Square Root. NOTE.—Pupils who have mastered the work on the preceding page will readily understand the following process. See rule on page 212. 1. Find the square root of .6. Operation. Observe 1. That in grouping deci."6000(.774 49 mals for the purpose of extract ing the square root it is 70 x 2 = 140 | 1100 7 1029 necessary to begin at the deci mal point. 770 x 2 1540 7100 2. That the square root of 924 any number of hundredths is a number of tenths; the square root of any number of ten-thousandths is a number of hundredths, etc. 1 2. Find the square root of 54264.25. 54264.'25(232.946 4 3 129 2 924 2320 x 2 4640 | 440.25 9418.41 23290 x 2 46580 21.8400 4 18.6336 232940 x 2 465880 3.206400 6 2.795316 .411084 Observe that the trial divisor is always 2 times 10 times the part of the root already found. Square Root. 312. The following numbers are perfect squares. Find their square roots by both the factor method and the method given on the four preceding pages. (1) 6889 (4) 1849 (2) 841 (5) 729 (3) 71824 (a) Find the sum of the six results. 576 (7) out (9) 2 (12) 82% 6 2 5 (13) .81 (15) .04 (18) .0004 313. MISCELLANEOUS. or or or 1. The square of a number represented by one digit gives a number represented by digits. 2. The square of a number represented by two digits gives a number represented by digits. 3. The square of a number represented by three digits gives a number represented by digits. 4. The square root of a perfect square represented by one or two digits is a number represented by — digit. The square root of a perfect square represented by three or four digits is a number represented by digits. 6. The square root of a perfect square represented by five or six digits is a number represented by digits. Square Root. 314. MISCELLANEOUS PROBLEMS. 1. What is one of the two equal factors of 9216 ? 3. If a body of 7921 soldiers were arranged in a solid square, how many soldiers would there be on each side ? 4. How many rods of fence will enclose a square field whose area is 40 acres ? 5. How many rods long is one side of a square piece of land containing exactly one acre ? + 6. If the surface of a cubical block is 150 square inches, what is the length of one edge of the cube ? 7. How many rods of fence will enclose a square piece of land containing 4 acres 144 square rods? 8. Find the side of a square equal in area to a rectangle that is 15 ft. by 60 ft. 9. Compare the amount of fence required to enclose two fields each containing 10 acres : one field is square, and the other is 50 rods long and rods wide. 10. Find the area of the largest possible rectangle having a perimeter of 40 feet. 11. If a square piece of land is of a square mile, how much fence will be required to enclose it? 12. Find the squares of numbers from 10 to 20, inclusive, and memorize them. * To find one of the four equal factors of a number (the 4th root) extract the square root of the square root. Why? What is the fourth root of 81 ? † Find the answer to problem 5, true to hundredths of a rod. Algebra. X = 315. SQUARE ROOT AND AREA. 1. If a piece of land containing 768 square rods is three times as long as it is wide, how wide is it? * Let the width, then 3 x = the length, the area. 16 X = 2. If a certain room is twice as long as it is wide, and the area of the floor 968 square feet, what is the length and the breadth of the room ? 3. One half of the length of Mr. Smith's farm is equal to its breadth. The farm contains 80 acres. How many rods of fence will be required to enclose it? 4. Each of four of the faces of a square prism is an oblong whose length is twice its breadth. The area of one of these oblongs is 72 square inches. What is the solid content of the prism.t 5. The width of a certain field is to its length as 2 to 3. Its area is 600 square rods. The perimeter of the field is how many rods? 6. If of the length of an oblong equals the width and its area is 768 square inches, what is the length of the oblong? 7. If to 24 times the square of a number you add 15 the sum is 375. What is the number? *To solve this problem arithmetically, one must discern that this piece of land can be divided into three equal squares, the side of each square being equal to the width of the piece. + Let no pupil attempt to solve this problem without first bringing into corsciousuess an image of the prism. Algebra. 316. SQUARE ROOT AND PROPORTION. When the same number forms the second and the third term of a proportion it is called a mean proportional, of the first and the fourth term; thus, in the proportion 3:6::6:12, 6 is a mean proportional of 3 and 12. EXAMPLE In the proportion 12 : x : :X : 75, find the value of x. Since the product of the means equals the product of the extremes, x times x equals 12 times 75, or, x2 900. 30. X Find the value of x in each of the following proportions : 1. 9:x ::x: 16. 4. 12:x :: X:48. 3. 8: x ::x: 32. 6. 36 :x : :x: 49. 7. An estate was to be divided so that the ratio of A's part to B's would equal the ratio of B’s part to C's. If A received $8000 and C received $18000, how much should B receive? 8. Find the mean proportional of and 13. 9. The ratio of the areas of two squares is as 4 to 9. What is the ratio of their lengths ? 10. The ratio of the lengths of two squares is as 9 to 16. What is the ratio of their areas ? 11. The area of the face of one cube is to the area of the face of another cube as 16 to 25. What is the ratio of the solid contents of the cubes ? |