APPLICATIONS OF CUBE ROOT.

466. Since the volume of a cube is the product of the three equal factors that represent its edges, it is evident that the cube root of the volume gives the length of the edge.

1. A cubical box contains 54,872 cubic inches. What is the length of each side?

2. How deep is a cubical cistern containing 2744 cu. ft.? 3. What is the number of square inches in one face of a cubical block whose contents are 185,193 cubic inches?

4. What are the dimensions of a cubical box which contains as much as a rectangular box 5 ft. 4 in. long, 4 ft. 6 in. wide, and 2 ft. 8 in. deep?

5. What must be the depth of a cubical bin that will contain exactly 1200 bushels ?

6. A cubical cistern holds 400 barrels of water. How deep is it?

7. How much will it cost, at 35 cents per square yard, to plaster the bottom and sides of the cistern?

8. A bin that is just twice as long as it is wide or high holds 500 bushels of grain. What is its length?

9. Which has the greater surface and how much; a cube whose solid contents are 4096 cubic feet, or a rectangular solid having the same contents, whose width is twice its height, and whose height is one third its length?

SIMILAR VOLUMES.

The following principles are proved by geometry: 467. PRINCIPLES.-1. Similar solids are to each other as the cubes of their like dimensions. Hence,

2. The corresponding dimensions of similar solids are to each other as the cube roots of their volumes.

1. If a globe 4 inches in diameter weighs 8 lb., what will be the diameter of a similar one that weighs 125 lb. ?

EXPLANATION.-Since the correspond

4:x:: V8: V125 (1) ing dimensions of similar solids are pro

4:x:: 2: 5 (2) x= 10, inches in diam.

portional to the cube roots of these volumes, we have the diameter of the smaller globe 4 inches the diameter of the larger globe x: the cube root of the weight of the smaller globe 8: the cube root of the weight of the other globe 125 (1). Extracting the cube root of 8 and 125, we have (2). Whence, solving, the diameter is 10 inches.

2. There are two cubes whose dimensions are 4 inches and 16 inches respectively. The larger is how many times. the smaller?

3. If a ball 3 inches in diameter weighs 7 pounds, what will be the weight of a similar ball 5 inches in diameter?

4. A cubical bin 5 feet long will hold 100.44 bushels. How much will a cubical bin 10 feet long hold?

5. The height of a cubical vessel is 1 foot 6 inches. How high must another cubical vessel be to hold four times as much?

6. If a globe of gold 1 inch in diameter is worth $120, what is the diameter of a globe of gold worth $6400?

7. If a man 5 ft. 6 in. high weighs 140 pounds, what is the weight of a man of similar build whose height is 6 ft.?

8. There are two balls whose diameters are 4 inches and

5 inches respectively. What is the diameter of a ball whose contents are equal to them both?

9. If a haystack 13 feet in diameter contains 17 tons, what is the diameter of a similar stack which contains 136 tons?

10. A bushel measure is in the form of a cylinder 18 in. in diameter, and 8 in. deep. What will be the dimensions of a peck measure of similar shape?

GENERAL REVIEW EXERCISES.

ORAL EXERCISES.

468. 1. Two boys have twice as much as the other.

2. If a man can do

together 45 cents, but one has How many cents has each? of a piece of work in a day, how

long will it take him to do one half of it?

3. If 5 men can do a piece of work in 12 days, how long will it take 8 men to do it?

4. A man bought sheep at $3 a head. a head they would have cost $16 more. buy?

Had he paid $5 How many did he

5. In what time can 40 men do a piece of work that 50 men can do in 8 days?

6. A can do a piece of work in 3 days and B in 4 days. In what time can they together do it?

7. James and Henry can hoe a field in 5 days. James can do it alone in 9 days. In how many days can Henry hoe the field alone?

8. A can make a door in of a day, and B in & of a day. How many doors can they together make in a day?

9. How long will it take A to finish a door after B has worked on it half a day?

10. A, B, and C can do a piece of work in 4 days. A can do it alone in 12 days, and B alone in 15 days. How long will it take C to do it alone?

11. If I lose of my money, and spend of the remainder, what part have I left?

12. Three boys, Peter, George, and Jacob, can do a piece of work in 3 days. Peter can do it alone in 12 days, and Peter and Jacob can do it in 8 days. How long will it take each of them to do it?

13. If I gain of a cent apiece by selling eggs at 8 cents a dozen, how much apiece will I gain by selling them at 10 cents a dozen ?

14. If I sell my apples at 6 cents a dozen, I lose 15 cents; but if I sell them at 9 cents a dozen, I gain 12 cents. How many have I, and what did they cost me?

15. If of A's money equals of B's, what part of B's equals of A's?

16. Five times of a number is 14 less than the number. What is the number?

17. I sold a bureau to A for more than it cost me. He sold it for $6, which was less than it cost him. What did it cost me?

18. A man agreed to work 16 days for $24 and board, but he was to pay $1 a day for his board for every day he was idle. He received $14. How many days did he work?

19. B engaged to work 20 days for $40, and agreed to forfeit $1 for every day he was idle. How many days was he idle, if he received $291 ?

20. Two persons share $150 in the ratio of and . What is the share of each?

21. A and B engaged in a business in which A invested $36 and B received $5 out of the $8 which they gained. How much did B invest?

22. Three persons are to in the ratio of 1, 1, and 1.

share a certain sum of money The second receives $9 more

than the third. What is the share of each?

23. If a man can earn g of a dollar in of a day, how much can he earn in & of a day?

24. If of the value of a carriage is equal to of the value of a horse, and the value of the carriage is $20 more than the value of the horse, what is the value of each?

25. In an orchard of the trees bear apples,

bear pears, and the remainder, 300, bear peaches. How many trees are there in the orchard?

26. A man can saw 2 cords of wood per day, or he can split 3 cords of wood when sawed. How much must he saw that he may be occupied the rest of the day in splitting it?

27. A man spent one half of his money and half a dollar for a coat, one half of what he had left and half a dollar for a hat, one half of what was left and half a dollar for shoes, and had a dollar left. How much had he at first?

28. A merchant, after selling from a cask of vinegar 15 gallons more than of the whole, found that he had left just 4 times as much as he had sold. How many gallons did the cask contain at first?

29. Three boys had together earned 150 cents. James had earned as much as John, and Henry as much as James and John. What sum had each earned ?

30. A tree 129 feet high was broken in a storm. of the part broken off was equal to § of the part standing. What was the length of each part?

31. Wheat sold at $1.50 per bushel pays a profit of one half the cost. If it is sold at $2 per bushel, what part of the cost will be gained?

32. If a merchant sells of an article for what of it cost, what is his gain per cent?

33. I sold some goods at a discount of 40%, and 10% off for cash. What was the total % discount?

34. If goods are bought at 80% discount, and 20% off for cash, what is the entire % discount?