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Year Miles

1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 4000 4500 2500 2000 1500 1600 1800 2200 3800 3500

3. Draw a diagram to represent the temperature record given in the table below. Each unit on XX should represent 1 hour, beginning with 7 A.M., while each unit on YY should represent 1°, beginning with 22°.

Time

17 A.M.8 A.M. 9 A.M. 10 A.m. 11 A.m. 12 m. 1 P.M. 2 P.M. 3 P.m. 4 P.M. 5 P.M. 6 P.M.

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4. Draw a diagram similar to that for Ex. 3 to represent the temperature record below. Observe that some of the temperatures here are negative. Wherever this happens, draw the corresponding heavy line downward, instead of upward.

Time

7 A.M. 8 A.M. 9 A.M. 10 A.M. 11 A.M. 12 m. 1 P.M.2 P.M. (3 P.m. 4 P.M. 5 P.M. 6 P.M.

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5. The table for dry measure as learned in arithmetic is

2 pints=1 quart, 8 quarts=1 peck, 4 pecks=1 bushel. Draw a diagram for this, placing the words pint, quart, peck, bushel, each one unit apart, along XX, and letting each unit on YY represent 1 quart.

6. Draw a diagram for the table learned in arithmetic for liquid measure. Proceed as in Ex. 5, taking each unit on YY to represent 1 quart. It is interesting to compare the two diagrams.

118. The Graph. Thus far we have considered diagrams consisting of a series of heavy upright lines. In practice, we do not always draw in the full length of such lines, but we simply mark (by a cross or a dot) their end points, and then draw a smooth line or curve through all such points. This gives what is known as a graph. It will be illustrated by the following examples.

EXAMPLE. The average weight of boys at different ages beginning at 6 years and continuing up to 15 years is given in the table below.

Age | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 Weight 50 53 57 62 67 | 72 | 78 85 93 105

Draw the graph of this table.

SOLUTION. We begin just as in preparing the diagrams in $ 117, letting each unit on XX here represent a year, beginning with 6, and letting each unit on Y Y represent 10 pounds, beginning with 50. But instead of drawing in the heavy upright lines, we simply mark (by the cross) their upper end points, and then draw, free hand, a smooth curve through all these points, giving the result shown in Fig. 63. This curve is the graph of the above table.

119. Use of the Graph. The graph in Fig. 63 not only gives us a picture of the weight of boys at exactly 6 years, 7 years, etc., but also at intermediate ages such as 7.5 years, 8.75 years, etc. For example, at 7.5 years the weight is seen to be about 55 pounds, because when we are at the point 7.5 on XX the height of the curve (as measured on YY) is seen to be about 55. Likewise, for any age whatever between 6 and 15 years the graph gives us instantly a good idea of the corresponding average weight. By the use of such a curve it is easy to tell whether a boy is under or over average weight.

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Note also that the graph grows steeper as it goes to the right, thus indicating that the rate of growth steadily increases as the boy gets to be near 15 years of age.

EXERCISES - GRAPHS

1. The table below shows the average weight of girls between the ages of 6 and 15 years. Draw the graph for this table similar to that for boys shown in § 118.

Age | 6 7 8 9 10 11 12 13 14 Weight 43 | 47 | 52 | 57 | 62 | 69 | 78 | 89 | 98

15 106

2. On the figure which you have drawn for Ex. 1 sketch in also the curve for boys (as given in Fig. 63). In other words, draw both graphs on the same sheet, using the same lines XX and YY for both. Upon comparing the two curves you will see that they cross each other at a certain point, owing to the fact that the curve which starts lowest rises fastest. What is thus brought out about the comparative growths of boys and girls ?

3. Draw the graph showing how one's savings increase if he saves $1.00 a month for 12 months.

(HINT. Each unit on XX represents a month and each unit on YY represents $1.00. The graph turns out to be a straight line.]

4. Draw a graph showing how one's savings increase if he saves $1 the first month, $2 the second month, $3 the third month, and so on for a year. Compare your result with that for Ex. 3 and state what facts are thus brought out to the eye.

5. The water in a glass tube was at 60°. Heat was applied and the temperature of the water was then measured at intervals of five minutes, giving the results shown in the table below.

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Draw the corresponding graph and from it answer (approximately) the following questions : (a) What was the temperature at the end of 12 minutes ? (6) At the end of 27 minutes ?

6. Below are given the times of sunrise at intervals of 15 days for a certain period of the year. Draw a curve (similar to that in § 118) to show its change during that period. Represent the dates along the line XX with 1 unit for every 15 days. Take each unit on the line YY equal to 25 minutes.

Dec. 1 Dec. 16 Dec. 31 Jan. 14 Jan. 29 Feb. 13

Date Nov. 17
Time of
Sunrise

6:54

7:09

7:24 | 7:30

7:27

7:16

| 6:59

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DESCARTES

(René Descartes, 1596-1650) Profound student and ranked as one of the greatest leaders of all time in both mathematics and philosophy. He invented representation by graphs and was thus led to the discovery and development of the branch of mathematics called Analytic Geometry. He was also much interested in medicine and surgery.

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