More Exercises for the Slate 8. What is the cube root of 9663597? 20. What is the cube root of 491? A. 17. $261 If the root be a surd, reduce it to a decimal before its root is extracted, as in the square root, 23. What is the length of one side of a cubical block, which contains 1728 solid or cubic inches? A. 12. 24. What will be the length of one side of a cubical block, whose contents shall be equal to another block 32 feet long, 16 feet wide, and 8 feet thick ? 16 feet, Ans. 32X16X8: 25. There is a cellar dug, which is 16 feet long, 12 feet wide, and 12 feet deep; and another, 63 feet long, 8 feet wide, and 7 feet deep; how many solid or cus bic feet of earth were thrown out, and what will be the length of one side of cubical mound, which may be formed from said earth? A. 5832; 18. t 26. How many solid inches in a cubical block, which measures 1 inch on each side? How any in one measuring 2 inches on each side? 3 inches on each side? 4 inches on each side? 6 inches on each side? 10 inches on each side? 20 inches on each side? A. 1, 8, 27, 64, 216, 1000, 8000. 27. What is the length of one side of a cubical block, which contains 1 solid or cubic inch? 8 solid inches? 27 solid inches? 64 solid inches? 125 solid inches? 216 solid inches? 1000 solid inches? 8000 solid inches? A. 1, 2, 3, 4, 5, 6, 10, 20. By the two preceding examples we see that the sides of the cube are as the cube roots of their solid contents, and their solid contents as the cubes of their sides. It is likewise true, that the solid contents of all similar figures are in proportion to each other as the cubes of their several sides or diameters. Note. The relative length of the sides of cubes, when compared with their solid contents, will be best illustrated by reference to the cubical blocks, accompanying this work. 28. If a ball, 3 inches in diameter, weigh 4 pounds, what will a ball of the same metal weigh, whose diameter is 6 inches? 33634: 32: Ratio, 23 X4 32 lbs., Ans. 29. If a globe of silver, 3 inches in diameter, be worth $160, what is the value of one 6 inches in diameter ? 3363 $160: $1280, Ans. 30. There are two little globes; one of them is 1 inch in diameter, and the cther 2 inches; how many of the smaller globes will make one of the larger? A. 8 31. If the diameter of the planet Jupiter is 12 times as much as the diameter of the earth, how many globes of the earth would it take to make one as large as Jupiter? A. 1728. 32. If the sun is 1000000 times as large as the earth, and the earth is 8000 miles in diameter, what is the diameter of the sun? A. 800000 miles. Note. The roots of most powers may be found by the square and cube roots only; thus the square root of the square root is the biquadrate, or 4th root, and the sixth root is the cube of this square root ARITHMETICAL PROGRESSION. ↑ LXXXVIII. Any rank or series of numbers more than 2, increasing by a constant addition, or decreasing by a constant subtraction of somo given number, is called an Arithmetical Series, or Progression. The number which is added or subtracted continually is called the common difference. "When the series is formed by a continual addition of the common difference, it is called an ascending series; thus, 2, 4, 6, 8, 10, &c., is an ascending arithmetical series; but 10, 8, 6, 4, 2, &c., is called a descending arithmetical series, because it is formed by a continual subtraction of the common difference, 2. The numbers which form the series are called the terms of the series or progression. The first and last terms are cailed the extremes, and the other terms the means. In Arithmetical Progression there are reckoned 5 terms, any three of which being given, the remaining two may be found, viz. 1. The first term. 2. The last term. 3. The number of terms. The first term, the last term, and the number of terms, being given, to find the common difference;--- 1. A man had 6 sons, whose several ages differed alike; the youngest was 3 years old, and the oldest 28; what was the common difference of their ages? The difference between the youngest son and the eldest evidently shows the increase of the 3 years by all the subsequent additions, till we come to 28 years; and, as the number of these additions are, of course, 1 less than the number of sons, (5), it follows, that, if we divide the whole difference (28-3), 25, by the number of additions, (5), we shall have the difference between each one separately, that is, the common difference. Thus, 28-325; then, 25 ÷ 55 years, the common difference. A.5yrs. Hence, to find the common difference ; Divide the difference of the extremes by the number of terms, less 1, and the quotient will be the common difference. 