29. How many cubic feet of gas will be required to fill a spherical balloon whose diameter is 50 feet? 30. How many cubic miles of volume has the moon, allowing its diameter to be 2100 miles ? SIMILAR SURFACES. 408. Similar Surfaces are surfaces having the same form, without regard to size. 409. Similar surfaces are to each other as the squares of their corresponding dimensions. Hence, The corresponding dimensions of similar surfaces are to each other as the square roots of their areas. 31. A triangle whose base is 12 feet has an area of 72 square feet. What is the base of a similar triangle whose area is 32 square feet? Solution. — 72 : 32 = 122 : x?, or 72 : 32 = 12 : 8. Ans. 8 ft. 32. A side of one of two similar triangles is 12 feet, and the corresponding side of the other is 8 feet. If the larger of the triangles has an area of 72 square feet, what is the area of the other ? 33. The areas of two similar rectangular fields are, respectively, 9 and 6.25 acres, and the first is 120 rods long. What is the length of the second ? 34. The diameters of two circles are, respectively, 100 feet and 50 feet. How much larger is the one than the other ? 35. If a pipe whose diameter is 1.5 inches fills a cistern in 5 hours, in what time will another whose diameter is 35 inches fill it? 36. If it costs $ 30 to pave the bottom of a cellar whose width is 16 feet, what will it cost to pave a similar one whose width is 24 feet ? 37. A gentleman has a park, in the form of a right-angled triangle, containing 105.55 square meters, the longest side of which is 15 meters. He wishes to lay out another in the same form, whose longest side shall be 3 times the length of the first. Required the area. SIMILAR SOLIDS. 410. Similar Solids are solids having the same form without regard to size. 411. Similar solids are to each other as the cubes of their corresponding dimensions. Hence, 412. The corresponding dimensions of similar solids are to each other as the cube root of their contents or volumes. 38. If a cone 3 feet in height contains 1539 cubic feet, what are the contents of a similar cone 2 feet in height ? Solution. — 33 : 28 = 1539 : x, or 27 : 8= 1539 : 456. 39. Two similar bins contain, respectively, 400 and 600 bushels. If the first is 4 feet deep, how deep is the second ? 40. A sugar loaf which is 12 inches high weighs 16 pounds. How many inches may be broken from the base, that the rest may weigh 8 pounds ? 41. If a sphere 6 inches in diameter weighs 16.50 kilograms, what is the weight of a sphere of the same material 12 inches in diameter ? 42. Suppose the diameters of the earth and the moon to be, respectively, 8000 and 2000 miles, how many times larger is the earth than the moon ? 43. If the weight of a well-proportioned man 5 ft. 6 in. in height, is 140 pounds, what should be the weight of Barnum’s Chinese giant who is 8 ft. 3 in. in height? REVIEW. ORAL EXERCISES. 413. 1. If 6 barrels of flour cost $33, what will 11 barrels cost? 2. If 9 tons of coal cost $ 54, how many cords of wood at $4 a cord would cost as much as 5 tons of coal ? 3. If 6 horses in a certain time consume 14 bushels of oats, how many bushels will 10 horses consume in the same time? 4. If of a barrel costs $43, what will £ of a barrel cost ? 5. If A can do a piece of work in 3 days, and B can do the same in 4 days, what part of the work can they together do in 1 day? 6. If A and B can in 1 day perform of a piece of work, how long will it take them to do the whole ? 7. Divide 75 into two numbers that shall be to each otber as 7 to 8 ? 8. The first three terms of a proportion are 7, 8, and 9. What is the fourth term ? 9. How long will 40 bushels of oats last 3 horses, if they consume 8 bushels in 2 weeks ? 10. If 7 horses consume 3 tons of hay in 6 weeks, how many tons will 6 horses consume in 4 weeks? 11. A and B run a race in which A gains 3 rods in running 11 rods. How far must he run to gain 7 rods? 12. A box of tea lasted a man and his wife 9 months. When the man was absent it would last his wife 12 months. How long would it have lasted the man alone ? 13. Smith and Jones hire a pasture together. Smith pays $ 16 and Jones $ 20. Smith puts in 12 sheep. How many does Jones put in ? 14. A furnished 2 loaves for a lunch, and B furnished 3, while C contributed 30 cents to be divided between A and B. How much should each receive ? 15. Three men trade together. A puts in $3 as often as B puts in $4 and as often as C puts in $5. They gain $87. What was each man's share of the gain ? 16. A hare is 60 rods before a hound, and runs 3 rods while the hound runs 6. How many rods must the hound run to overtake the hare ? 17. What is the time of day when } of the time past midnight equals the time to noon ? 18. At what time between 4 and 5 o'clock are the hour and the minute hand together? 19. A can do a piece of work in 2 days, B can do it in 4 days, and C in 3 days. In what time can they do it together ? 20. If I pay $ 15 four months before it is due, how long after it is due may I keep $ 20 to balance it? 21. If 24 men can mow 66 acres of grass in 2 days, how many acres can 14 men mow in 7 days ? 22. What is the interest of $ 800 for 36 days at 10 % ? 23. I bought 12 cords of wood that was to have been 4 ft. long. It proved, however, to be only 42 in. What was my per cent of loss ? 24. William Simonds receives a bequest of $ 840, with which he buys U. S. 4 per cents at 120. What is his annual income? 25. In what time will $ 400 yield $ 72 interest at 4% ? 26. What is the perimeter of a square 10-acre field ? 27. My semi-annual income from an 8 % stock is $ 84. How many shares do I own ? 28. What is the longest straight line that can be drawn on a sheet of 6 x 8 paper ? 29. A lawyer earns $ 275 collecting money at 5 % commission. What amount does he collect ? 30. The diagonal of a garden 8 rods long is 10 rods. Re. quired the area of the garden. 31. The interest of $ 400 is $ 70 for 2 y. 6 mo. What is the rate ? 32. What is the cube root of the square of z? 33. Which is the larger, the square of for its cube, and how much? 34. Find a mean proportional between 3 and 48. 35. What is the inverse ratio of 3 to 4 ? 36. Received a discount for cash of 12} % on a bill of goods amounting to $128. How much did I pay ? 37. Which is the better investment, 6 % stock at 120 or 8 % bonds at 144 ? 38. Find the missing term in the proportion 2] : x = : x=5:71 39. A note for $ 800, which matures May 31, is discounted April 1. Required its proceeds. 40. In selling cloth at $6 per yard, I gain 20 %. What % should I gain if I sold it at $ 6.40 ? 41. A bank renewed $ 100000 worth of U. S. 6 per cents at 3.1 %. What is the annual loss of interest ? 42. What is the interest of the present worth of a debt of $ 520 due in a year, money being worth 4 % ? 43. When exchange on London is quoted at $ 4.80, what will a £ 200 draft cost? 44. I lent Willard Aldrich $ 1500 for three months. How long should he lend me $ 1000 to requite the favor ? 45. What is the ratio of the square of 4 to the cube root of 125 ? 46. What is the inverse ratio of a decigram to a kilogram? 47. A party of pleasure-seekers hired a sailboat for $4, each paying as many cents as there were persons in the party. What did each pay ? 48. A man with more money than learning willed 1 his property to his wife, } to his son, and to his daughter. He left $ 13000 ; how ought it to be equitably divided ? |