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9. The length of a room being 20 feet, its breadth 14 feet 6 inches, and height 10 feet 4 inches, how many yards of painting are in it, deducting a surplus of 4 feet by 4 feet 4 inches, and 2 windows, each 6 feet by 3 feet 2 inches ?
Ans. 73 yards. 10. Required the solid contents of a wall 53 feet 6 inches long, 10 feet 3 inches high, and 2 feet thick.
Ans. 1096ft. 9'. 11. There is a house with four tiers of windows, and 4 windows in a tier; the height of the first is 6 feet 8 inches; the second, 5 feet 9 inches; the third, 4 feet 6 inches; the fourth, 3 feet 10 inches ; and the breadth is 3 feet 5 inches ; how many square feet do they contain in the whole ? Ans. 283ft. 7in.
12. How many square feet of paper would it require to line 15 boxes, each of which is 7 feet 9 inches long, 3 feet 4 inches wide, and 2 feet 10 inches high ; and how many cubic yards would the boxes contain ? Ans. 1717ft. lin. 40315 cubic yds.
13. A mason has plastered 3 rooms; the ceiling of each is 20 feet by 16 feet 6 inches, the walls of each are 9 feet 6 inches high, and there are to be 90 yards deducted for doors, windows, &c. How many yards must he be paid for?
Ans. 25lyd. 1ft. 6in. 14. How many feet in a board which is 17 feet 6 inches long, and 1 foot 7 inches wide ?
Ans. 27ft. 8' 6". 15. How many feet in a board 27 feet 9 inches long, 29 inches wide ?
Ans. 67ft. O'g. 16. How many feet of boards will it take to cover the side of a building 47 feet long, 17 feet 9 inches high ?
Ans. 834ft. 3'. NOTE. — A board to be merchantable should be 1 inch thick ; therefore to reduce a plank to board measure, the superficial contents of the plank should be multiplied by its thickness.
17. How many feet, board measure, are in a plank 18 feet 9 inches long, 1 foot 6 inches wide, and 3 inches thick ?
Ans. 84ft. 4' 6". 18. How many feet, board measure, are in a plank 20 feet long, 1 foot 6 inches wide, and 21 inches thick ? Ans. 75ft.
19. How many feet in a plank 40 feet 6 inches long, 30 inches wide, and 2 inches thick ? Ans. 278ft. 5' 3".
Note. -A pile of wood that is 8 feet long, 4 feet high, and 4 feet wide, contains 128 cubic feet, or a cord, and every cord contains 8 cord-feet; and as 8 is 1 of 128, every cord-foot contains 16 cubic feet; therefore, dividing the cubic feet in a pile of wood by 16, the quotient is the cord-feet; and if cord-feet be divided by 8, the quotient is cords.
20. How many cords of wood in a pile 18 feet long, 6 feet high, and 4 feet wide ?
Ans. 3; cords. 21. How many cords in a pile 10 feet long, 5 feet high, 7 feet wide ?
Ans. 2 cords, 94 cubic feet. 22. How many cords in a pile 35 feet long, 4 feet wide, 4 feet high ?
Ans. 43 cords. 23. How many cords in a pile that measures 8 feet on each side ?
Ans. 4 cords. 24. How many cords in a pile that is 10 feet on each side ?
Ans. 71% cords. Note. — When wood is “ corded " in a pile 4 feet wide, by multiplying its length by its height, and dividing the product by 4, the quotient is the cord-feet; and if a load of wood be 8 feet long, and its height be multiplied by its width, and the product divided by 2, the quotient is the cord-feet.
25. How many cords of wood in a pile 4 feet wide, 70 feet 6 inches long, and 5 feet 3 inches high ? Ans. 1116 cords. Note. — Small fractions are rejected.
26. How many cords in a pile of wood 97 feet 9 inches long, 4 feet wide, and 3 feet 6 inches high ? Ans. 10177 cords.
27. Required the number of cords of wood in a pile 100 feet long, 4 feet wide, and 6 feet 11 inches high. Ans. 2158.
28. Agreed with a man for 10 cords of wood, at $5.00 a cord; it was to be cut 4 feet long, but by mistake it was cut only 46 inches long. How much in justice should be deducted from the stipulated price ?
