per M. 32. IIow many yards of brussels carpeting, $ of a yard wide, laid lengthwise of the room, will be required to cover a room 22 ft. by 17 ft. 4 in., if the waste in matching be 6 in. on each strip ? REMARK.—When the width of the room is not exactly divisible by the width of the carpet, drop the fraction in the quotient and add 1 to the whole number. The waste in such cases is either cut off or turned under in laying. 33. What will it cost, at 21¢ per sq. yd., to plaster the sides and ceiling of a room 24 ft. by 31} ft. and 10 ft. high, if one-sixth of the surface of the sides is taken up by doors and windows ? 34. A street 4975 ft. long and 40 ft. wide was paved with Trinidad asphaltum, at $2.65 per square yard. What was the cost ? 35. A skating rink, 204 ft. by 1964 ft., was floored with 2 in. plank, at $23.50 What was the cost of the lumber? 36. What will be the cost of the carpet border for a room 16ft. by 21 ft., if the price be 62.14 per yard ? 37. How many single rolls of paper, 8 yd. long and 18 in. wide, will it take to cover the ceiling of a room 56 ft. long and 27 ft. 4.in. wide, if there be no waste in matching ? REMARK.— When no allowance is made for waste in matching, divide the surface to be papered by the number of square feet in one roll of the paper. 38. How many yards of carpeting, of a yard wide, will be required to carpet a room 32 ft. long and 25 ft. wide, if the lengths of carpet are laid crosswise of the room, and 8 inches is lost on each length in matching the pattern ? Ilow many yards if the lengths are laid lengthwise and 6 in. is lost in matching ? If the carpet is laid in the most economical way, what will be the cost, at $2.55 per yard ? 89. How many sheets of tin, 20 in. by 14 in., will be required to cover a roof 60.5 ft. wide and 156.25 ft. long? 40. What is the difference between four square feet and four feet square ? 41. What will it cost, at $1.15 per yard, to carpet a flight of stairs 11 ft. 4 in. high, the tread of each stair being 10 in. and the riser 8 in. ? 42. How many shingles, averaging 4 in. wide and laid 5 in. to the weather, will cover the roof of a barn, one side of the roof being 24 ft. wide and the other 42 ft. wide, the length of the barn being 60 ft. ? 43. Divide an acre of land into 8 equal sized lots, the length of each of which shall be twice its frontage. What will be the dimensions of cach lot ? 44. How many granite blocks, 12 in. by 18 in., will be required to pave a mile of roadway 42 ft. in width ? 45. What will be the cost, at 20¢ per sq. yd., for plastering the ceiling and walls of a room 22 ft. wide, 65 ft. long, and 15 ft. high, allowance being made for 8 doors 4 ft. 6 in. wide by 11 ft. 6 in. high, and 10 windows each 42 in. wide by 8 ft. high? 46. I wish to floor and ceil a room 27} yd. long and 15 yd. 2 ft. wide, with matched pine. What will be the cost of the material, at $26.40 per M? INVOLUTION. 425. A Power of a number is the product arising from multiplying a number by itself, or repeating it several times as a factor. 426. A Perfect Power is a number that can be exactly produced by the involution of some number as a root; thus, 64 and 16 are perfect powers, because 8 X 8=64, and 2 X 2 X 2 X 2= 16. 427. The Square of a number is its second power. 428. The Cube of a number is its third power. 429. Involution is the process of finding any power of a number; and a number is said to be involved or raised to a power, when any power of it is found. EXAMPLES FOR PRACTICE. 430. 1. What is the square of 1 ? 5. What is the square of 9 ? 2. What is the square of 3 ? 6. What is the square of 10 ? 3. What is the square of 4 ? 7. What is the square of 99 ? 4. What is the square of 5 ? 8. What is the square of 256 ? REMARK.–From the solution of the above examples the pupil will observe: 1st. That the square of any number expressed by one figure cannot contain less than 1 nor more than 2 places. 2d. That the addition of 1 place to any number will add 2 places to its square. EVOLUTION. 431. Evolution is the process of extracting the root of a number considered as a power. It is the reverse of Involution, and each may be proved by the otlicr. 432. A Root of a number is one of the equal factors which, multiplied together, will produce the given number; as, 4 X 4 X 4 = 64; 4 is the root from which the number 64 is produced. SQUARE ROOT. 433. The Square Root of a Number is such a number as, multiplied by itself, will produce the required number. 434. The operation of finding one of the two equal factors of a square, or product, is called extracting the square root. REMARK.— The square root of any number, then, is one of its two equal factors, the given number being considered a product. 433. In practical operations, a surface and one of its dimensions being given, the wanting dimension is found by dividing the surface by the given dimension. e The accompanying diagram is a square 14 feet by 14 feet. Its square feet, or area, is by inspection found to be made up of: 1st. The tens of 14, the number representing the length of one side, or 10 squared 100 square feet, as shown by the square within the angles a, b, c, d. 2d. Two times the product of the tens by the units of the same number, or 2 (10 X 4) = 80 square feet, b clf as shown by the surfaces within the angles e, f, g, h, i I w and i, j, k, l. 3d. The square of the units, 4 feet, or the product of 4 ft. by 4 ft. 16 square feet, as shown by the КІХ square within the angles w, x, y, 2. 2 (10 X 4) = 80 square feet. 14 Ft:10 Ft.& 4 Ft. 196 square feet. Or, the square of 14 is made up of or equals the square of 10, plus twice the product of 10 by 4, plus the square 4, the number to be squared. 436. General Principles.—The square of any number composed of two or more figures is equal to the square of the tens, plus twice the product of the tens multiplied by the units, plus the square of the units. 437. Units and Squares Compared. UNITS. SQUARES. REMARK.-Squaring the numbers from 1 to 10 inclusive, shows: 1° = 1 1st. That the square of any number will contain at least one place, or 22 4 one order of units. 