2. What are the solid contents of the frustum of a cone whose upper base is 5 feet in diameter, whose lower base is 8 feet in diameter, and whose altitude is 7 feet?

3. A tree was 3 feet in diameter at the butt and its diameter at a height of 40 feet was 1 foot. What were the cubical contents of that portion of the tree?

4. A vat whose inside measurements were as followsdiameter of the bottom 12 feet, diameter of the top 10 feet, height 9 feet-was filled with water. How many gallons did it contain?

609. To find the volume or contents of a sphere.

A sphere may be regarded as composed of pyramids whose bases form the surface of the sphere, and whose altitude is the radius of the sphere. Hence the following is the

RULE.-1. Multiply the convex surface by one-third of the radius; or,

2. Multiply the cube of the diameter by .5236.

EXERCISES.

610. 1. The diameter of a sphere is 5 feet. How many cubic feet does it contain?

2. Find the contents of a sphere whose diameter is 8 feet. 3. The circumference of a sphere is 9.4248. What are its cubical contents?

4. A cubic foot of cast-iron weighs about 450 pounds. What is the weight of a cannon-ball whose diameter is 18 inches?

5. What are the cubical contents of a spherical vessel the diameter of which is 2 feet?

6. How many cubic feet are there in a spherical body whose diameter is 25 feet?

MISCELLANEOUS EXAMPLES.

611. 1. If 5 men can do a piece of work in 12 days, how long will it take 6 men to do the same work?

2. If 5 barrels of apples cost $7.50, what will 8 barrels cost at the same rate?

3. It required 20 men to load a vessel in 6 days, how many men would it require to load it in 14 days?

4. A steamboat sailed 42 miles in 2 hours. How far did she sail in 20 minutes?

5. If 6 men can dig 28 rods of ditch in 1 day, how many men will it require to dig 56 rods in & of a day?

6. If of a yard of broadcloth cost $33, what will of a yard cost?

7. If it costs $50 to support a family of 8 persons for 21 weeks, what will it cost to support 10 persons 3 weeks?

8. If 3 pounds of tea are worth 14 pounds of coffee, and 5 pounds of coffee are worth 18 pounds of sugar, and 21 pounds of sugar are worth 60 pounds of flour, how many pounds of flour are equal in value to 7 pounds of tea?

9. A farmer sold 12 firkins of butter, each containing 56 pounds, for 23 cents a pound, and received in payment 5 pounds tea at 85 cents per pound, 60 pounds sugar at 13 cents per pound, 15 yards cloth at $1.12 per yard, and the balance in money. How much money did he receive?

10. A regiment of soldiers consisting of 1100 men, was furnished with bread sufficient to last it 8 weeks, allowing each man 15 oz. per day. If of it was found to be unfit for use, how many ounces per day shall each man receive so that the balance may last 8 weeks?

11. A man being asked how many sheep he had, replied, "If I had 3 times as many as I have and 5 sheep, I would have 185." How many had he?

12. A man paid of his money on a debt, of the remainder for a suit of clothes, of the remainder for provisions, and lost of the remainder, when he had $5 left. How much had he at first?

13. Three men engage to reap a field of wheat. A can do it in 15 days, B in 18 days and C in 20 days. In what time can they do it together?

14. A farmer was offered $1.45 per bu. for his wheat, but determined to have it ground and sell the flour. It cost to take it to the mill 24 cents per bu.; the miller took for grinding; it took 448 bu. to make a barrel of flour; he paid 45 cents apiece for barrels, and it cost 25 cents per barrel commission to sell it. 75 bbl, sold for $550 and 25 bbl. for $165. If the refuse was sold for $100, did he make or lose, and how much per hundred barrels?

15. A farmer being asked how many apple-trees he had, replied, "If I had 3 times as many and 5 trees more, I would have 1358." How many had he?

16. of A's money is equal to of B's, and the difference is $8. How much has each?

17. A, B and C hire a pasture for $170. A puts in 70 sheep for 6 months, B 24 cattle for 4 months, C 10 cattle and 35 sheep for 5 months. If 2 cattle eat as much as 7 sheep, how much should each pay?

18. If a pole 10 feet long, casts a shadow 13 feet long, what is the length of a pole which will cast a shadow 621 feet long at the same time?

19. A's weight is that of B, and C's weight is as much as A's and B's together. The sum of their weights is 490 pounds. What is the weight of each?

20. of A's money is equal to of B's, and the difference is $5. How much money has each?

21. The ages of A, B and C, are to each other as 3, 4 and 5, and their sum is 136 years. What is the age of each?

22. A boy bought a certain number of apples at the rate of 4 for 5 cents, and sold them at the rate of 3 for 4 cents. He gained 60 cents. How many did he buy?

A and B can do

23. A, B and C agree to build a house. the work in 32 days, B and C in 28 days, and A and C in 26 days. How long will it take them to do it by working together? How long would it take each to do it alone?

24. A can build a wall in 10 days, by working 12 hours a day, B can build it in 9 days, by working 10 hours a day. In how many days can both build it, by working 8 hours a day?

25. A pair of horses is sold for $390. One of them is worth as much as the other. What is the value of each?

26. A hind wheel of a carriage 4 feet 6 inches high, revolved 720 times in going a certain journey. How many revolutions did the fore wheel make, which was 4 feet high?

27. The shadow of a pole 6 feet long is 9 inches, and the shadow of a steeple at the same time is 9 feet long. What is the height of the steeple?

28. What is the bank discount on a note for $245.30, due in 90 days, if discounted at 6%?

29. If a man takes 2 steps of 30 inches each in 3 seconds, how long will it take him to walk 10 miles?

30. It cost $150 to support 4 grown persons and 3 children 8 weeks. What will it cost to support 3 grown persons and 8 children for the same time, if 3 children cost as much as 2 grown persons?

31. A man bought 20 bushels of wheat and 15 bushels of corn for $36, and 15 bushels of wheat and 25 bushels of corn for $32.50? What did he pay per bu. for each?

32. A fox has 120 rods the start of a hound. If the hound runs 30 rods while the fox runs 26, how far will the hound run before he overtakes the fox?

33. A starts on a journey at the rate of 3 miles an hour. 6 hours afterward, B starts after him at the rate of 4 miles an hour. How far will B travel before he overtakes A?

34. One-fourth of a certain number is 10 more than of it. What is the number?

35. If to a certain number you add

of itself and of itself, the sum will be 105. What is the number?

36. If to a certain number you add 15 more than of itself, the sum will be 40. What is the number?

37. How many days will it take 30 men to do a piece of work, which 20 men can do in 45 days?

38. If a man can earn of a dollar in of a day, how much can he earn in § of a day?

39. How many yards of silk

yard wide, will it take to line 41 yards of broadcloth 13 yards wide?

40. If 14 ounces of wool make 24 yards of cloth 1 yard wide, how much will it take to make 6 yards 14 yards wide?

41. How many tiles 14 inches long, will it take to make a drain which is of a mile long?

42. If $300 placed at interest yields an income of $18 in 9 months, how much must be placed at interest at the same rate to yield an income of $115 in 6 months?

43. If to a certain number you add of itself, the result will be 20 less than double the number. What is the number? 44. At what time between 5 and 6 o'clock will the hour and minute hands of a clock be exactly, together?

45. Two soldiers start together for a certain fort. One, who travels 12 miles per day, after traveling 9 days, turns back as far as the other had traveled during those 9 days. He then turns and pursues his way toward the fort, where both arrive together 18 days from the time they set out. At what rate did the other travel?

46. A man agreed to execute a piece of work in 60 days,