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5. Multiply 161ft. 8' 6" by 7ft. 10'. Ans. 1266ft. 8'7". 6. Multiply 87ft. 1' 11" by 5ft. 7' 5".
Ans. 489ft. 8' 0" 21" 7'. 7. What are the contents of a board 18ft. long and ift. 10in. wide ?
Ans. 33ft. 8. What are the contents of a board 19ft. 8in. long and 2ft. llin. wide ?
Ans. 57ft. 4' 4". 9. What are the contents of a floor 18ft. 9in. long and 10ft. 6in wide ?
Ans. 196ft. 10' 6". 10. How many square feet of surface are there in a room 14ft. 9in. long, 12ft. 6in. wide, and 7ft. 9in. high ?
Ans. 791ft. 1' 6". 11. John Carpenter has agreed to make 12 shoe-boxes of boards that are one inch thick. The boxes are to be 3ft. 8in. long, 1ft. 9in. wide, and Ift. 2in. high. How many square feet of boards will it require to make the boxes, and how many cubic feet will they contain ? Ans. 280
feet ; 66 cubic feet, 864 inches. 12. My garden is 18 rods long and 10 rods wide ; a ditch is dug round it two feet wide and three feet deep, but the ditch not being of a sufficient breadth and depth, I have caused it to be dug one foot deeper and ift. bin. wider. How many solid feet will it require to be removed ?
Ans. 7540 feet.
Note 1. 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 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.
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.
Note 2. Small fractions are rejected in the operation.
13. How many cords of wood in a pile 56 feet long, 4
feet wide, and 5 feet 6 inches high? Ans. 9 cords. 14. How many cords of wood in a pile 23 feet 8 inches long, 4 feet wide, and 3 feet 9 inches high ?
Ans. 2. cords
15. How much wood in a pile 97 feet long, 3 feet 8 inches wide, and 7 feet high?
Ans. 19 cords 33 feet. 16. If a pile of wood be. 8 feet long, 3 feet 9 inches wide, how high must it be to contain one cord ?
Ans. 445 feet. 17. If a board be 1 foot 7 inches wide, how long must it be to contain 20 square feet?
Ans. 12 feet 713 inches. 18. From a board 19 feet 7 inches long, I wish to slit off one square yard; how far from the edge must the line be drawn ?
Ans. 51; } inches. 19. I have a shed 19 feet 8 inches long, 14 feet 6 inches wide, and 7 feet 6 inches high ; how many cords will it contain ?
Ans. 16 cords 5 s feet +. 20. I have a room 12 feet long, 11 feet wide, and 71 feet high; in it are 2 doors, 6 feet 6 inches high, and 30 inches wide, and the mop-boards are 8 inches high ; there are 3 windows, 3 feet 6 inches wide, and 5 feet 6 inches high ; how many square yards of paper will it require to cover the walls ?
Ans. 2525 square yards.
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,
3 = 3 is the first power of
3= 31. 3 X3
9 is the second power of 3 = 32. 3 x 3 x 3 - 27 is the third power of 3 = 33. 3 x 3 x 3 x 3= 81 is the fourth power of 3 = 34.
The number denoting the power is called the index, or exponent, of the power. Thus, the fifth power of 2 is 32, or 25; the third power of 4 is 64, or 43.
To raise any number to any power required, we adopt the following
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. 1. What is the 3rd power of 5 ? 5x5x5= 125 Ans. 2. What is the 6th power of 4 ?
Ans. 4096. 3. What is the 4th power of 3 ?
Ans. 81. 4. What is the 1st power of 17 ?
Ans. 17. 5. What is the 0 power of 63 ?
OR THE EXTRACTION OF ROOTS.
Evolution, or the reverse of involution, is the extraction or finding the roots of any given power.
The root is a number, whose continued multiplication into itself produces the power, and is denominated the square, cube, biquadrate, or second, third, fourth, &c., power, equal to that power.
Thus, 4 is the square root of 16, because, 4 x 4 = 16; and 3 is the cube root of 27, because, 3x3 x3= 27; and so on.
Roots, which approximate, are surd roots; and those, which are perfectly accurate, are called rational roots.
EXTRACTION OF THE SQUARE ROOT.
1. What is the square root of 625 ?
To illustrate this question, we will suppose, that we
have 625 tile, each of which is one foot square ; we wish to know the side of a square room, whose floor they will pave or cover. If we find a number multiplied into itself, that will produce 625, that number will be the side of a square room, which will require 625 tiles to cover its floor. We perceive that our number (625) consists of three figures, therefore, there will be two figures in the root ; for the product of any two numbers can have, at most, but just so many figures, as there are in both factors, and, at least, but one less. We will, therefore, for convenience, divide our number (625) into two parts,
called periods, writing a point
over the right hand figure of 625 (25 Ans. each period ; thus, 625. We 400
now find, that the greatest square 45) 225
number in the left hand period, 225
6 (hundred), is 4 (hundred);
and that its root is 2, which we write in the quotient (see operation). As this 2 is in the place of tens, its value must be 20 and its square 400.
Let this be represented by a square, whose sides measure 20 feet each, and whose contents will, therefore, be 400 square feet. (See figure 1.) subtract 400 from 625, and there remains 225 square feet, to be arranged on two sides of figure 1, in order that its form may remain square.
We therefore double the root 20, one of the sides, and it gives the length of the two sides to be enlarged ; viz. 40. We then inquire, how many times 40, as a divisor, is contained in the dividend, and find it to be 5 times ; this we write in the root, and also in the divisor.
This 5 is the breadth of the addition to our square. (See figure 2.) And this breadth, multiplied by the length of the two additions (40) gives the contents of the two figures, E and F, 200 square feet, which is 100 feet for each.
There now remains the figure G, to complete the square, each side of which is 5 feet ; it being equal to
20 20 400
the breadth of the additions E
5 account of this last addition, that
100 the last figure of the root is placed in the divisor. If we now multiply the divisor, 45, by the last figure
20 in the root (5), the product will
100 be 225, which is equal to the remaining feet, after we have formed our first square, and equal to the additions E, F, and G, in fig
D contains 400 square feet.
do. do. ure 2. We therefore perceive, F that figure 2 may represent a
do. do. floor 25 feet square, containing Proof. 625 625 square feet. From the above, we infer the following
25 x 25 = 625.
do. do. do.
1. Distinguish the given number into periods of two figures each, by putting a point over the place of units, another over the place of hundreds, and so on, which points show the number of figures the root will consist of.
2. Find the greatest square number in the first or left hand period, place the root of it at the right hand of the given number, (after the manner of a quotient in division,) for the first figure of the root, and the square number under the period, and subtract it therefrom, and to the remainder bring down the next period for a dividend.
3. Place the double of the root already found, on the left hand of the dividend for a divisor.
4. Seek how often the divisor is contained in the dividend, (except the right hand figure,) and place the answer in the root for the second figure of it, and likewise on the right hand of the divisor. Multiply the divisor with the figure last annexed by the figure last placed in the root, and subtract the product from the dividend. To the remainder join the next period for a new dividend.
5. Double the figures already found in the root for a new