29. Find the three cube roots of 1. Suggestion.—Let x = √1, then x3 = 1, or x3 — 1 = 0. 30. Find the three cube roots of 31. Find the four fourth roots of 1. . 1. 1. The product of two consecutive numbers is 156. What are the numbers? 2. The sum of two numbers is 17, and their product is 42. Required the numbers. 3. The difference of two numbers is 8, and their product is 105. What are the numbers ? 4. There are 900 trees in an orchard, and the number in one row exceeds twice the number of rows by 5. How many rows are there? 5. A man bought some cloth for $90; had he bought 15 yards more for the same money, he would have paid $1 a yard less. How many yards did he buy? 6. The sum of two numbers is 30, and their quotient is the less number. Required the numbers. 7. A lot that is 2 rods longer than wide contains 48 square rods. What are its dimensions? 8. If a train would increase its speed 5 miles an hour, it would go 360 miles one hour sooner. What is the rate of the train? 9. A can do a piece of work in 2 days less than B, and they together can do it in 2 days. In what time can each alone do it? 10. One pipe can fill a cistern 3 hours sooner than another can empty it, and if they run together the cistern will be filled in 13 hours. In what time could the first fill it? 11. A certain number exceeds its square root by 30. Required the number. Suggestion.-Let x2 equal the number. 12. If the circumference of a wheel were increased by 4 feet, the wheel would make 110 revolutions less in going a mile. What is the circumference of the wheel? 13. A man sold a horse for $75, and thereby gained as many per cent as there were dollars in the cost. Required the cost. 14. A sold his farm at $48 an acre, and thereby lost one half as many per cent as there were dollars in the cost. Required the cost per acre. 15. A man increased his capital stock by $500 without increasing his gain, which was $500, in consequence of which his rate of gain was lowered 5%. What was his original stock? 16. How much must be added to both the length and the width of a rectangle, 18 by 20 inches, to make it contain 483 square inches? 17. Eggs rose 5 cents a dozen, in consequence of which 3 eggs less could be purchased for 25 cents. What was the price per dozen before the rise? 18. A and B together earned $432. B earned $6 a month more than A, and the number of months they worked was one third of the number of dollars A earned in a month. How much did each earn per month, and how many months did they labor? 19. A boy rowed 3 miles down a river and back again in 11⁄2 hour. The rate of the current was 2 miles an hour. Determine his rate of rowing in still water. 20. A and B are 320 miles apart. If A travels 8 miles a day more than B, they will meet in one half as many days as B travels miles per day. How far does each travel per day? 21. Twice the length of a rectangle equals three times the width, and if 2 feet be added to the length and 3 feet to the width, the area will be 56 square feet. the dimensions ? 22. A and B have each a debt of $150 to pay. $3 a week more than B, and pays his debt 8 How much does A pay per week? sooner. What are A pays weeks 23. A and B undertook to earn $640. A earned $8 a week more than B, and the number of weeks required was one fourth of the number of dollars that B earned in a week. What were the weekly wages of each? 24. Around a flower-bed, 18 feet by 12, is a gravel-walk whose area equals that of the flower-bed. What is the width of the walk? 25. A farmer sold 7 pigs and 12 lambs for $50, and found that he had sold 3 more pigs for $10 than he sold lambs for $6. Required the price of each. 26. One person husked 48 shocks of corn in a day; another husked the same number two hours sooner, and husked 2 shocks per hour more than the first. How many shocks per hour did each husk ? 27. A and B were engaged at different rates of wages. A worked a certain number of days and received $24, and B, who worked 6 days fewer, received $132. If A had worked 6 days fewer and B 6 days more, they would have received the same sum. How many days did each work? Quadratic Equations of Two Unknown Quantities. Definitions. 213. A quadratic equation of two unknown quantities is complete when it contains all the second degree and all the first degree terms possible; as, ax2+ bxy+cy2+dx+ey+f=0. 214. A quadratic equation of two unknown quantities is pure, or homogeneous, if all the terms containing unknown quantities are of the second degree; as, a x2+ bxy+cy2 = 215. Any equation containing two unknown quantities is symmetrical if the unknown quantities may change places without destroying the equation; as, 2x2+3 x y +2 y2 = 12 and 2 y2 + 3y x+2x2 = 12. 216. The solution of two simultaneous quadratic equations of two unknown quantities often involves the solution of a bi-quadratic equation. Given Illustration.- x — y2 = 3 |