Ans. $ 2.081. 29. If a load of wood be 8 feet long, 3 feet 8 inches wide, and 5 feet high, how much does it contain ?
Ans. 94 cord-feet. 30. If a load of wood be 8 feet long, 3 feet 10 inches wide, and 6 feet 6 inches high, how much does it contain ?
Ans. 124. cord-feet. 31. If a load of wood be 8 feet long, 3 feet 6 inches wide, how high should it be to contain 1 cord? Ans. 4ft. 6 104".
32. If a load of wood be 12 feet long, and 3 feet 9 inches wide, how high should it be to contain 2 cords?
Ans. 5ft. 8' 33". 33. D. H. Sanborn's parlour is 17ft. Iin. long, 14ft. 8in. wide, and 8ft. 9in. high. There are two doors 3ft. 4in. wide, and 7ft. high, and four windows 5ft. 3in. high, and 3ft. 4in. wide ; the mop-boards are 9in. high. B. Gordon, a first-rate mason, will charge 10 cents per square yard for plastering the room. The paper for the room is 20 inches wide, and costs 64 cents per yard. E. Eaton will “ paper" the room for 4 cents per square yard. Each window has 12 lights of 10in. by 14in. glass, the price of which is 124 cents per square foot. The painter's bill for setting the glass is 8 cents per light, and for painting the floor, mop-boards, and doors is 25 cents per square yard. What is the amount of Mr. Sanborn's bill ?
Ans. $ 33.72,3%.
INVOLUTION. INVOLUTION is the raising of powers from any given number, as a root.
A power is a quantity produced by multiplying any given number, called a root, a certain number of times continually by itself ; thus,
2= 2 is the root, or 1st power of 2 = 21. 2 x 2 = 4 is the 2d power, or square of 2 = 22. 2 X 2 X 2= 8 is the 3d power, or cube of 2= 2 X 2 X 2 X2= 16 is the 4th power, or biquadrate of 2 = 24.
The number denoting the power is called the index or exponent of the power. Thus, the fourth power of 3,=81, is expressed by 34, and 4 is the index or exponent; and the second power of 7, = 49, is expressed by 72.
To raise a number to any power required. Rule. - Multiply the given number continually by itself, till the number of multiplications be one less than the index of the power to be found, and the last product will be the power required.
EXAMPLES 1. What is the 5th power of 4?
4 X 4X4 X4 X4 = 1024 Ans. 2. What is the 3d power of 8?
Ans. 512. 3. What is the 10th power of 7 ? Ans. 282475249. 4. What is the 6th power of 5 ?
Ans. 15625. 5. What is the 3d power of ?
Ans. 41. 6. What is the 5th power of 1?
Ans. 3] • 7. What is the 4th power of 2 ?
Ans. 504 6.
8. What is the 6th power of ik?
Ans. 161319 9. What is the 4th power of .045 ? Ans. .000004100625. 10. What is the O power of 1728?
OR THE EXTRACTION OF ROOTS.
EVOLUTION is the reverse of Involution, and teaches to find the roots of any given powers.
The root is a number whose continual multiplication into it. self produces the power, which is denominated the 2d, 3d, 4th, &c., power, according to the number of times which the root is multiplied into itself. Thus, 4 is the square root of 16, because 4 X4= 16; and 3 is the cube root of 27, because 3 X3 X3 = 27; and so on.
Although there is no number of which we cannot find any power exactly, yet there are many numbers of which precise roots can never be determined; but, by the help of decimals, we can approximate towards the root to any assigned degree of exactness.
The roots which approximate are called surd roots; and those which are perfectly accurate are called rational roots.
Roots are sometimes denoted by writing the character ✓ be. fore the power, with the index of the root over it; thus, the 3d root of 36 is expressed 36, and the second root of 36 is → 36, the index 2 being omitted when the square root is designed.
If the power be expressed by several numbers with the sign + or - between them, a line is drawn from the top of the sign over all the parts of it; thus, the 3d root of 42+22 is 42+22, and the second root of 59 — 17 is 59 — 17, &c.
Sometimes roots are designated like powers with fractional indices. Thus the square root of 15 is 153, the cube root of 21 is 215, and the 4th root of 37 — 20 is 37 — 201, &c.
It sometimes will happen that one root is involved in another, thus :
125 — 5+ 19+6, or v161 — 147.