3. What is the square root of 516961 ? Ans. 719. 4. What is the square root of 182329 ? Ans. 427. 5. What is the square root of 23804641 ? Ans. 4879. 6. What is the square root of 10673289 ? Ans. 3267. 7. What is the square root of 20894041 ? Ans. 4571. 8. What is the square root of 42025 ? Ans. 205. 9. What is the square root of 1014049 ? Ans. 1007. 10. What is the square root of 538 ? Ans. 23.194+. 11. What is the square root of 71 ? Ans. 8.426+ 12. What is the square root of 7? Ans. 2.645+. 13. What is the square root of .1024? Ans. .32. 14. What is the square root of .3364? Ans. .58. 15. What is the square root of .895 ? Ans. .946+ 16. What is the square root of .120409 ? Ans. .347. 17. What is the square root of 61723020.96 ? Ans. 7856.4. 18. What is the square root of 9754.60423716 ? Ans. 98.7654. 282. If it is required to extract the square root of a common fraction, or of a mixed number, the mixed number must be reduced to an improper fraction ; and in both cases the fractions must be reduced to their lowest terms, and the root of the numerator and denominator extracted. Note. — When the exact root of the terms of a fraction cannot be found, it must be reduced to a decimal, and the root of the decimal extracted. EXAMPLES FOR PRACTICE. 1. What is the square root of ? 2. What is the square root of ? Ans. 3. What is the square root of 425? 4. What is the square root of 1283439 1849. ? Ans. 12. 5. What is the square root of 6016 ? 6. What is the square root of 2887 ? Ans. 53. Ans. 25 Ans. $7 Ans. 77. 282. What do you do when it is required to extract the sqyare root of a common fraction, or of a mixed number? APPLICATION OF THE SQUARE ROOT. 283. The square root may be applied to finding the dimensions and areas of squares, triangles, circles, and other surfaces. 1. A general has an army of 226576 men; how many must he place rank and file to form them into a square ? Ans. 476. 2. A gentleman purchased a lot of land in the form of a square, containing 640 acres; how many rods square is his lot? Ans. 320 rods. 3. I have three pieces of land; the first is 125 rods long, and 53 wide ; the second is 621 rods long, and 34 wide ; and the third contains 37 acres ; what will be the length of the side of a square field whose area will be equal to the three pieces ? Ans. 121.11+ rods. 4. W. Scott has 2 house-lots ; the first is 242 feet square, and the second contains 9 times the area of the first; how many feet square is the second ? Ans. 726 feet. 5. There are two pastures, one of which contains 124 acres, and the area of the other is to the former as 5 to 4; how many rods square is the latter? Ans. 157.48+ rods. 6. I wish to set out an orchard containing 216 fruit-trees, so that the number of trees in length shall be to the number of trees in breadth as 3 to 2, and the distance of the trees from each other 25 feet; how many trees will there be in a row each way, and how many square feet of ground will the orchard cover ? Ans. 18 in length; 12 in breadth ; 116875sq. ft. 284. A Triangle is a figure having three sides and three angles. A Right-angled Triangle is a figure having three sides and three angles, one of which is a right angle. 283. To what may the square root be applied ? - 284. What is a triangle? What is a right-angled triangle ? In the triangle, A B C, the angle at B is a right angle ; the longest side, A C, the hypothenuse; the side, A B, on which the triangle stands, the base; and the side, B C, perpendicular to the base, the perpendicular. Hypothenuse Perpendic. A Base. 285. In every right-angled triangle, the square of the hypothenuse is equal to the sum of the squares of the base and perpendicular. с It will be seen, by examining this diagram, that the large square, formed on the hypothenuse A C, contains the same number of sınall squares as the other two counted together. B 286. To find the HYPOTHENUSE, the base and perpendicular being given. RULE. — Add the square of the base to the square of the perpendicular, and extract the square root of the sum. 287. To find the PERPENDICULAR, the base and hypothenuse being given. RULE. — Subtract the square of the base from the square of the hypothenuse, and extract the square root of the remainder. 288. To find the BASE, the hypothenuse and perpendicular being given. RULE. Subtract the square of the perpendicular from the square of the hypothenuse, and extract the square root of the remainder. EXAMPLES FOR PRACTICE. 