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Geometry.

345. To find the solid content of a cylinder or of a right prism.*

Observe that in any cylinder or right prism the number of cubic units in one layer 1 unit high (as indicated in the diagrams) is equal to the number of square units in the area of the base. Thus, if there are 41 square units in the area of the base there are 43 cubic units in one layer. The content of the entire solid is as many times the cubic units in one layer, as the solid is linear units in height. Hence the rule as usually given : "Multiply the area of the base by the altitude."

TO THE TEACHER.—This rule must be carefully interpreted by the pupil. He must not be allowed the misconception that area multiplied by any number can give solid content, except through such interpretation as is suggested in the above observation.

PROBLEMS.

1. Find the solid content of a square right prism whose base is 6 in. by 6 in., and whose altitude is 8 inches.

2. Find the approximate solid content of a cylinder 6 inches in diameter and 10 inches long.

* A right prism is a solid whose bases, or ends, are similar, equal, and parallel plane polygons, and whose lateral faces are perpeudicular to its bases.

346. MISCELLANEOUS PROBLEMS.

1. Find the solid content of an octagonal right prism the area of whose base is 24 square inches and whose altitude is 15 inches.

2. What is the solid content of a cylinder or of any right prism, the area of whose base is 30 square inches and whose altitude is 12 inches ?

3. How many cubic feet of earth must be removed to dig a well 6 feet in diameter and 20 feet deep ? *

4. Find the approximate number of feet of 14-in. lumber required to make the lining of the sides of a cylindrical silo that is 20 feet in diameter and 30 feet deep.

5. Find the approximate number of cords of rough stone in a cylindrical pile that is 16 feet in diameter and six feet deep.

6. Find the approximate number of brick necessary for a solid cylindrical foundation that is 9 feet in diameter and 4 feet high.

7. If the average specific gravity of the brick and mortar used in the foundation described in problem 6, is 1.9, how much does the entire foundation weigh?

8. Find the weight in kilograms of a column of water 1 decimeter square and 10 meters deep.

9. Find the weight in pounds of 1000 feet of white pine 1inch boards, the specific gravity being .6.

10. Find the weight of a load (1 cubic yard) of wet sand, the specific gravity being exactly 2.

* The exact number of cubic feet cannot be expressed in figures. An approxi. mation that will answer many practical purposes may be obtained by regarding the circle (base) as of its circumscribed square. If an answer more nearly accurate is required use .78 instead of .

DENOMINATE NUMBERS.

Capacity.

347. The standard unit of capacity used in measuring liquids is a gallon. A gallon equals 231 cubic inches.

Liquid Measure.
4 gills (gi.) = 1 pint (pt.).
2 pints

1 quart (qt.).
4 quarts = 1 gallon (gal.).
314 gallons 1 barrel (bbl.).

Observe that 1 cubic foot nearly 7 gallons.
Observe that 4,2 cubic feet nearly 1 barrel.

A kerosene barrel contains about 52 gallons. It equals nearly 7 cubic feet.

PROBLEMS.

1. Find the capacity (approximate or exact), in galions, of a rectangular tank 3 ft. by 4 ft. by 8 ft.

2. Find the approximate capacity, in gallons, of a cylindrical tank 4 feet in diameter and 4 feet deep.

3. Find the approximate capacity, in barrels (31+ gal.), of a rectangular tank 2 ft. by 4 ft. by 12 ft.

4. Find the approximate capacity, in barrels (31+ gal.), of a cylindrical cistern 6 ft. in diameter and 6 ft. deep.

5. Find the approximate capacity, in barrels (31) gal.), of a cylindrical cistern 12 ft. in diameter and 6 ft. deep.

6. Find the approximate capacity, in barrels (314 gal.), of a cylindrical cistern 12 ft. in diameter and 12 ft. deep.

Denominate Numbers-Capacity.

348. The standard unit of capacity used in measuring grain, fruits, vegetables, lime, coal, etc., is a bushel. A bushel equals 2150.4 cubic inches.

Note.-In measuring large fruits, vegetables, lime, and coal, the unit is the “ heaped bushel." A heaped bushel equals about 144 "stricken bushels."

DRY MEASURE.

2 pints (pt.) 1 quart (qt.).
8 quarts = 1 peck (pk.).
4 pecks = 1 bushel (bu.).
A bushel is nearly 12 cubic feet.

A “heaped bushel”' is about 1} cubic feet.
A “dry gallon” (4 quarts dry measure) equals 268.8 cubic inches.

Enough “ear corn” to make, when shelled, one bushel, occupies about 24 cubic feet. If the corn is inferior in quality it will occupy more space than this—sometimes 2; cubic feet.

PROBLEMS.

1. Find the capacity in bushels of a wheat bin 8 ft. by 8 ft. by 10 ft. *

2. Give the dimensions of the smallest bin in which 1000 bushels of oats may be stored.

3. In a bin 12 feet square, there is rye to the depth of 7+ feet. How many bushels ?

4. How many bushels of potatoes (without heaping the bin) may be stored in a bin that is 8 ft. by 4 ft. by 6 ft.?

5. If the corn is of excellent quality, how many bushels of “shelled corn” may be expected from a crib of ear corn, 8 ft. by 10 ft. by 80 ft. ?

* For many practical purposes the approximate ratio (11) of the bushel to the cubic fout will give in such problems as these, results sufficiently accurate.

Denominate Numbers-Weight.

349. The standard unit of weight in common use is a pound Avoirdupois.

AVOIRDUPOIS WEIGHT. 16 ounces (oz.) 1 pound (lb.).

2000 pounds 1 ton (T.). The abbreviation for 1 hundredweight (100 lb ) is owt.

62lb.

MISCELLANEOUS WEIGHTS. 1 gallon of water

about 8} lb. 1 gallon of milk

about 8 6 lb.
1 gallon of kerosene

about 61 lb.
1 cubic foot of water
1 bushel of wheat

60 lb.
1 bushel of beans

60 lb. 1 bushel of clover seed

60 lb. 1 bushel of potatoes

60 lb. 1 bushel of shelled corn

56 lb. 1 bushel of ear corn

70 lb.* 1 bushel of rye

56 lb. 1 bushel of barley

48 lb. 1 bushel of oats

= 32 lb. 1 barrel of flour

= 196 lb. 1 barrel of beef or pork = 200 lb.

PROBLEMS.

Find the cost –
1. Of 2650 lb. coal at $5.50 per ton.
2. Of 2650 lb. oats at 24¢ a bushel.
3. Of 3330 lb. wheat at 80¢ a bushel.
4. Of 4650 lb. potatoes at 42¢ a bushel.

(a) Find the sum of the four results.

* This means the amount of ear corn required to make 1 bushel of shelled

corn.

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