10. What is the square root of 17 ? NOTE.-14. 11. What is the square root of 101% ? 14. What is the square root of 94 15. What is the square root of % of 31? OPTIONAL EXAMPLES. Ans. Ans. 31. Ans. .86602+. Ans. 2.8635+. Ans. 3.02334+. Ans. 1. NOTE. - Extract the root in the following to five places. 392. PRACTICAL EXAMPLES. 1. There is a field of corn having an equal number of rows and hills in a row, which contains 1020100 hills in all; what is the number of rows in the field? Ans. 1010 rows. 2. A body of troops, consisting of 2601 men, has an equal number in rank and file; how many are there in each? Ans. 51 men. 3. A company of persons spent $3.24; each person spending as many cents as there were persons, how many cents did each spend? Ans 18 cents. 4. What is the length of one side of a square farm containing 302 acres, 2 roods of land? Ans. 220 rods. 5. What is the length of a square park which contains 2 square miles? Ans. 1.4142+ miles. 6. There is a circular lot which contains 3 acres; what is the length of a square lot whose area is the same? Ans. 21.9089+ rods. 7. What is the size of a square lot whose area is thirty times that of the above? Ans. 120 rods. 8. What is the cost of fencing a square lot which contains 1 acre, at $5 per rod? Ans. $252.98. 9. The side of a square is 8 ft. 6 in.; what is the side of a square having 25 times the area ? 10. A owned a lot of land 51 rods by 80 rods, and another 180 rods by 100 rods, which he bartered with B for a square lot containing 138 acres; how many rods less of fencing are there in the square lot than in the other two? Ans. 228 rods nearly. 11. I have two square lots of land, the larger of which contains 270 acres; the ratio of the smaller to the larger is as 5 to 6; what is the length of one side of the smaller? Ans. 189.73+ rods 12. On a roof there are laid 5000 slates,—the number in the length being twice the number in the breadth; what is the number each way? NOTE. It is evident that the slates are laid in two equal squares; hence the square root of of 5000 (V of 5000) will equal the breadth. Ans. 50 slates in breadth; 100 slates in length. 13. Suppose the above roof to have had 10000 slates, and the breadth to have been one third of the length, what would have been the number of slates in the length and breadth? Ans. 173.205+ length; 57.735+ breadth. 14. What is the difference between the fencing of a 34-acre lot, whether it be a square or a rectangular lot, twice as long as it is wide? Ans. 17.89 rods. 15. My orchard contains 5400 trees; the number of trees in width is to the number in length, as 2 to 3; what is the number each way? NOTE.—of the trees will be a square, whose square root will be the number of trees in the width of the orchard. 16. Suppose, in the above orchard, the outer rows of trees to stand upon the boundary line, and all to stand 30 feet apart, what is the area covered by the orchard? Ans. 108119 acres. 17. There is a rectangular court paved with 1728 paving. stones 15 inches square; the length of the court is to the width as 4 to 3; what is the number of stones each way? 18. How many square feet in the superficial contents of the above court? 19. What is the side of a square that will contain as many square feet as a rectangle whose sides are 150 and 70 feet? 20. What is the mean proportional between 6 and 24? (Art. 373.) APPLICATION OF SQUARE ROOT TO RIGHT-ANGLED TRIANGLES DEFINITIONS. 393. An Angle is the opening between two lines that meet each other. 394. A Right Angle is the angle formed by two lines that are perpendicular to each other. (Art. 191.) 395. A Triangle is a figure having three angles, and bounded by three straight lines. 396. A Right-angled Triangle is a triangle having one of its angles a right angle. 397. The Hypothenuse of a Right-angled Triangle is the side opposite the right angle. 398. The Base of a Right-angled Triangle is the side upon which it is supposed to stand. 399. The Perpendicular of a Right-angled Triangle is the side perpendicular to the base. 400. TO FIND EITHER SIDE OF A RIGHT-ANGLED TRIANGLE, THE OTHER TWO SIDES BEING KNOWN. Suppose the figure A B C to be a right-angled triangle, whose sides are 3, 4 and 5 feet respectively. A square formed upon the hypothenuse, A C, will contain 25 square feet; one formed upon the base, B C, will contain 16 square feet, and one formed upon the perpendicular, A B, will contain 9 square feet. Thus, it appears that the square upon the line A C is equal to the two squares upon A B and B C; and generally, A B C The square upon the hypothenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. Hence, RULE I. To find the hypothenuse, the base and perpendicular being given: Square the base and perpendicular, and extract the square root of their sum. RULE II. To find the base or perpendicular, the hypothenuse and other side being given: Square the hypothenuse and the given side, and extract the square root of their difference. 401. EXAMPLES. 1. The base of a right-angled triangle being 30 feet, the perpendicular 40 feet, what is the hypothenuse? Ans. 50 feet. 2. The hypothenuse of a right-angled triangle being 32.5 feet, the base 30 feet, what is the perpendicular? Ans. 12.5 feet. 3. What must be the height of the eaves of a house that can be reached by a ladder 30 feet long, the foot of the ladder standing 18 feet from the underpinning of the house? Ans. 24 feet. 4. How far from the foot of a post 15 feet high can a horse feed that has a rope fastened around his neck and attached to the top of the post, the distance being 37 feet to the neck, and the horse feeding two feet beyond the end of the rope in a direct line with the rope? Ans. 36 feet. 5. G. W. Bailey had a tree, which being partially broken off 24 feet from the ground, the top struck the ground 10 feet from the foot of the tree, and on a level with it; what was the height of the tree? Ans. 50 feet. 6. What must be the length of a ladder to reach to the top of a chimney 48 feet high, the foot of the ladder being 20 feet from the chimney? Ans. 52 feet. 7. If the top of the ladder mentioned above be lowered 6 feet, how far will the foot stand from the chimney? 8. Two vessels start at the same point, and sail, one due south 6 degrees, and the other due east 8 degrees; how many miles apart are they, reckoning 69 miles to a degree? 9. What is the width of a street from a point in which a ladder 32 feet long will reach a window 26 feet high on one side, and one 24 feet high on the other side? Ans. 40.85+. 10. What is the width of a common, on which stands a flagstaff 195 feet high, from the top of which to one side of the common is 675 feet, and to the other 360 feet? 11. How far from the foot of a flagstaff 24 feet high, must a ladder 23 feet long be placed that a person may ascend to within 5 feet of the top? 12. My house is 40 feet wide, and the ridge-pole is 15 feet above the middle of the beam which connects the eaves; what is the length of the rafters? 13. Provincetown, Erie, and Elmira are in nearly the same latitude; suppose Elmira to be 243 miles directly north of Washington, Erie to be 305 miles north-westerly, and Provincetown 380 miles north-easterly, how far is Provincetown from Erie? 14. Four persons, Messrs. Ames, Barnes, Carnes, and Doane, are residing around Cincinnati, as follows: Ames, 20 miles north; Barnes, 60 miles east; Carnes, 27 miles south; and Doane 36 miles west of the city; what is the shortest distance one of these persons must travel to visit all the rest, and reach his own home? 15. What is the length of the diagonal, that is, the distance from one corner to the opposite corner, of a square lot which contains 16 square rods? Ans. 5.6568+ rods. |