2. If the extremes be 3 and 23, and the number of terms 11, what is the common difference? A. 2. 3. A man is to travel from Boston to a certain place in 6 days, and to go only 5 miles the first day, increasing the distance travelled each day by an equal excess, so that the last day's journey may be 45 miles; what is the daily increase, that is, the common difference? A. 8 miles. 4. If the amount of $1 for 20 years, at simple interest, be $2,20, what is the rate per cent. ? In this example we see the amount of the first year is $1,06, and the last year $2,20; consequently, the extremes are 106 and 220, and the number of terms 20. A. $,06 = =6 per cent. 5. A man bought 60 yards of cloth, giving 5 cents for the first yard, 7 for the second, for the third, and so on to the last; what did the last cost? Since, in this example, we have the common difference given, it will be easy to find the price of the last yard; for, as there are as many additions as there are yards, less 1, that is, 59 additions of 2 cents, to be made to the first yard, it follows, that the last yard will cost 2 X 59118 cents more than the first, and the whole cost of the last, reckoning the cost of the first yard, will be 1185 $1,23. A. $1,23. Hence, when the common difference, the first term, and the number of terms, are given, to find the last term ; Multiply the common difference by the number of terms, less 1, and add the first term to the product. 6. If the first term be 3, the common difference 2, and the number of terms 11, what is the last term? A. 23. is 7. A man, in travelling from Boston to a certain place in 6 days, travelled the first day 5 miles, the second 8 miles, travelling each successive day 3 miles farther than the former; what was the distance travelled the last day? A. 20. 8. What will $1, at 6 per cent., amount to, in 20 years, at simple interest? The common difference is the 6 per cent.; for the amount of $1, for 1 year, $1,06, and $1,06 + $,06 $1,12 the second year, and so on. A. $2,20. 9. A man bought 10 yards of cloth, in arithmetical progression; for the 1st yard he gave 6 cents, and for the last yard he gave 24 cents; what was the amount of the whole ? In this example it is plain that the cost of the first and last yar la will be the average price of the whole number of yards; thus, Gets. - 24 els. - 06 2. 15 cts., average piice; then, 10 yds. X 15 -- 159 cts, un21,50, synolo cost. S. $1,50 Hence, when the extremes, and the number of terms are given, to find the sum of all the terms;— Multiply the sum of the extremes by the number of terms, and the product will be the answer. 10. If the extremes be 3 and 273, and the number of terms 40, what is the sum of all the terms? A. 5520. 11. How many times does a regular clock strike in 12 hours? A. 78. 12. A butcher bought 100 oxen, and gave for the first ox $1, for the second $2, for the third $3, and so on to the last; how much did they come to at that rate? A. $5050. 13. What is the sum of the first 1000 numbers, beginning with their nature order, 1, 2, 3, &c.?, A. 500500. 14. If a board, 18 feet long, be 2 feet wide at one end, and come to a point at the other, what are the square contents of the board? A. 18 feet. 15. If a piece of land, 60 rods in length, be 20 rods wide at one end, and at the other terminate in an angle or point, what number of square rods does it contain? A. 600. 16. A person, travelling into the country, went 3 miles the first day, and increased every day's travel 5 miles, till at last he went 58 miles, in one day; how many days did he travel? We found, in example 1, the difference of the extremes, divided by the number of terms, less 1, gave the common difference; consequently, if, in this example, we divide (583) 55, the difference of the extremes, by the common difference,5, the quotient, 11, will be the number, of terms, less 1; then, 1+11 = 12, the number of terms. A. 12. Hence, when the extremes and common difference are given, to find the number of terms;— Divide the difference of the extremes by the common difference, and the quotient, increased by 1, will be the answer. 