26. What is the square root of 70? 27. What is the square root of 90? 28. What is the square root of 60? 29. What is the square root of 200? 30. There is a field in the form of a square, containing 1 acre; how many rods does it measure on a side? 31. There is a right angled triangle; its hypotenuse measuring 60 rods. What is the sum of the squares of the two legs? (See Sec. XVII. Part I.) 32. There is a right angled triangle; the squares of its legs added together are 81 rods; what is the length of the hypotenuse? 33. There is a right angled triangle; its legs measure, one 25, the other 30 rods; how long is the hypotenuse? 34. Two men start from the same place; one travels 8 miles east; the other, 15 miles north; how far are they then apart? 35. A ladder 40 feet long stands against a house, the foot resting on the ground, on a level with the foundation of the house, and 20 feet distant from it; how far up will it reach? 36. The floor of a room measures 16 feet in length, and 14 feet in width; how long a line will reach diagonally from corner to corner? 37. The two parts of a carpenter's square, one 12, the other 24 inches long, may be regarded as the legs of a right angled triangle; how long would be the hypotenuse connecting their extremities? 38. There is a room 16 feet long, 14 feet wide, and 10 feet high; how long must a straight line be, reaching from a corner of the room, at the bottom, to the diagonal corner, at the top? 39. There is a room, the length, breadth, and height of which are each 10 feet; how far is it from a corner of the room at the bottom, to the diagonal corner at the top? 40. There is a room, the length, breadth, and height of which are equal; the distance from a corner, at the bottom, to the diagonal corner at the top, is 18 feet; what is the size of the room? 41. I have a cubic block, measuring 4 inches each way; how far apart are its diagonal corners ? 42. How large a cube can be cut from a sphere which is 1 foot in diameter ? SECTION XXVIII. EXTRACTION OF THE CUBE ROOT. [See Section XIX. Part I.] We will first consider those numbers the cube root of which is expressed by a single figure. Every exact cube of not more than three figures, will have for its root some number less than 10, and, consequently, it will be expressed by a single figure. This root can be found by successive trials. Examples. 1. What is the cube root of 125? 2. What is the cube root of 216? 3. What is the cube root of 512 ? 4. What is the cube root of 729 ? We will next take perfect cubes, the root of which consists of two figures. Operation. Rule. Place a period over the unit figure, and another over that of thousands. Find the greatest cube in the first period, whose root is expressible in tens. Set down this root as a quotient in division; find the cube of the root, and subtract it from the first period, and bring down the second period as a dividend. At the left hand of this set down three times the square of the root, and under this three times the root; add these together, for a trial divisor. Find, by trial, what the next figure of the root will be, and set it under the first part already found. Multiply, by this figure, three times the square of the first part of the root, setting the product under the dividend. Multiply, by the square of this figure, three times the first part of the root, setting the product underneath the other; under these set the cube of the root figure last found. Add these three numbers together, and subtract their sum from the dividend. If the work be correct, there will be no remainder. Add together the two parts of the root for the answer. 6. What is the cube root of 2744? 7. What is the cube root of 3575? 8. What is the cube root of 4913 ? 9. What is the cube root of 9261? 10. What is the cube root of 13824? 11. What is the cube root of 46656? We will next consider the case where there are more than two figures in the root. The number of figures in the root can always be determined by the number of periods placed over the sum, beginning with units, and placing a period over every third place. If there are more than three periods in the cube, regard, first, only the two left hand periods, obtaining the first and second figures of the root, just as if they constituted the whole root. Then, after bringing down the figures of another period, add the two parts of the root, and consider their sum as the first part of the root, and proceed to find the next part. To indicate this, you must annex a cipher to the figures of the root already found. 12. What is the cube root of 1953125? 13. What is the cube root of 2406104? 14. What is the cube root of 3796416? If there are decimals in the given sum, point off both ways from the units' place, adding ciphers, if necessary, to the decimal, in order to make the period complete. 15. What is the cube root of 15.625 ? 16. What is the cube root of 35.937 ? If the number given is not a perfect cube, add periods of ciphers, and carry out the root in decimals as far as may be desired. 17. What is the cube root of 10? 18. What is the cube root of 20? 19. What is the cube root of 50? 20. What is the cube root of 100? 21. A bushel, even measure, contains 2152 solid inches; what would be the inside measure of a cubic box containing 12 bushels? 22. A gallon, wine measure, contains 231 cubic inches; what must be the inside measure of a cubic cistern containing 10 barrels ? 23. What would be the measure of a cubic pile of wood, containing one cord? SECTION XXXIX. PROPORTION. [See Section XX. Part I.] Several changes that may be made in the terms of a Proportion, are exhibited in page 105. In continuing the subject, we will first state some further changes that may be made in the terms without destroying the proportion. 1. Multiply all the terms by the same number. 2. Divide all the terms by the same number. 3. Add the terms of the first ratio for the first antecedent, and the terms of the second ratio for the second antecedent. 4. Add the terms of the first ratio for the first consequent, and the terms of the second ratio for the second consequent. 5. Instead of the sum of the terms in the third case above, take the difference of the terms. 6. Instead of the sum of the terms in the fourth case above, take the difference of the terms. 7. Raise each term to the same power, as second or third power. 8. Extract of each term the same root. The result, after each of these operations, will still be a proportion, and may be proved to be so, by multiplying the extremes together, and finding the product, equal to that of the means. Take the proportion, 4:16:: 9:36, and perform on it the first change, using any number you please for a multiplier, and then prove the proportion. Perform on the same proportion the second change. Perform the fourth change. Perform the sixth change. Perform the seventh change, raising to the second power. Perform the eighth change, extracting the square root. Finally, 7, you may, in any case, invert the whole proportion; or, invert the terms of each ratio; or invert the means, or the extremes. Practical Questions. 1. If 7 lbs. of flour cost 31 cents, what will 196 lbs. cost? As the smaller quantity is to the larger quantity, so is the price of the smaller quantity to the price of the larger. 2. If 3 cwt. of hay cost 2 dollars, what will 35 cwt. cost? 3. If 4 qts. of molasses cost 38 cents, what will 10 qts. cost? 4. If a horse travels 19 miles in 3 hours, how far will he travel in 11 hours? 5. If the freight of 7 cwt. cost 2 dollars, what will the freight of 20 cwt. cost? 6. If 11 dollars buy 3 cords of wood, how many cords will 50 dollars buy? 7. If 7 bushels of oats last a horse 2 months, how long will 23 bushels last him, at the same rate? 8. A man bought a horse for 84 dollars, and sold him for $93; what did he gain per cent.? As the whole outlay is to 1 dollar, so is the whole gain to the gain on a dollar. 9. A merchant buys flour at $4.35 a barrel, and sells it for $4.63; what is his gain per cent.? 10. A and B form a partnership in trade; A puts in $500, and B $300, for the same time; they gain $180; what ought each to share? As the whole stock is to each one's share, so is the whole gain to each one's gain. 11. C and D trade in company; C puts in 750 dollars, and D $450, for the same time; they gain 240 dollars; how much gain ought each to receive? 12. Two men buy a lot of wood in company for 340 dollars; one takes away 42 cords, the other the remainder, which was 34 cords; what ought each to pay? 13. Two men hire a sheep-pasture in company for 20, dollars; one keeps 30 sheep in it 14 weeks; the other 24 sheep, 16 weeks; what ought each to pay? Find how many weeks' pasturing for a single sheep each one had. |