5. A boy's height each year between the ages of 5 yr. and 15 yr. was found in inches to be as follows: Age 3 14 15 Height 42 44 46 49 52 54 56 58 61 63 68 5 6 7 8 9 10 11 12 13 Draw a graph and from it determine in what part of this period the boy was growing most rapidly. 6. In a certain mill the steam gauge on the boiler showed at 8 A.M. a pressure of 122 lb. to the square inch. The pressure varied through the day as shown by this graph. Write the pressure for every hour from 8 A.M. to 5 P.M. and also for 12.30 P.M. Why was the pressure so high at 8 A.M.? 50 40 30° 122 120 118 116 114 112 110 8 10 12 2 4 7. When the temperature is below zero this fact is indicated by a minus sign. That is, -5° means 5° below zero. This graph shows the changes in temperature from noon on a day in January to noon on the following day. Write the approximate 20° temperatures for every hour of this 10 period, and a statement telling when 0° the temperature fell the most rapidly -10° and when it rose the most rapidly. 12 2 4 6 8 10 12 2 4 6 8 1012 8. The population of the earth by continents in millions is approximately as follows: Africa, 143; North America, 140; South America, 56; Asia, 873; Australasia, 16; Europe, 465. Draw a graph, using that kind of graph which you think shows the facts most effectively. III. ADVANCED MENSURATION Square and Square Root. If a square has a side of 4 units, it has an area of 16 square units. Therefore 16 is called the square of 4, and 4 is called the square root of 16. Square Root applied to Areas. Hence, considering the numbers as abstract, The side of a square is the square root of the area. Symbols. The square of 4 is written 42, and the square root of 16 is written √16. Perfect Square. Such a number as 16 is called a perfect square, but 10 is not a perfect square. We may say, however, that 10 equals 3.16+, because 3.162 nearly equals 10. Square Root of a Perfect Square. Square roots of perfect squares may often be found by simply factoring That is, we separate 441 into its factors and then separate these factors into two equal groups, 3 × 7 and 3 × 7. Hence we see that 3 × 7, or 21, is the square root of 441. By factoring, find the square roots of the following : 11. 121. 15. 729. 17. 1296. 19. 12.25. 16. 576. 18. 1089. 20. 40.96. Find the sides of squares whose areas are as follows: 33. From a corner of a square piece of land that contains 57,600 sq. ft. there is cut a square lot containing 6400 sq. ft. Draw a plan of the lots and find the perimeter of each lot. 34. A square lot contains 18,225 sq. ft. What is the perimeter of the lot? What is the perimeter of a square lot of four times this area? The second perimeter is how many times the first? 35. Two square building lots front on the street. The area of the two together is 8900 sq. ft., that of the larger being 6400 sq. ft. What is the frontage of each lot? 36. A square lot has an area of 28,900 sq. ft. What is the perimeter of the lot? What is the perimeter of a square lot that contains nine times this area? The perimeter of the second lot is how many times the perimeter of the first? Square of the Sum of Two Numbers. Since 4740 +7, the square of 47 may be obtained as follows: This relationship is conveniently seen in the annexed figure. Every number consisting of two or more figures may be regarded as composed of tens and units. Therefore, The square of a number contains the square of the tens, plus twice the product of the tens and units, plus the square of the units. This principle, important in square root, should be clearly understood both from the multiplication and from the illustration. Separating into Periods. Since 1= 12, 100=102, 10,000 = 1002, it is evident that the square root of any number between 1 and 100 lies between 1 and 10, and that the square root of any number between 100 and 10,000 lies between 10 and 100. In other words, the square root of any integral number of one figure or two figures is a number of one figure; the square root of any integral number of three or four figures is a number of two figures; and so on. Therefore, if an integral number is separated into periods of two figures each, from the right to the left, the number of figures in the square root is equal to the number of the periods of figures. The last period at the left may have one figure or two figures. Finding the Square Root. Required the square root of 2209. If the figures of the number are separated into periods of two figures each, beginning at the right, we see that there are two integral places in the square root of the number. The first period, 22, contains the square of the tens' number of the root. Since the greatest square in 22 is 16, then 4, the square root of 16, is the tens' figure of the root required. 22 09 (47 16 80 6 09 87 6 09 Subtracting the square of the tens, the remainder contains twice the tens x the units, plus the square of the units. Dividing by twice the tens (that is, by 80, which is 2 x 4 tens), we find approximately the units. Dividing 609 by 80 (or 60 by 8), we have 7 as the units' figure. Since twice the tens x the units, plus the square of the units, is equal to (twice the tens + the units) x the units, that is, since 2 × 40 × 7+72 = (2 × 40 +7) x 7, we add 7 to 80 and multiply the sum by 7. The product is 609, thus completing the square of 47. Checking the work, we find that 472 is 2209. FINDING SQUARE ROOTS 3. 3969. Find the square roots of the following: 1. 3249. 2. 3721. Find the sides of squares that have the following areas: 4. 5041. Find the square roots of the following fractions by taking the square roots of the terms of the fractions: 11. 1. 12. 961 13. 1089 1156' 14. 1936 5041 |