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483. To find the convex surface of a frustum of a pyramid or of a cone.

1. What is the convex surface of a frustum of a square pyramid, whose slant height is 7 ft., each side of the greater base 4 ft., and of the less base 18 inches?

SOLUTION.-The perimeter of the greater base is 16 ft., of the less, 6 ft. 16 ft.+6 ft. x 7÷2 = 77 sq. ft., convex surface.

2. Find the convex surface of a frustum of a cone whose slant height is 15 ft., the circumference of the lower base 30 ft., and of the upper base 16 ft.

FORMULA. Sum of perimeters

slant height = conv. surf.

To find the entire surface, add to this product the areas of both ends.

3. How many square yards in the convex surface of a frustum of a pyramid, whose bases are heptagons, each side of the lower base being 8 ft., and of the upper base 4 ft., and the slant height 55 ft. ?

484. To find the volume of a frustum of a pyramid or

cone.

1. Find the volume of the frustum of a sq. pyramid, whose altitude is 10 ft., each side of the lower base 12 ft., and of the upper base 9 ft.

SOLUTION.—122 +92 = 225; (225+ √/144 × 81) × 10÷3 = 1110; hence,

1110 cu. ft. is the volume.

2. How many cubic feet in the frustum of a cone whose altitude is 6 ft. and the diameters of its bases 4 ft. and 3 ft. ?

RULE. To the sum of the areas of both bases add the square root of the product, and multiply this sum by one-third of the altitude.

3. How many cu. ft. in a piece of timber 30 ft. long, the greater end being 15 in. square, and that of the less 12 in.?

4. How many cu. ft. in the mast of a ship, its height being 50 ft., the circumference at one end 5 ft. and at the other 3 ft.?

485. To find the surface of a sphere.

1. Find the surface of a sphere whose diameter is 9 in. SOLUTION.-92 × 3.1416 254.4696; hence, 246.4696 sq. in., surface.

==

FORMULA: Diameter x 3.1416 surface of sphere.
2. What is the surface of a globe 3 feet in diameter ?
3. Find the surface of a globe whose radius is 1 foot.

486. To find the volume of a sphere.

1. Find the volume of a globe whose diameter is 18 in. SOLUTION.-183 × .5236 = 3053.6352, hence, 3053.6352 cu. ft., volume. FORMULA: Diameter3 × .5236 = volume of sphere.

2. Find the volume of a globe whose diameter is 30 in. 3. Find the solid contents of a globe whose radius is 5 yd.

487. Gauging is the process of finding the capacity or volume of casks and other vessels.

For ordinary purposes the diagonal rod is used, which gives only approximate results.

A cask is equivalent to a cylinder having the same length and a diameter equal to the mean diameter of the cask.

To find the mean diameter of a cask (nearly).

RULE.-Add to the head diameter, or, if the staves are but little curved, .6, of the difference between the head and bung diameters.

To find the volume of a cask in gallons:

RULE.-Multiply the square of the mean diameter by the length (both in inches) and this product by .0034.

1. How many gallons in a cask whose head diameter is 24 in., bung diameter 30 in., and its length 34 inches?

SOLUTION.—24+ (30—24 × 3)

282 x 34 x .0034

=

= 28 in., mean diameter.

90.63 gal., capacity.

2. What is the volume of a cask whose length is 40 in., the diameters 21 and 30 in. respectively?

3. How many gallons in a cask of slight curvature, 3 ft. 6 in. long, the head diameter being 26 in., the bung diam. 31 in.?

METRIC SYSTEM.

488. The metric system of weights and measures is based upon the decimal notation.

489. The meter is the base of the system, and is one tenmillionth part of the distance on the earth's surface from the equator to the north pole, and is equal to 39.37079 inches.

490. The metric system has three standard units; viz., the me'ter (mee'ter) or unit of length, the li'ter (lee'ter), or unit of capacity or volume, and the gram, or unit of weight. The ar and ster are the units of square and cubic measures. Each of these units has its multiples and sub-multiples.

491. The multiple units, or higher denominations, are formed by prefixing to the standard unit the Greek numerals, děk'a (10), hěk'to (100), kilo (1000), mýr'ia (10000).

492. The sub-multiple units, or lower denominations, are formed by prefixing to the standard unit the Latin. ordinals, dec'i (1), cent'i (780), mil'lı (7000).

Hence, it is apparent from the name of a unit, whether it is greater or less than the standard unit, and also how many times.

Thus, 1 deka-meter (Dm.) denotes 10 meters; 1 hekto-meter (Hm.), 100 meters; 1 kilo meter (Km.), 1000 meters; and 1 myria-meter (Mm.)' 10000 meters. Also, 1 deci-meter (dm.) denotes (1) of a meter; 1 centimeter (cm.), 1ʊ (.01) of a meter; 1 millimeter (mm.), Todd (.001) of a

meter.

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494. The metric system being based upon the decimal scale, the denominations correspond to the orders of the Arabic notation, and hence are written like United States money, the lowest denomination at the right. Thus,

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The number is read, 67015.638 meters. It may be expressed in different denominations by placing the decimal point at the right of the required denomination, and writing the name or abbreviation after the figures.

Thus, the above may be read, 670.15638 Hm.; or 67.015638 Km.; or 670156.38 dm.; or 6701563.8 cm.; or it may be read,

6 Mm. 7 Km. 0 Hm. 1 Dm. 5 m. 6 dm. 3 cm. 8 mm.

Write 3672.045 meters, and read it in the several orders; read it in kilometers; in hektometers; in dekameters; in decimeters; in centimeters.

The names mill, cent, and dime, used in U. S. money, correspond to milli, centi, and deci in the metric system. Hence, the eagle might be called a deka dollar, since it is 10 dollars; the dime, a deci-dollar, since it is of a dollar, etc.

MEASURES OF LENGTH.

495. The me'ter is the unit of length, and is equal to 39.37 in., or 1.0936 yd. +.

Metric Denominations.

4 in. 1 dm.

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Symbols. U.S.Values.

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2 One Deci-meter = 10 Centi-meters = 100 Milli-meters = 3.973 inches.

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1 Mỹr′ia-mē'ter. Mm. 6.214

Units of length form a scale of tens; hence, in writing numbers expressing length, one decimal place must be allowed for each denomination.

Thus, 9652 mm. may be written 965.2 cm., or 96.52 dm., or 9.652 m., or .9652 Dm.

1. The meter is used in measuring cloths and short distances.

2. The kilo-me' ter is commonly used for measuring long distances, and is about of a common mile.

3. The cen'ti-me'ter and milli-me'tcr are used by mechanics and others for minute lengths.

4. In business, dec'i me'ters are usually expressed in cěn ti meters.

5. The děka-mēter, hěkto-mēter, and myr'ia-meter are seldom used, but their values are expressed as ki’lo-mē'ters.

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Write the following, expressing each in three denominations: 2. 24379 dm.; 15032036 cm.; 2475064 mm.; 30471 Dm. 3. 6704 Hm.; 85 Km.; 120000 m.; 780109 cm.; 75 m.

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