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. a foot rule, measure its circumference.

By rolling it upon

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

its diameter.

times

4. The circumference of a circle is nearly 3 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 diameter 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 of the length of a pendulum that vibrates once a second at the level of the sea in the latitude of London. It is of a meter.

1 fathom (used in measuring the depth of the sea)

1 knot (used in navigation)

1 league (used in navigation)

I hand (used in measuring the heights of horses)

1 chain (used by civil engineers)

1 chain (used by land surveyors)

1

pace (used in measuring approximately)

1 barleycorn (used in grading length of shoes)

1 furlong (a term nearly obsolete)

Denominate Numbers-Linear Measure.

EXERCISE.

1. Mont Blanc is 15810 feet, or about

2. Mt. Everest is 29000 feet, or about

miles high.

miles high.

3. Commodore Dewey opened fire on the enemy at a dis

tance of 5000 yards, or about

or

miles.

4. My horse, measured over the front feet, is 16 hands,

5. The vessel seemed to be about three leagues, or

miles distant.

6. On sounding, they found the depth of the water to be 15 fathoms, or

feet.

7. The cruiser made 20 knots or about

hour.

8. The length of the lot was 36 paces, or about

miles, an

- rods.

1. A seven-foot drive wheel of a locomotive makes how many revolutions to the mile?

2. Which is the longest distance, 5 miles 319 rods 16 feet 6 inches, 5 miles 319 rods 5 yards 1 foot 6 inches, or 6 miles?

3. Reduce 40 rd. 4 ft. 5 in. to inches.

4. Reduce 1100 inches to yards, feet, and inches.