EXAMPLES. XLVIII.

Find by duodecimals the areas of rectangles having the following dimensions:

5.

6.

7.

5 feet 7 inches, 4 feet 10 inches.

5 feet 11 inches, 4 feet 7 inches.

4 feet 3 inches 4 twelfths, 3 feet 3 inches.

8. 4 feet 8 inches 5 twelfths, 3 feet 4 inches.

9. 5 feet 4 inches 8 twelfths, 2 feet 7 inches 3 twelfths.

10. 6 feet 8 inches 7 twelfths, 3 feet 4 inches 5 twelfths.

Find by duodecimals the volumes of rectangular parallelepipeds having the following dimensions:

11. 3 feet, 3 feet, 1 foot 6 inches.

12. 5 feet, 3 feet, 2 feet 3 inches.

13. 4 feet, 3 feet 4 inches, 3 feet 3 inches.

14. 5 feet, 4 feet 8 inches, 3 feet 2 inches.

15. 6 feet 3 inches, 5 feet 3 inches, 3 feet 9 inches. 16. 7 feet 5 inches, 6 feet 7 inches, 3 feet 10 inches.

XLIX. METRICAL SYSTEM.

436. The French system of measures called the metrical system is frequently used in English scientific works; so that we shall here explain the system.

437. The standard of length is the metre which is equal to 39 37079 English inches. The metre was intended to be one-ten-millionth part of the distance from the pole to the equator measured on the Earth's surface; recent investigations shew that the metre is about of an inch shorter than it should have been to correspond to this intention.

1

208

The standard of area is the are, which is 100 square metres.

The standard of volume is the stere, which is a cubic metre.

All the multiples and the sub-divisions of any measure are decimal and are formed in the same manner; the multiples by syllables derived from the Greek, and the subdivisions by syllables derived from the Latin. Thus

Similarly a hectare = 100 ares, a centiare=.

1

dekastre 10 steres, a decistere:

stere.

10

1

are, a

100

For liquid measures the standard is a litre, which is a cubic decimetre.

For weight the standard is a gramme, which is the weight of a cubic centimetre of water: it is equal to 15.432 English grains.

EXAMPLES. XLIX.

1. The diameter of a circle is 15 metres: find the circumference.

2. Find the area of a rectangle which is 407.75 metres long, and 304 metres wide.

3. The parallel sides of a trapezoid are 157'6 metres and 94 metres; and the perpendicular distance between them is 72 metres: find the area.

4. Find in cubic metres the volume of a wall which is 48 metres long, 3'4 metres high, and 45 of a metre thick.

5. Find to the nearest cubic decimetre the volume of a right cone, the height being 24 metres, and the radius of the base 4 of a metre.

6. A vessel when empty weighs 1.67 kilogrammes, and when full of water weighs 6.9 kilogrammes: find the capacity of the vessel in cubic decimetres.

7. A cistern is 85 metres long, 6 metres wide, and 9.2 metres deep: find how many hectolitres of water it will hold.

8. The weight of water which a certain cylinder will hold is 36 kilogrammes; the radius of the cylinder is 15 centimetres: find the height of the cylinder to the nearest centimetre.

9. A telegraph wire is 35 kilometres long, and 2 millimetres in diameter: find the volume in cubic decimetres.

10. The diameter of an iron ball is 2 of a metre: find its weight in kilogrammes, supposing that any volume of iron is 7.5 times as heavy as an equal volume of water.

11. Find to the nearest centimetre the area of the whole surface of a right cone, the radius of the base being 3 of a metre, and the slant height 8 of a metre.

12. Find to the nearest centimetre the area of the surface of sphere of which the diameter is 9 of a metre.

13. Shew that an acre contains about 40'47 ares. 14. Shew that a cubic yard contains about 764.5 cubic decimetres.

15. Shew that a gallon contains about 4:54 litres.

The following examples involve the extraction of the cube root:

16. A hollow sphere holds a litre: find the radius of the sphere.

17. A vessel in the form of a right circular cylinder with its height equal to the diameter of its base holds a litre find the height.

18. The volume of a right circular cone with its height equal to the radius of its base is a cubic decimetre: find the height.

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MISCELLANEOUS EXAMPLES.

1. THE top of a may-pole, being broken off by a blast of wind, struck the ground at a distance of 15 feet from the foot of the pole: find the height of the whole may-pole, supposing the length of the broken piece to be 39 feet.

2. Find how many square feet there are in 1200 square inches.

3. The area of two squares is 100 acres, and the side of one is three times as long as the side of the other: find the area of each.

4. A square contains 2533 feet 64 inches: find its side. 5. The perimeter of a rectangle is 144 yards, and the length is three times the breadth: find the area.

6. Find the expense of carpeting a room 21 feet long and 20 feet broad, with carpet 27 inches wide at 4s. 6d. per yard.

7. A room is 16 feet 2 inches long, 15 feet 3 inches broad, and 12 feet high: find the expense of covering the walls with paper, 9 inches wide, at 2 d. per yard.

8. A building has 63 windows: 40 of them contain 12 panes each 20 inches by 16; the others contain 9 panes each 16 inches square: find the cost of glazing the whole at 2s. 3d. per square foot.

9. The sides of a triangle are 890, 990, and 1000 links: find the area.

10. The area of a trapezium is 475 square feet; the perpendicular distance between the two parallel sides is 19 feet: find the two parallel sides, their difference being 4 feet.

11. Two roads cross at right angles; two men start at the same time from the point where the roads meet, one man walking along one road at the rate of 4 miles an hour,