For each of the following problems draw a diagram similar to the one on p. 71, showing the longitude and the time of each place. Differences in solar time are to be found. Refer to the table on p. 71 for the longitudes of the cities named.

What is the difference in time between :

1. Greenwich and Manila?

2. Washington and Berlin?

3. San Francisco and Chicago?

4. Berlin and Tokyo?

5. Boston and Paris?

6. Manila and Tokyo?

7. When it is noon in Berlin, what time is it in Manila?

8. When it is 9:30 p.m. in Hongkong, what time is it in Paris? 9. When it is 4 p.m. in Berlin, what time is it in Boston? 10. When it is 1 p.m. in New York, what time is it in Denver? 11. When it is 2:30 p.m. in Chicago, what time is it in Honolulu ?

12. What time is it in New York when it is 2:35 p.m. in San Francisco?

13. What time is it in Manila, when it is 11: 20 a.m. in Bombay?

88. Written.

What is the longitude of Melbourne if its time is 1 hr. 36 min. 4 sec. later than the time of Manila?

Since 15° of longitude correspond to 1 hr. of time, the difference in longitude between two places is 15 times as great as the difference in time. The 1 hr. 36 min. 4 difference in longitude between Melbourne and Manila is 24° 1'.

sec.

15

1. The time of Rome is 49 min. 56 sec. later than that of Greenwich. What is the longitude of Rome?

2. The time of Moscow is 2 hr. 20 min. 56 sec. later than that of Paris. Find the longitude of Moscow.

3. When it is noon in Hongkong, it is 28 min. 39 sec. past 5 a.m. in Vienna. What is the longitude of Vienna ?

4. What is the longitude of a place where it is 44 min. 52 sec. past 8 p.m. when it is 10 p.m. in Tokyo?

5. What is the longitude of a place where it is 14 min. 33 sec. past 8 p.m. when it is 1 p.m. in Boston?

6. When it is noon in New York, it is 55 min. 11 sec. past 10 a.m. in St. Louis. What is the longitude of St. Louis?

7. If a person travels from Tokyo to Manila, will his watch be too fast or too slow, and how much?

8. When it is 44 min. past 11 a.m. in Naples, 14° 15′ 30′′ E., what time is it in Moscow, 37° 34′ 15'' E.?

9. When it is noon, Jan. 1, in San Francisco, what time is it in Manila?

10. When it is half past 3 p.m. in Denver, what time is it in Chicago?

11. The time in Manila is 5 hr. 33 min. 33 sec. faster than in Moscow. What is the longitude of Moscow ?

12. When it is midnight in New York, what time is it in Washington?

13. The time in Boston is 11 min. 44 sec. faster than in New York. What is the longitude of Boston?

14. When it is 2 p.m. in Hongkong, it is 27 min. 8 sec. past 2 p.m. in Manila. in Manila. What is the longitude of Manila?

15. The longitude of Vienna is 16° 20' 15" E., and Vienna time is 11 min. 46 sec. faster than Berlin time. What is the longitude of Berlin ?

89. Oral.

PRACTICAL MEASUREMENTS

Parallel lines are lines which are the same distance apart throughout their length.

A right angle is the angle of a square, or 90°. Its sides are perpendicular to each other.

A polygon is a plane figure bounded by straight lines.

A triangle is a polygon of three sides.

A trapezoid is a polygon of four sides, two of which are parallel.

A parallelogram is a polygon of four sides, opposite sides being parallel and equal.

A rectangle is a parallelogram whose angles are right angles.

The base is the side on which the figure is assumed to stand.

The altitude of a figure is the distance between the base and the highest point opposite.

The diagonal of a figure is a straight line joining opposite angles:

The mere repetition of these definitions will not acquaint the pupils with the figures. The teacher should draw on the board many figures, varying the size and proportions, and test the ability of the pupils to recognize them by name. The pupils should be given practice in drawing them until the mention of a name calls forth an appropriate drawing.

The area of a parallelogram is equal to the product of its base and altitude. The base and altitude must be expressed in units of the same kind.

The diagonal of a parallelogram divides it into two equal triangles, whose bases and altitudes are equal respectively to the base and altitude of the parallelogram.

The area of a triangle is equal to one half the product of its base and altitude.

Give the areas of the following parallelograms:

11. I bought a piece of land in the form of Its length was 244 m and its width 175 m. P 125 a hektar.

12. The area of a rectangle is 6600 sq. cm. 75 cm. What is its length?

a parallelogram. Find the cost at

Its altitude is

13. At P150 a hektar, what will a farm in the shape of a parallelogram cost if its length is 480 m and its width 225 m? 14. I have a piece of land in the form of a parallelogram which contains 3.6 Ha. It is 225 m long. What is its width ?

91. Written.

A trapezoid, such as ABCD, may be divided into two triangles by a diagonal, as AC. The altitude of each triangle is equal to the distance between the parallel sides of the trapezoid, represented by EC. The area of triangle ACD = } of AD × EC. The area of triangle ABC= } of BC × EC. The area of the trapezoid of (AD + BC) × EC.

=

The area of a trapezoid is equal to one half the sum of the parallel sides times the altitude.

Have the pupils draw figures for each of the following examples and problems, approximating the given proportions.

Find the areas of the following trapezoids:

1. Parallel sides 50 cm and 70 cm, altitude 24 cm.

2. Parallel sides 48 m and 52 m, altitude 38 m.

3. Parallel sides 120 m and 160 m, altitude 75 m. 4. Parallel sides 125 m and 140 m, altitude 68 m. 5. Parallel sides 232 m and 188 m, altitude 155 m.

6. Find the area of a field in the form of a trapezoid whose altitude is 120 m and whose parallel sides are 130 m and 180 m.

7. At P145 a hektar, what is the value of a piece of land in the form of a trapezoid whose altitude is 75 m and whose parallel sides are 115 m and 125 m respectively?

8. One side of a field is 142 m long. The side parallel to it is 188 m long. The perpendicular distance between the sides is 82 m. Find the area.

9. What is the area of a board in the shape of a trapezoid, 5.4 m long, 30 cm wide at one end, and 40 cm wide at the other? 10. The area of a trapezoid is 497 sq. m. The altitude is 14 m, and one of the parallel sides is 28.5 m. Find the side to which it is parallel.