c., corresponding with the operation of the rule. They may be made in a few minutes, from a small strip of pine board, with a common penknife, at the longest, in less time than the teacher can make the pupil comprehend the reason, from merely seeing the picture on paper. This method of demonstrating the rule will be an amusing and instructive exercise, both to teacher and pupil, and may be comprehended by any pupil, however young, who is so fortunate as to have progressed as far as this rule. It will give him distinct ideas respecting the different dimensions of square and cubic measures, and indelibly fix on his mind the reason of the rule, and consequently the rule itself. But, for the convenience of teach gs, Hondasdastrative of the oneration. the orgong example, accompany this work quotient figure, 2, by 300, thus 2 × 2-4 × 300-1200; then the divisor, 1200 (the square contents) is contained in 16389 (solid contents) 9 times, that is, 9ft. is the thickness of each block marked B. This quotient figure, 9, we place at the right of 16389, and then 1200 square feet × 9 feet, the thickness, 10800 s. ft. If we now examine the block, thus increased by the addition of the 3 Bs, we shall see that there are yet three corners not filled up; these are represented by the three blocks, each marked C, and each of which, you will perceive, is as long as either of the Bs, that is, 20 ft., being the length of A, which is 20 in the quotient. Their thickness and breadth are the same as the thickness of the Bs, which we found by dividing, to be 9 feet, the last quotient figure. Now, to get the solid contents of each of these Cs, we multiply their thickness (9 feet) by their breadth, (9 feet,)=81 square feet; that is, the square of the last quotient figure, 9-81; these square contents must be multiplied by the length of each, (20 feet,) or, as there are 3, by 3×20=60; or, which is easier in practice, we may multiply the 2, (tens) in the root, 20, by 30, making 60, and this product by 92-81, the square contents = 4860 solid feet. We do the same in the operation, by multiplying the 2 in 20 by 30—60 × 9 × 9=4860 solid feet, as before; this 4860 we write under the 10800, for we must add the several products together by and by, to know if our cube will contain all the required feet. By turning over the block with all the additions of the blocks marked B and C, which are now made to A, we shall spy a little square space, which prevents the figure from becoming a complete cube. The little block for this corner is marked D, which the pupil will find, by fitting it in, to exactly fill up this space. This block D, is exactly square, and its length, breadth and thickness are alike, and, of course, equal to the thickness and width of the Cs, that is, 9 feet, the last quotient figure; hence 9ft. x 9ft. × 9ft.=729 solid feet in the block D; or, in other words, the cube of 9, (the quotient figure,) which is the same as 93-729, as in the operation. We now write the 729 under the 4860, that this may be reckoned in with the other additions. We next proceed to add the solid contents of the Bs, Cs, and D, together, thus, 10800 x 4860 × 729-16389, precisely the number of solid feet which we had remaining after we deducted 8000 feet, the solid contents of the cube A. If, in the operation, we subtract the amount, 16389, from the remainder, or dividend, 16389, we shall see that our additions have taken all that remained, after the first cube was deducted, there being no remainder. The last little block, when fitted in, as you saw, rendered the cube complete, each side of which we have now found to be 20+9=29 feet long, which is the cube root of 24389 (solid feet); but let us see if our cube contains the required number of solid feet. 9. Proof.-8000 s. ft. in A+ 10800 s. ft. in 3 Bs+4860 s. ft. in 3 Cs × 729 s. ft. in D=24389 s. ft. in the given sum which because they are equal to 293 form a perfect cube, then, 29 is the length of the required side; therefore,— 10. If by Involution the cube of the root found from the operation be equal to the given sum, the operation is correctly performed. 11. By reasoning similar to that employed in xCIX. 9, it may be shown that the product of any three numbers into each other never has more figures than all its factors, nor fewer than that same number less two. 12. We infer also from the same reasoning, that if we point off any sum into periods of three figures each, the number of periods will equal the number of figures in its root. Hence the direction in the rule RULE. 13. Divide the given number into periods of three figures each, by placing a point over the unit figure, and over every third one from the place of units to the left in whole numbers, and to the right in decimals. Q. Of how many figures will every root consist? 12. What is the reason for it' 11. What is the rule for pointing off the given number? 13. 14. Find the greatest cube in the first left hand period, and piace its root in the quotient. Subtract the cube thus found from this period, and to the remainder bring down the next period, and the result will be the dividend. 15. Multiply the square of the root or quotient by 300 for a divisor. Divide the dividend by the divisor for the next figure in the root. 16. Multiply the divisor by the quotient figure; multiply the former quotient figure or figures by 30 times the square of the last quotient figure; finally, cube the last quotient figure; then add these three results together for a subtrahend. 17. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before, and so on till all the periods are brought down.* 18. NOTE. When the subtrahend happens to be larger than the dividend, the quotient figure must be made one less, and we must find a new subtrahend. The reason why the quotient figure will be sometimes too large, is, because this quotient figure merely shows the width of the three first additions to the original cube; consequently, when the subsequent additions are made, the width (quotient figure) may make the solid contents of all the additions more than the cubic feet in the dividend, which remain after the solid contents of the original cube are deducted. 19. When we have a remainder after all the periods are brought down, we may continue the operation by annexing periods of ciphers, as in the square root. When it happens that the divisor is not contained in the dividend, a ci pher must be written in the quotient (root,) and a new dividend formed by bringing down the next period in the given sum. |