Here slant surface (1.25+.75) 3.14159

1.5

=4.1888 square ft.

Ex. 6. If the earth were a sphere 7912 miles in diameter, what would be its surface?

Surface 79122 × 3.1415926=196662892.5123

=

square miles.

Ex. 7. The diameter of a sphere is 3 feet 6 inches. What is the convex surface of a segment, the height of which is 9 inches?

Surface 3.1416 × 42 × 9=1187.5248 square inches.

414. EXAMPLES OF SOLID MEASURE.

Ex. 1. How many gallons of distilled water will a cistern contain, the length being 8 feet, the breadth 3 feet 6 inches, and the depth 4 feet 2 inches?

Contents of the cistern is 96 × 42 × 50=201600 cubic inches. And 277.274 cubic inches:

.. 201600 cubic inches=

1 gallon,

201600 × 1

277.274

727 gallons nearly.

Ex. 2. A cylindrical vessel, the base of which is 3 yards in circumference, and height 5 yards, is three-quarters full of distilled water. The weight of the water is required.

The diameter of the base= 36 = 11.4 inches.

3.1416=

.. Surface of base = 36 × 114102.6 square inches.

3 × 60
4

· The vessel is full, the water in it= X 102.6= 4617 cubic inches, and .. 27.7274 cubic inches of distilled water weigh 1lb.

Ex. 3. Required the number of cubic feet of air in a room, the length of which is 32 feet, breadth 20 feet 6 inches, and height 9 feet 8 inches.

Content=$2 x 20.5 × 9.6-6341.3 cubic feet.

Ex. 4. The sides of the base of a triangular pyramid are 4.5, 4, 3.5 feet, its altitude is 8.5 feet; find its solid content.

Ex. 5. What is the solidity of the frustum of a square pyramid, the sides of bases being 8 inches and 6 inches, and the altitude 15 inches.

Here side of middle section: =

8+6
2

=7 inches.

.. Content=15×(8×8+6×6+ 4×7×7)=740 inches. Ex. 6. The diameter of the moon is 2180 miles, find her solid content.

Solidity=21803×.5236=5424617475 cubic inches.

415. EXERCISES.

1. The height of the largest pyramid is 477 feet, and a side of the base, which is square, is 720 yards; find its solidity. 2. Required the content of a bale, the length being 5 feet 6 inches, breadth 4 feet 4 inches, and thickness 3 feet inches. 3. How often may a conical glass, 2 inches deep, and 11⁄2 inches in diameter at the top, be filled out of an imperial gallon?

4. A cistern is 6 feet 4 inches long, 5 feet 6 inches broad, and 4 feet 8 inches deep, how many gallons of water does it contain?

5. The interior diameter of a metal pipe is 8 inches, the thickness of the metal is .6 inch, and the length of the pipe is 35 yards, how much metal in the pipe?

6. How many times would a roller 3 feet 8 inches long and 1 foot 10 inches in diameter, turn on a walk 84 yards long, and 6 feet wide, supposing it never to pass twice over the same ground?

7. What length must be cut off from a board 9 inches wide, to contain 2 square feet?

8. The wheel of a carriage turns round 1640 times a distance of 4 miles, what is the diameter ?

9. What length of a wire, of an inch in diameter, may be drawn out of a cubic inch of metal?

10. A roof, which is 36 feet 9 inches by 16 feet 8 inches, is to be covered with lead, at 84lbs. per square foot, find the price at 14s. per cwt.

11. What will the diameter of a sphere be, when the solidity and superficial area are expressed by the same quantity?

12. The ball on the top of St. Paul's Cathedral is 6 feet in diameter, what will the gilding of it cost at 24d. per square inch?

13, How many gallons of water will be required to fill a ditch, mile long, 10 feet deep, 25 feet broad at the top and 18 feet at the bottom?

14. What is the surface of a sphere, the volume of which is 480 square yards?

15. Express the side of the cube equivalent in solidity to a sphere 2.5 feet in diameter.

16. Knowing that globes are to each other as the cubes of their diameters; if the diameter of the Earth be 1, that of the Sun is 109.93, of Mercury .39, of Venus .97, of Mars .56, of Jupiter 11.56, of Saturn 9.61, of Uranus 4.26, of Neptune 3.91, of the Moon, .27; it is required to find the solidity of these bodies, the earth's solidity being supposed unity.

17. The Sun weighs 354936 times as much as the Earth, Jupiter 332, Saturn 101, Neptune 29, Uranus 20, Venus .87, Mercury .18, Mars .14, and the Moon ; the solidity of these bodies being determined by the previous example, find their density or their weight under the same bulk.

18. Supposing the weight of a cubic inch of cast iron weighs 4.3 oz. avoirdupois, what is the weight of an iron ball of 10.5 inches diameter.

TO FIND THE MAGNITUDE AND THE WEIGHT OF A BODY BY ITS SPECIFIC GRAVITY.

416 When solids are too irregular to admit of the application of the rules of mensuration, we can find the content and also the weight by reference to their specific gravity.

The specific gravity of a body is its weight as compared with an equal bulk of some other body.

417. A cubic foot of rain water weighs just 1000 ounces avoirdupois, this has been taken in this country, as the specific gravity of one cubic foot of water, and comparing other bodies with it, the following table, determined by careful experiments contains the weight in ounces of a cubic foot of each.

419. Ex. 1. What is the solidity of 1 ton of oak?

By the table. 925 ounces of oak are the weight of one cubic foot.

925

.. 1 ton or 35840 ounces of oak are the weight of 35840 3118.

=

Ex 2. Required the weight of a block of marble whose length is 2.3 yards, the breath 1.25 yards, and the thickness .97 yard. Here the solidity is 2.3 × 1.25 ×.97 2.78875 cubic yards. .. By the table 1 cubit foot of marble weighs 2700 ounces. .. 1 cubic yard weighs 27 x 2.78875 × 2700 ounces=5 tons, 13 cwt., 1 qr., 22 lb, 4 oz.

420. EXERCISES.

1. A waggon is loaded with 24 bars of iron, 2 yards long, 8 inches, broad and 8 inches thick; find the weight of the load.

2. The weight or pressure of the atmosphere at the earth's surface, is equivalent to the weight of a column of mercury, the height of which is 294 inches. Supposing that the surface of the human body is 3.4 square yards, what is the pressure to which it is subjected? 3. Archimedes discovered the following principle: a body immersed in a fluid, loses as much weight as an equal bulk of the fluid weighs. From this it is required to find the weight of a cast iron ball 6.4 inches in diameter, when immersed in water.

4. What is the weight of the air contained in a room, the dimensions of which are 17 feet 6 inches in length, 14 feet 8 inches in breath, and 10 feet 4 inches in height?

5. What will be the weight of a foot of lead pipe, 3 inches diameter in the bore, and .6 inch in thickness?

6. How many cart loads of clay will there be in a drain 100 feet long, 2 feet broad and 6.4 feet deep, allowing 1 tons for each load?

7. Required the weight of an oak board, 5 yards long, 2.7 feet broad, and 4.4 inches thick?

8. What is the diameter of a gold ball of the same weight as a silver ball whose radius is 1.5 inches?