SQUARE ROOT-GENERAL METHOD 209 The Extraction of the Square Root of Any Number. methods so far used for finding square root are sufficient for the easier cases; the following method is more general: 1. Point off the number into periods of two figures each, beginning at the right (at the decimal point in a decimal). 2. By inspection find the largest integer whose square is not greater than the left period. (In Example A it is 9.) Root (A) 92 3. Use this integer as the first digit of the root. Subtract its square from the left period. (In Number 84'64 Example A this square is 81.) 4. Bring down the next period. (In Example A this makes 364.) 18 81 364 182 364 5. Multiply the part of the root already found by 2. This number is called the trial divisor. (18 in Example A.) 6. Divide the remainder (omitting the right digit) by the trial divisor and use the digit found as the next digit of the root. (In Root Example A, 36 ÷ 18 = 2.) 7. Annex this digit to the trial divisor. This forms the complete divisor. (182 in A.) 8. Multiply the complete divisor by the digit of the root just found and subtract. NOTE.-It may happen that the product to be subtracted is larger than the number from which it is to be subtracted. This indicates that the trial divisor led to too large a digit. Try the next smaller digit. (B) 3 0.69 Number 9'41'.87'61 6 9 41 60 4187 606 3636 612 55161 55161 (C) 6129 Root 1 4. 1 7+ 9. Repeat the steps 4 to 8 until all the Number 2'00. periods have been brought down. If the last remainder is zero, as in Example B, the process is ended, the given number is a perfect square, and its root has been found exactly. If the last remainder is not zero, as in (C), the process may be continued as far as desired by supplying zeros. the given number. Square Root of Fractions. Either reduce the fraction to a decimal or extract the square root of both terms. Written. Extract the square root of each number to two decimal places: C b 12 13. Find the length of c, according to a05 the figure. Also, find a, if c were .17 in. and b were .08 in. Find the other side of the following right-triangles; if there is a decimal part, carry the value to one decimal place: 19. Measure the length and width of a book cover and compute the diagonal to .1 in. Test the result by measurement. 20. Find the length to .01 ft. of a diagonal of a square 1 ft. on a side. 21. This is the picture of a cube 1 ft. on an edge. Using the result of Exercise 20, state 40 ft C 40ft the length of AB. A the length of the rafter R used for this roof. GRAPHICAL PROBLEMS 211 The approximate number of acres to each inhabitant in various parts of the world is shown in the following diagram: For example, there are 81 acres of American territory to each inhabitant of America. 1. What is the square root of the number in each square? 2. Measure a side of each square. Taking in. as the unit, what number represents the side of each square? Are the squares correctly drawn? Written. 3. China has approximately a population of 350 millions, British India 278 millions, and Japan 40 millions. Extract to the nearest unit the square roots of 350, 278, and 40, and construct squares to represent these populations. 4. Find to the nearest mile the length of the sides of squares having the same area as: Using in. to represent 1 mi. draw lines to represent the lengths found; then draw squares to represent the areas. 5. In 1900, 26,120,000 of the inhabitants of the United States were under 15 years of age; 44,800,000 were from 15 to 59 years of age, and 4,870 were 60 years of age or By extracting the square roots of 2,612, 4,480, and 487 to the nearest unit, find the lengths of the sides of squares to represent these numbers. Draw the squares. over. FORM STUDY AND MEASUREMENT VOLUMES OF SPHERES 1. Henry measured the diameter of a croquet ball and found it to be 4 in. Explain from the figure how it was done. 2. He filled a rectangular dish of base 11 in. by 16 in. with water to a depth of 5 in. When the ball was immersed in the water the depth was found to be 5 in.; what was the volume of the ball? 3. If convenient, perform a similar experiment. 4. Louise modeled from clay a cylinder and a sphere with the dimensions given. The clay cylinder weighed 30 oz. and the sphere 20 oz.; the weight of the sphere was what part of that of the cylinder? 5. If convenient, perform a similar experiment. 3 IN 3IN 3IN 6. Find the volume of a cylinder whose base has a diameter 2r and whose altitude is 2r. 7. According to Exercise 4, what part of this volume is the volume of a sphere of diameter 2r? What represents the volume of the sphere? The volume of any sphere of radius r is 8. What is the value of Tr3, taking r = 8 and π = 3.1416? 9. What is the volume of a sphere whose radius is 7 in.? 10. Find the volume of a sphere of radius 3 ft.; 1 yd.; 2 in.; 4 in.; 5 ft. 9. Regarding the earth as a sphere of radius 4,000 miles, find its volume. (Use 22 as an approximate value for π.) 10. The gilding of a ball on a church spire cost $126.72 at $1.12 per square foot. Find the diameter of the ball. II. The inner surface of the dome of the capitol at Washington is covered by a fresco painting. Regarding the surface as a hemisphere 165 ft. in diameter, how many square feet are there in the painting? 12. A balloon has the form of a sphere 10 yd. in diameter; how many square yards of material are needed to make the balloon? 13. How many cubic feet of gas are required to inflate the balloon? 14. What is the cost of inflating the balloon with gas costing $1.40 per 1,000 cu. ft.? 15. A kettle is in the form of a hemisphere 28 in. in diameter; how many cubic inches of water does it hold? 16. On a church spire there is a ball 2 ft. in diameter; what will it cost to gild the ball at $1.20 per square foot? 17. What represents the volume of a hemisphere of radius r? What represents the volume of of a sphere of radius r? |