456. A sphere (Fig. 59) is a body generated by the revolution of a semi-circle (or circle) about its diameter regarded

M

FIG. 59.

as fixed. Hence, it is a body bounded by a surface all points of which are equally distant from a point within called the

centre.

457. The straight line joining the centre to any point on

the surface is a radius of the sphere; and a straight line passing through the centre and terminated by the surface is a diameter of the sphere.

458. All sections of a sphere made by a plane passing through the centre are circles which have the same centre and the same radius as the sphere, and are called great circles.

459. All sections of a sphere made by a plane that does not pass through the centre are circles, but they have a smaller radius than that of the sphere, and are called small circles.

F

460. If the circle PAQB (Fig. 60) turns about its diameter PQ, the point A, which is 90° from P and also from Q, describes a great circle AGBH, called the equator; and the point C describes a small circle CEDF parallel to the equator, and called a parallel of latitude. The circle PAQB, which represents any position of the generating circle during its rotation, is called a meridian of longitude.

FIG. 60.

461. The area of the surface of a sphere is equal to four times the area of a great circle of a sphere.

462. The volume of a sphere is equal to one-third of the product of the area of the surface by the radius.

FORMULAS.

463. If V stands for volume, S for lateral surface, B for base, T for total surface, P for perimeter of base, R for radius of base, H for altitude, and L for slant height, the formulas for surfaces and volumes will be as follows:

1. Find the total surface of a rectangular parallelopiped whose length is 15 ft., breadth 9 ft., height 6 ft.

2. Find the total surface of a triangular prism whose height is 6 ft., side of base 3 ft.

3. Find the volume of a prism whose base is a square 8 ft. on a side, and whose length is 40 ft.

4. Find the volume of a right prism 32 ft. long, if its ends are trapezoids the parallel sides of which are 12 ft. and 8 ft., and the perpendicular distance between these parallel sides is 6 ft.

5. A right cylinder is 10 ft. high, and measures 7 ft., 4 in. around the base. Find the lateral surface and the volume. (34)

6. Reckoning 7 gallons to the cubic foot, if 375 gals. are pumped out of a cylindrical cistern 7 ft. in diameter, how many inches will the surface of the water fall in consequence?

7. A marble column measures 7 ft. 4 in. in circumference and is 15 ft. long. Find the expense of polishing the entire surface at $1.50 a square foot.

8. Find the total surface of a regular pyramid when each side of its triangular base is 6 ft. and its slant height is 18 ft.

9. Find the volume of a regular pyramid whose base is an equilateral triangle measuring 4 ft. on a side, and whose height is 15 ft.

10. How many square feet of canvas is required for a conical tent the altitude of which is 8 ft. and the diameter of the base is 7 ft.? (T=3.1416.)

11. A conical tent whose slant height is 12 ft. requires 132 sq. ft. of canvas to cover it. Find how many feet of ground the tent covers. (34)

12. Find the volume of a right cone the height of which is 15 ft. and the circumference of the base 14 ft.

13. Find the total surface of the frustum of a pyramid whose bases are 18 ft. sq. and 15 ft. sq. respectively, and whose slant height is 30 ft.

14. How many square feet of tin will be required to make a funnel if the diameters of the top and bottom are to be 28 in. and 14 in. respectively, and the height is 24 in.?

15. Find the volume of the frustum of a regular square pyramid whose height is 24 ft., and the sides of its square ends are 9 ft. and 4 ft. respectively.

16. Find the volume of the frustum of a right cone if the radii of the circular ends are 3 ft. and 3 ft. 10 in. respectively, and the slant height 2 ft. 2 in.

17. Find the surface of a sphere whose diameter is 10 in. 18. The circumference of a dome in the shape of a hemisphere is 66 ft. How many square feet of lead are required to cover it?

19. The ball on the top of St. Paul's cathedral in London. is 6 ft. in diameter. What will it cost to gild it at

7 cts. a square inch?

20. Find the volume of a sphere if the diameter is 14 in. 21. Find the volume of a sphere whose circumference is 45 ft. (3.1416.)

22. How many gallons of water will a hemispherical bowl hold whose diameter is 21 in., reckoning 7 gals. to the cubic foot?

CHAPTER XVI.

MISCELLANEOUS PROBLEMS.

DECIMAL FRACTIONS.

1. Four men together paid $20,000 for some land. The first puts in $2350, the second $5820.35, the third $7640.75. How much must the fourth man pay?

2. What will be the cost of uniforms for a base-ball nine at $2.87 for each uniform?

3. At $15.87 a ton, what will be the value of 637 tons of hay?

4. If peaches are worth $1.25 a basket, and it takes 3 dozen for a basket, what is the value of 2892 dozen peaches?

5. If 964 baskets of peaches are sold for $1301.40, what is the price per basket?

6. If 324 men contribute together $2647.08, what is the contribution of each?

7. A boy picks blueberries in a pasture, giving to the owner of the pasture for the privilege 1 quart out of every 8 quarts. In 2 days he picks 48 quarts, and sells his share of the berries for $3.78. What did he get a quart?

8. If 150 men work on a railroad at the same price per

day, and if, at the end of the week, they all together receive $1575, what price per day does each man receive?