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Algebra.

327. METRIC UNITS IN ALGEBRAIC PROBLEMS. 1. I am thinking of a rectangular surface. Its length is 5 times its breadth. Its area is 45 square decimeters. How long and how wide is the surface ?

NOTE.-Let x = the number of decimeters in the breadth of the surface.

2. I am thinking of a triangular surface. Its base is three times its altitude. Its area is 8.64 square meters. What is the length of its base?

3. I am thinking of a cube whose entire surface is 150 square centimeters. What is the length of one of its edges ?

4. The perimeter of a certain rectangle is 20.4 meters. Its length is twice its breadth. (a) Find its length and breadth. (b) Find its area.

5. The difference in the weight of two lead balls is 24 grams. The united weight of the two balls is 1 kilogram. (a) Find the weight of each ball. (b) Does the heavier ball weigh more or less than 1 pound?

6. A merchant had three pieces of lace. In the second piece there were twice as many meters as in the first. In the third piece there were 6 meters more than in the second. In the three pieces there were 106 meters. (a) How many meters in each piece? (b) Were there more or less than 43 yards in the second piece?

7. John weighs 3.6 kilograms more than Henry. Together they weigh 83.6 kilograms. (a) Find the weight of each boy.

(b) Does John weigh more or less than 90 pounds ?

Algebra.

328. METRIC UNITS IN ALGEBRAIC PROBLEMS.

1. A ball rolling down a perfectly smooth and uniformly inclined plane rolls 3 times as far the 2nd second as the 1st ; 5 times as far the 3rd second as the 1st ; 7 times as far the 4th second as the first. If in 4 seconds it rolls 192 decimeters (a) how far did it roll in the 1st second ? (b) in the 4th second ? (c) Did it roll more or less than 48 inches in the first second ?

2. I am thinking of a right-triangle. Its altitude is to its base as 3 to 4. The sum of its altitude and base is 14 centimeters. (a) Find the altitude. (b) Find the base. (c) Find the area. (d) Find the hypothenuse. (e) Is the hypothenuse more or less than 4 inches ?

3. A freely falling body falls three times as far the 2nd second of its fall as it does the 1st second. In two seconds it falls 19.6 meters.* (a) How far does it fall in the 1st second ? (b) in the 2nd second ?

4. A freely falling body falls 3 times as far the 2nd minute of its fall as it does the 1st minute. In 2 minutes it falls 70560 meters. * (a) How far does it fall in the 1st minute? (b) In the 2nd minute? (c) 70560 meters equals how many kilometers ? (d) 70560 meters equals (approximately) how

many miles ?

5. A freely falling body falls 3 times as far the 2nd halfsecond as it does the 1st half-second. In one second it falls 4.9 meters. (a) How far does it fall in the 1st half-second? (b) in the 2nd half-second?

* In such problems the resistance of the air is not considered.

Geometry.

329. THE CIRCUMFERENCE OF A CIRCLE. 1. Cut a 3-inch circle from cardboard. By rolling it upon a foot rule, measure its circumference.

2. Measure the diameter of a bicycle wheel; then by rolling it upon the ground or upon the school-room floor, measure its circumference.

3. In a similar manner measure the diameters and the circumferences of other wheels until you are convinced that the circumference of a circle is a little more than times its diameter.

4. The circumference of a circle is nearly 34 times the diameter; more accurately, it is 3.141592+ times the diameter.

NOTE.—It is a curious fact that the diameter of a circle being given in numbers it is impossible to express in numbers its exact circumference. The circumference being given in numbers it is impossible to express in numbers its exact diameter. In other words, the exact ratio of the circumference to the diameter is not expressible.

5. Find the approximate circumference of a 5-inch circle; of a 7-inch circle ; of a 10-inch circle. *

6. Find the approximate diameter of a circle that is 6 ft. in circumference. *

7. The circumference of a 6-inch circle is how many times the circumference of a 3-inch circle ?

8. The diameter of a circle whose circumference is 12 inches is what part of the diameter of a circle whose circumference is 24 inches ?

*In the solution of such problems as these the pupil may use, as the approxi. mate ratio of the circumference to the diameter, 3.14.

330. MISCELLANEOUS REVIEW.

1. Find the approximate circumference of a circle whose dianieter is 3.4 meters.

2. Find the approximate diameter of a circle whose circumference is 3.4 decimeters.

3. Find the approximate diameter in yards of a circular 1-mile race track; of a half-mile track.

4. Find the approximate diameter in meters of a circular 1-kilometer race track ; of a half-kilometer track.

5. A 28-inch bicycle wheel will make how many revolutions in running one mile ?

6. A 70-centimeter bicycle wheel will make how many revolutions in running 1 kilometer ? *

See table on page 226 and give approximate answers to the fol. lowing:

7. Forty meters are how many yards? 8. Forty yards are how many meters ? 9. Forty kilometers are how many miles ? 10. Forty miles are how many kilometers ?

11. Forty ars are how many acres ? 12. Forty acres are how many ars?

13. Forty sters are how many cords?
14. Forty cords are how many sters?
15. Forty liters are how many quarts ?
16. Forty quarts are how many liters ?
17. Forty kilograms are how many pounds ?
18. Forty pounds are how many kilograms?

*The exact answer to such questions as this cannot be found: but the approxi mation is practically correct.

DENOMINATE NUMBERS.

Linear Measure.

Note.—In part to provide for ready reference, and in part to give further application of the principles presented on the preceding pages, the remaining pages of this book are devoted chiefly to denominate numbers.

331. The English and United States standard unit of length is the Imperial yard arbitrarily fixed by Act of Parliament and afterward adopted in the United States. It is about list of the length of a pendulum that vibrates once a second at the level of the sea in the latitude of London. It is 639 of a meter.

TABLE.

16] feet

12 inches (in.) 1 foot (ft )
3 feet

1 pard (yd.)
54 yards 1 rod (rd.)

1 rod.
320 rods

1 mile (mi.)
1760 yards =1 mile.
5280 feet

1 mile.
1 fathom (used in measuring the depth of the sea) = 6 feet.
1 knot (used in navigation)

1.15 + miles. 1 league (used in navigation)

3 knots. 1 hand (used in measuring the heights of horses) 4 inches. 1 chain (used by civil engineers)

100 feet. 1 chain (used by land surveyors)

66 feet. 1 pace (used in measuring approximately)

Í of a rod. 1 barleycorn (used in grading length of shoes) - } of an inch. 1 furlong (a term nearly obsolete)

= ; of a mile.

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