47. There are three numbers in arithmetical progression. The difference of the second powers of the first and second is 7; and the difference of the squares of the second and third is What are the numbers?

9.

Let x the middle term, and y = the common difference; then,

48. There are three numbers in geometrical progression: the mean is 6, and the sum of the first and second is 13. What are the numbers?

Let

Then

y2

xy = the middle term, [SEE PAGE 116, PROP.] xy = 6

49. There are three numbers in Geometrical progression. The mean is 8, and the sum of the squares of the first and third is 272. What are the numbers?

50. Find two numbers, whose difference multiplied by the difference of their squares, is 32; and whose sum multiplied by the sum of their squares, gives 352.

Let
Then (1)
And (2)

(3)x

- x y2

x2y + y3

32

=272

(4) x2 + x y2+x2y+y3

Add (3) and (4,) and divide by 2, and we have

Subtract (3) from (4,) and divide by 2, and we have

Multiply this last equation by 3, and add the result to equation (5,) and we have

(8)x+3xy+3xy2+y=512

Take the third root of this last equation, and we have

Substitute this value of x+y in equation (7,) and we have

Take equations (5) and (11,) and find the values of x and y. 51. The difference of two numbers multiplied by the difference of their squares, is 256; and their sum multiplied by the sum of their squares, is 2176. What are the numbers?

52. A and B own, each, a square piece of land. They made an exchange; but, as A had more land than B, B engaged to pay as many dollars per square rod as there were rods in the side of one square more than there were in the side of the other. Upon making an estimate, B had to pay A 625 dollars. They afterward sold both their lots to C for 8125 dollars, and found that they had received as many dollars per square rod, as there were rods in a side of each of the squares. What were the lengths of the sides of the squares?

53. The product of two numbers, multiplied by the sum of their squares, is 1248; and the difference of their squares is 20. What are the numbers?

x3y+xy3 1248
x2-y2 = 20

Multiply (4) by xy, and we have

=

(5) xy-xy3 20 x y

Square each member of (3) and (5,) and we have

(6) x® y2+2x+y+x2 y2 = 1557504

x® y2 — 2 x1 y2 + x2 ya = 400 x2 y3

Substitute this value of xy in equation (1,) and we have

Take equations (2) and (8,) and find the values of x and y.

54. The product of two numbers multiplied by the sum of their squares, is 19968; and the difference of their squares is 80. What are the numbers?

55. There are two square lots of ground; one contains 20000 square poles more than the other. Each is sold for asmany shillings per square pole, as there are poles in the side. of the other; and the whole, at that rate, came to 2000000000 shillings. What are the sides of the squares?

56. Twelve times the sum of two numbers are equal to their product; and, the second power of half the sum, is equal to 25 more than their product. What are the numbers?.

57. There is a rectangular piece of land; twelve times the number of rods round it, are equal to the number of square rods in it. A square of the same perimeter will contain 100 square rods more than the rectangle. What are the length and breadth of the rectangle?

SECTION LXXXVIII.

Questions: Producing Equations of the Third Degree. 1. A number multiplied by of its second power, gives 9. What is the number?"

2. A man sold a square lot of ground for as many dollars per square rod as there were rods in the side of the square; the whole came to 1000 dollars.. What is the side of the square?

3. There are two numbers, one of which is of the other; and the sum multiplied by the sum of their second powers, is 520. What are the numbers?

4. There are two numbers; the first multiplied by the square of the second, gives 18; and the second multiplied by the square of the first, gives 12. What are the numbers?

5. A man commenced trade with a certain sum of money; the first year he gained a sum per cent..equal to his capital. The second year he traded on what he had gained, and gained the same per cent. that he did the first year; and found that the gain of the second year was 100 dollars. With what sum did he commence trading?

6. A number of men agreed to raise 108 dollars for a charitable purpose; at first, each one furnished a number of dollars

equal to the number of men in the company; when it was found, that if each one had furnished as much as the whole had furnished, they would have raised just the sum proposed. How many persons were there in the company?

7. A man purchased a square lot of ground for 512 dollars; upon making an estimate, he found that he had paid as many dollars per square rod, as there were rods in the side of the square. What was the length of one side of the square?

8. A has a rectangular piece of ground, the length of which is four times its breadth. B has a square piece of the same perimeter. A sells his for as many dollars per square rod as it is rods in length. B sells his for as many dollars as there are rods in one side of the square; when it appeared that B had received 24 dollars less than A. What quantity of land had each?

9. A man asked his friend how much money he had; his friend answered, that if the number of dollars he had, were multiplied by four times the number, and that product multiplied by half of itself, the product would be twenty-seven times the number. What is the number?

10. If the difference of two numbers be multiplied by the second power of the greater, and the sum of the two numbers by the second power of the greater, the sum of the two products will be 432; and the difference of the products 280. What are the numbers?

11. There are three numbers; the first multiplied by the square of the second, gives 48; the first by the square of the third, gives 75; the second by the square of the third, gives What are the numbers?

100.

12. A number x, multiplied by the second power of a number y, gives a; and the second power of x, multiplied by y, gives b. What are the numbers?

13. There are three numbers; the first multiplied by the square of the second, gives a ; the second by the square of the third, gives b; and the first by the square of the third, gives c. What are the numbers?

14. Four times a certain number is equal to the part of its fourth power. What is the number?

15. A man sold a number of bushels of wheat for a number of dollars equal to of the second power of the number of If, then, the whole price be multiplied b

bushels of wheat.

the number of bushels, the product will be equal to 216 dollars. What number of bushels of wheat did he buy?

16. The third power of a number added to nine times its second power, twenty-seven times the number and 27 more, is equal to 125. What is the number?

17. A and B commenced playing at cards, each having the same sum of money; after a certain number of games, A had the third power of the number of shillings which he had at first, wanting thirty-six times its second power. B had four hundred and thirty-two times as much as he had at first, wanting 1728; then the sum of what A and B had, was equal to 343. What sum had A and B at first?

SECTION LXXXIX.

Questions Producing Equations of the Third Degree.

1. How many roots has a cubic equation?

2. What cubic equations have the following roots?

3. Find the values of x in the following equations of the

third degree:

4. There are three numbers; the second is 2, and the third 3 more than the first, and their continued product is 40. What are the numbers?

5. A has 4 dollars more than B; but, if the number A has be multiplied by the second power of the number B has, the product will be 225. What number has each?

6. The sum of two numbers is 12, and the sum of theirr third powers is 578. What are the numbers?