1. What must be the length of a ladder to reach to the top of a house 40 feet in hight, the bottom of the ladder being placed 9 feet from the sill ? Ans. 41 feet. 284. What is the longest side called? What the other two ? — 285. How does the square of the hypothenuse compare with the base and perpendicular ? How does this fact appear from Fig. 2? - 286. The rule for finding the hypothenuse ?— 287. What for finding the perpendicular?— 288. What for finding the base ? 2. Two vessels sail from the same port; one sails due north 360 miles, and the other due east 450 miles; what is their distance from each other? Ans. 576.2+ miles. 3. The hypothenuse of a certain right-angled triangle is 60 feet, and the perpendicular is 36 feet; what is the length of the base ? Ans. 48 feet. 4. A line drawn from the top of the steeple of a certain meeting-house to a point at the distance of 50 feet on a level from the base of the steeple, is 120 feet in length ; what is the hight of the steeple ? Ans. 109.08+ feet. 5. The hight of a tree on an island in a certain river is 160 feet. The base of the tree is 100 feet on a horizontal line from the river, and is elevated 20 feet above its surface. A line extending from the top of the tree to the further shore of the river is 500 feet. Required the width of the river. Ans. 366.47+ feet. 6. On the edge of a perpendicular rock, whose base is 90 feet, on a level, from a certain road that is 110 feet wide, there is a tower 160 feet high ; the length of a line extending from the top of the tower to a point on the opposite side of the road is 300 feet. What is the elevation of the base of the tower above the road ? Ans. 63.6+ feet. 7. John Snow's dwelling is 60 rods north of the meetinghouse, James Briggs's is 80 rods east of the meeting-house, Samuel Jenkins's is 70 rods south, and James Emerson's 90 rods west of the meeting-house ; how far will Snow have to travel to visit his three neighbors, and then return home? Ans. 428.47+ rods. 8. A certain room is 24 feet long, 18 feet wide, and 12 feet high ; required the distance from one of the lower corners to an opposite upper corner. Ans. 32.3+ feet. 289. A Circle is a plane figure bounded by a curved line, every part of which is equally distant from a point called the center. The Circumference or Periphery of a circle is the line which bounds it. A The Diameter of a circle is a line drawn through the center, and terminated by the circumference; as A B. 289. What is a circle? The circumference of a circle? The diameter ? 290. AU CIRCLES are to each other as the squares of their diameters, semi-diameters, or circumferences. All similar TRIANGLES and other RECTILINEAL FIGURES are to each other as the squares of their corresponding sides. 291. To find the SIDE, DIAMETER, or CIRCUMFERENCE of a surface, which is similar to a given surface. RULE. — State the question as in Proportion, and square the given sides, diameters, or circumferences, and the square root of the fourth term of the proportion will be the required answer. 292. To find the AREA of a surface which is similar to a given surface. RULE. State the question as in Proportion, and square the given sides, diameters, or circumferences, and the fourth term of the proportion is the required answer. EXAMPLES FOR PRACTICE. Ex. 1. I have a triangular piece of land containing 65 acres, one side of which is 100 rods in length; what is the length of the corresponding side of a similar triangle containing 324 acres? Ans. 70.71+ rods. OPERATION. 6 5 : 324::10 02 : 5 0 0 0; ~ 5 000 = 70.7 1+ rods. 2. I have a board in the form of a triangle ; the length of one of its sides is 16 feet. My neighbor wishes to purchase one half the board ; at what distance from the smaller end must it be divided parallel to the base or larger end? Ans. 11.31+ feet. 3. There is a triangular piece of land, the length of one side of which is 11 rods ; required the length of the corresponding side of a similar triangle containing three times as much. Ans. 19.05+ rods. 4. The diameter of a circle is 6 feet, and its area is 28.3 feet ; what is the diameter of a circle whose area is 42.5 feet ? Ans. 7.35+ feet. 5. If an anchor, which weighs 2000lb., requires a cable 3 inches in diameter, what should be the diameter of the cable, when the anchor weighs 4000lb ? Ans. 4.24+ inches. 6. A rope 4 inches in circumference will sustain a weight of 1000lb. ; what must be the circumference of a rope that will sustain 5000lb.? Ans. 8.94+ inches. 290. What ratio do circles have to each other?—291. The rule for finding the side, diameter, &c., of a surface similar to a given surface ? — 292. The rule for finding the area of a surface similar to a given surface ? |