6. What is the effect upon the area of a circle of trebling its diameter ? [HINT. Let D be the first diameter. Then 3D will be the second diameter.) 7. Compare the volume of a sphere whose diameter is 1 inch with that of a sphere whose diameter is 2 inches. 8. If a bottle of a certain shape holds 1 pint, how much will a similar bottle half as high hold ? 9. A man whose eye is 5 ft. 6 in. above the ground sights over the top of a 12-foot pole and just sees the top of a tree. If he is 7 ft. from the pole and 63 ft. from the tree, how high is the tree? [Hint. First draw a figure.] 10. The figure represents a kind of compasses used by draftsmen. By adjusting the screw at 0, the lengths OA and OC, and the corresponding lengths OB and B A OD, may be changed proportionally. If OA=3 in. and OC=5 in., what part of the opening CD will the opening AB be? 116. Mean Proportional. If the means of a \ proportion are equal, either mean is called the mean proportional between the extremes. Fig. 57. · Thus, in the proportion 1:2 = 2:4, we have 2 as the mean proportional between 1 and 4. Likewise, in the proportion 18:6 = 6:2, 6 is the mean proportional between 18 and 2, and in the proportion 24: x3 = 23: x2, we have x: as the mean proportional between x4 and x2. EXERCISES — MEAN PROPORTIONAL SOLUTION. Let x be the mean proportional. Then, 6/x = x/24. Whence, x2=144, and x=12. Ans. Find the mean proportional between the two numbers given in each of the following exercises. 2. 3 and 27. 4. 4 a and 9 a. 6. 1 and (a+b)?. 8. In the semicircle ABC, suppose a line CD drawn perpendicular to AB. Then (as shown in geometry) the length of CD will be a mean proportional between the lengths AD and DB. If AD=2 inches and DB=18 inches, find CD. 9. In Fig. 58, suppose AB=29 feet, and AD=4 feet. What is Fig. 58. CD? 10. The figure shows a circle and a point P outside it from which are drawn two lines PT and PS. The first of these lines just touches the circle and is called a tangent, while the second line cuts through the circle at two points R and S and is called a secant. In all such cases, the tangent PT is a mean proportional be tween the whole secant PS and its Fig. 59. external part PR (as shown in ge ometry). Find the length of PT if PR=9}, and RS=50%. CHAPTER XII GRAPHICAL REPRESENTATION 117. Use of Diagrams. A few examples will show how diagrams are often used in everyday life to bring facts clearly before the eye. EXAMPLE 1. A branch of the Y. M. C. A. wished to let people know of its progress in collecting money for a new Fig. 60. building. It placed a large signboard on the street and after ten days the board had the appearance shown in Fig. 60. EXPLANATION. Two lines XX and Y Y had been drawn perpendicular to each other and each had been divided into equal units, beginning at the point where the lines cross. The points of division were numbered 1, 2, 3, etc., as on a yardstick. Each unit on XX represented one of the days during which the money had been coming in, while each unit on YY represented $100. Starting at the point marked 1 on XX, the secretary of the Y.M.C. A. had drawn a heavy line extending upward until its end was on the level with the point marked 2 on YY. This indicated that on the first day just $200 was received. Similarly, he had drawn a heavy line beginning at the point marked 2 on XX and extending upward 21 units as measured on YY. This indicated that on the second day the amount received was $250. In the same way, he had drawn a heavy line upward corresponding to each of the 10 days. Read for yourself (from the scale on YY) the amount received on each day after the second. EXAMPLE 2. Figure 61 shows the home expenses of a small family for a period of twelve months, beginning with January. Each unit on XX represents 1 month, while each unit on YY represents $1. Read for yourself the amount spent during each of the twelve months. EXAMPLE 3. Figure 62 shows the number of miles of railway built in the United States from 1881 to 1890. Each unit on XX represents a year (beginning with 1881), while each unit on YY represents 500 miles. Read off (as near as you can) the number of miles built during each of the nine years Miles Built 71 unit 5500 miles) T EXERCISES In the three Examples of § 117 a diagram was given each time and you were asked to read off from it the facts which it expressed. In the Examples below this is reversed; that is, the facts are given first in a table and you are asked to draw the corresponding diagram yourself. 1. The table below shows the money spent each month of the year by a small family for clothing. Draw a diagram (similar to Fig. 61) to represent the facts here given. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nov. Dec. $ 20 / $ 10 / $ 12 / $ 20 $ 15 | $8 | $9 $ 20 $7 | $20 | $28 $ 10 2. The miles of railway built in the United States from 1891 to 1900 is shown in the table below. Draw a diagram (similar to Fig. 62) to represent this table, indicating clearly the numerical values in the margin, as in Fig. 62. † The pupil will find it to his advantage to secure at this point paper ruled in small squares. Such paper is called squared paper, or coördinate paper. It may be found at most stationery stores. |