33 9 2d. That the square of no number represented by a single figure will 4° = 16 contain more than two places. If the number of which the square root is sought be separated into periods of two figures each, beginning at the 25 right, the number of periods and partial periods so made will represent 6 36 the number of unit orders in the root. Therefore, the square of any numny2 ber will contain twice as many places, or one less than twice as many, as 8° = 64 its root. 92 = 81 3d. Where the product of the left hand figure multiplied by itself is not 10% = 100 greater than 9, then the square will contain one less than twice as many places as the root. = 49 t u 438. EXAMPLE.—Find the square root of €25. OPERATION. EXPLANATION.—The number consists of one full and one partial period; hence its root will contain two places 6.25 ) 2 5 -tens ard units. The given number, C25, must be the 4 400 product of the root to be extracted multiplied by itself; 20 X 2 = 40 | 225 rem. thcrcfore, the first figure of the root, which will be the 5 | 225 highcst order of units in that root, must be obtained from the first left hand period, or highest order of units 45 0 rem. in the given number. Hence, the first or tens figure of the root will be the square root of the greatest perfect square in 6. 6 coming between 4, the square of 2, and 9, the square of 3, its root must be 2 tens with a remainder. Subtracting from the 6 hundreds or 6, the square of 2 (tens) = 400 or 4, gives 225 as a remainder. Having now taken away the square of the tens, the remainder, 225, must be equal to 2 times the tens multiplied by the square of the units, plus the square of the units. Since the tens are 2 or 20, twice the tens = 40. Observe, therefore, that 225 must equal 40 times the units of the root, together with the square of such units. If, then, 225 be divided by 40, the quotient, 5, will nearly, if not exactly represent the units of the root sought. Using 40, then, as a trial divisor, the second, or unit figure of the root is found to be 5. . The term, twice the tens multiplied by the units, is equal to 2 (20 x 5), or 200, and the units, or 5, squared = 25; the sum of these wanting terms, or 225, is the remainder, or what is left after taking from the power the square of the first figure of the root. Therefore, the square root of 625 is 25. Rule.-I. Beginning at the right, separate the given number into periods of two places each. II. Take the square root of the greatest perfect square contained in the left hand period for the first root figure; subtract its square from the left hand period, and to the remainder bring down the next period. III. Divide the number thus obtained, exclusive of its units, by twice the root figure already found for a second quotient, or root figure; place this figure at the right of the root figure before found, and also at the right of the divisor; multiply the divisor thus formed by the new root figure, subtract the result from the dividend, and to the remainder bring down the next period, and so proceed till the last period has been brought down, considering the entire root already found as so many tens, in determining subsequent root figures. REMARKS.—1. Whenever a divisor is greater than the dividend, place a cipher in the root and also at the right of the divisor; bring down another period and proceed as before. 2. When the root of a mixed decimal is required, form the periods from the decimal point right and left, and if necessary supply a decimal cipher to make the decimal periods of two places each. 3. A root may be carried to any number of decimal places by the use of decimal periods. 4. Any root of a common fraction may be obtained by extracting the root of the numerator for a numerator of the root, and the root of the denominator for the denominator of the root. 5. To find a root, decimally expressed, of any common fraction, reduce such common fraction to a decimal, and extract the root to any number of places. EXAMPLES FOR PRACTICE, 439. 1. Find the square root of 196. b 12. Find the square root of 106.413, carried to four decimal places. 15. What is the square root, decimally expressed, of 15, carried to three decimal places ? 16. What is the square root, decimally expressed, of '}, carried to two decimal places? 17. What is the square root of 30368921, carried to one decimal place. 18. What is the square root of 4698920043, carried to two decimal places. FIG. T. 440. A Triangle is a plane figure having three sides and three angles. 441. The Base is the side on which the triangle stands; as, a, c. 442. The Perpendicular is the side forming a right angle with the base; as, a, b, in fig. S. 443. The Hypothenuse is the side opposite the right angle; as, a, c, in fig. S. Fig. T. is a triangle, having angles at a, b, c. FIG. S. 444. A Right-angled Triangle is a triangle having a right angle. Fig. S is a right-angled triangle, the angle at b being a right angle. The line ab is the Perpendicular; the line b c is the Base; the line a c is the Hypothenuse. a с TRIANGLE. a REMARK.-It is a geometrical conclusion that the square formed on the hypothenuse is equal to the sum of the squares formed on the base and the perpendicular 445. To find the hypothenuse, when the base and perpendicular are given. RULE.—To the square of the base add the square of the perpendicular, and extract the square root of their sum. To find the base, when the hypothenuse and perpendicular are given. RULE. — From the square of the hypothenuse take the square of the perpendicular, and extract the square root of the remainder. To find the perpendicular, when the hypothenuse and base are given. RULE.—Take the square of the base from the square of the hypothenuse, and extract the square root of the remainder. FOR 446. 1. The base of a figure is 6 ft. and the perpendicular 8 ft. the hypothenuse. 2. The perpendicular is 17.5 ft. and the base is 46.6 ft. Find the hypothenuse to three decimal places. EXAMPLES FOR PRACTICE. |