17. If the extremes be 3 and 45, and the common difference 6, what is the number of terms? A. 8. 18. A man, being asked how many children he had, replied, that the youngest was 4 years old, and the eldest 32, the increase of the family having boen I in overy 4 years; how many had he?' A. 8. GEOMETRICAL PROGRESSION. ¶LXXXIX. Any rank or series of numbers, increasing by a constant multiplier, or decreasing by a constant divisor, is called Geometrical Progres sion. Thus, 3, 9, 27, 81, &c., is an increasing geometrical series; And 31, 27, 9, 3, &c., is a decreasing geometrical series. There are fiye terms in Geometrical Progression; and, like Arithmetical Progression, any three of them being given, the other two may be found, viz. 1. The first term. 2. The last term. 3. The number of terms. 4. The sum of all the terms. 5. The ratio; that is, the multiplier or divisor, by which we form the series. 1. A man purchased a flock of sheep, consisting of 9; and, by agreement, was to pay what the last sheep came to, at the rate of $4 for the first sheep, $12 for the second, $36 for the third, and so on, trebling the price to the last; what did the flock cost him? = We may perform this example by multiplication; thus, 4X3X 3X 3X3 X3X3X3X3 $26244, Ans. But this process, you must be sensible, would be, in many cases, a very tedious one; let us see if we cannot abridge it, thereby making it easier. In the above process we discover that 4 is multiplied by 3 eight times, one time less than the number of terms; consequently, the 8th power of the ratio 3, expressed thus, 38, multiplied by the first term, 4, will produce the last term. But, instead of raising 3 to the 8th power in this manner, we need only raise it to the 4th power, then multiply this 4th power into itself; for, in this way, we do, in fact, use the 3 eight times, raising the 3 to the same power as before; thus, 381; then, 81 X 81 6561; this, multiplied by 4, the first term, gives $90014, the same result as before. A. $26244. Hence, when the first term, ratio, and number of terms, are given, to find the last term; I. Write down some of the leading powers of the ratio, with the numbers 1, 2, 3, &c. over them, being their several indices. II. Add together the most convenient indices to make an index less by 1 than the number of terms sought. III. Multiply together the powers, or numbers standing under those indices; and their product, multiplied by the first term, will be the term sought. 2. If the first term of a geometrical series ba 4, and the ratio 3, what is the 11th term? Note. The pupil will notice that the series 3, 9, 27, 81, 243, powers.) does not commence with the first term, but with the ratio. 1, 2, 3, 4, 5, indices.) The indices 5+3+2= 10, and the powers under each, 243 X 27 X 9 59049; which, multiplied by the first term, 4, makos 236196, the 11th term required. A. 236196. 3. The first term of a series, having 10 terms, is 4, and the ratio 3; what is the last term? A. 78732. 4. A sum of money is to be divided among 10 persons; the first to have $10, the second $30, and so on, in threefold proportion; what will the last have? A. $196830. 5. A boy purchased 18 oranges, on condition that he should pay only the price of the last, reckoning 1 cent for the first, 4 cents for the second, 16 cents for the third, and in that proportion for the whole; how much did he pay for them? A. $171798691,84. 6. What is the last term of a series having 18 terms, the first of which is 3, and the ratio 3? A. 387420489. 7. A butcher meets a drover, who has 24 oxen. The butcher inquires the price of them, and is answered, $60 per head; he immediately offers the drover $50 per head, and would take all. The drover says he will not take that; but, if he will give him what the last ox would come to, at 2 cents for the first, 4 cents for the second, and so on, doubling the price to the last, ho might have the whole. What will the oxen amount to at that rate? A. $167772,16. 8. A man was to travel to a certain place in 4 days, and to travel at whatever rate he pleased; the first day he went 2 miles, the second 6 miles, and so on to the last, in a threefold ratio; how far did he travel the last day, and how far in all? |