The algebraic discussion, in brief, is as follows: Let t represent the tens of a root, and u the units. seen (page 310), a square number equals + 2t × u proceed to find from this expression its root, t + u. 4. Dividing 2t × u by 2t (twice the tens), we obtain u, the units of the root. 5. Adding u to the divisor, 2t, we have 2t+u. 6. Multiplying 2t + u by u, we have 7. Subtracting, we have remaining Therefore the root of t2 + 2t × u + u2 is t + u. Brief directions are: 2txu + u3. 0 1. Point off the number into two-figure periods. 2. Find in the first period the greatest square and its root. 3. Subtract and annex the next period for a remainder. 4. Divide the remainder by twice this root to find the second figure of the root. 5. Annex the quotient to both root and divisor. 6. Multiply by the units. 7. Apply (3), (4), (5), and (6) again, if necessary. To apply the rule : Find the square root of 53,361. 1. What is the square root of 1? √1 = 1; for 1 × 1 = 1. 2. What is the square root of ? √ }=}; for} ×}=}. Hence, √ Fraction 3. Find the square root of .25. ✓ Numerator = ✓ Denominator By rule: .40'00 (.632 123) 400 369 1262) 3100 2524 Hence, V.4.632+. In such cases annex ciphers and make periods from the point toward the right. SQUARES. Since the area of a square lot whose side is 12 rods equals 12 X 12 or 144 square rods, a side of the lot = V 144. Hence the formula : Side of Square ✓ Area. PROBLEMS. 1. What is the side of a square whose area is 1225 sq. ft.? 2. What is the side of a square whose area is 2025 sq. rd.? 3. What is the side of a square farm containing 40 A.? 4. A square plot of ground contains 320 A. How many feet long is each side? 5. A circular pond has an area of 529 sq. rd. What is the side of a square of equal area? 6. If an acre of land be laid out in a square farm, what will be the length of each side in rods? 7. To arrange 7225 men in the form of a square, how many men must be put in each line? 8. What would it cost to fence a square lot containing 640 A. at $4.00 per rod? 9. If it cost $312 to enclose a field 216 rd. long and 24 rd. wide, what will it cost to enclose a square field of equal area with a like kind of fence? 10. The attempt to form a square of 10,200 men excluded 200 of the men. How many men stood in each line of the square? 11. If the faces of a cubical box measure 23,064 sq. in., how many linear inches in one of its edges? 12. Which will cost the more to fence, a field measuring 40 by 80 rd. or a field of the same area in the form of a square? How much more at $1.333 per rod? TRIANGLES. A Triangle is a figure bounded by three straight lines. A Right Triangle has one right angle. h denotes the hypotenuse, the side opposite the right angle; p, the perpendicular; b, the base. These three lines are so related that h2 = b 1. The base of a right triangle is 10 feet, its perpendicular 15 feet. Find its hypotenuse. h. = √b2 + p2 = √100 + 225 = √325 = 18 very closely, 2. Find the sides indicated by x in the table, using formulæ 1, 2, and 3. No. 3. b Vh2 — p2 = = √64-49=1/153.87+. 3. The perpendicular of a right triangle is 30 ft. and the hypotenuse is 50 ft. What is the base? 4. A square floor contains 400 sq. ft. Find the length of the longest straight line that can be drawn thereon. 5. A tree 150 ft. high stood on the bank of a stream. A part broken off 125 ft. from the top exactly measured the distance to the opposite bank. How wide was the stream? 6. How far from a tower 40 ft. high must the foot of a ladder 50 ft. long be placed that it may exactly reach the top of the tower? |