599.

PROPOSITION IX.

How do you compute the volume of a pyramid? To what does the pyramid approach as its limit if the sides be increased indefinitely?

Can you prove that the volume of a cone equals one-third of the product of its base by its altitude?

Sug. Inscribe a pyramid within the cone. Pupil complete demonstration.

Cor.

600.

Show that if v =

volume of circular cone,

the formula vv h is true if r radius of base,

and h altitude of the cone.

=

601.

PROPOSITION X.

What are similar cones of revolution?

Compute the volumes of two cones whose altitudes are 6 inches and 3 inches and whose diameters are 8 inches and 4 inches.

Compare their volumes with the cubes of the altitudes; with the cubes of the radii of their bases; with the cubes of the diameters of their bases.

A A

(1)

(2)

Given (1) and (2), two similar cones of revolution with h, h' the respective altitudes and r, r' the respective radii.

Can you show that the volumes of these cones are to each other as the cubes of their altitudes, or as the cubes of the radii of their bases, or as the cubes of the diameters of their bases?

volume of (1) and v = volume of (2).

Sug. Let V

=

Write the conclusion, and call it Prop. X.

602.

PROPOSITION XI.

How do you find the volume of the frustum of any pyramid?

As the number of sides is indefinitely increased, to what does the frustum of the pyramid approach as a limit?

E

Let A EC DE' be an inscribed frustum of a pyramid. Its volume V'. Call h the altitude, B' the area of the lower base, 'the area of the upper base, and V B'b' the area of the mean base. [P. P.] Call V the volume of the cone. Sug. 1. §519, V' { h ( B' + b2 + √B′ · b').

Increase the lateral faces of the frustum of the pyramid indefinitely. To what does B'? b? V'? Is the above equation always true for any number of sides? Apply law of limits and complete the demonstration.

EXERCISE.

501. If r and r' denote the radii of the base, can you

prove that v = {} π h ( r22 + (r')2 +rr').

502. If B base, b
=

EXERCISES.

=

upper base, C= circumference

of base, c = circumference of upper base, c' circumference of mid-section, r = radius of base, r' = radius of upper base, diameter, s = lateral area, S total area,

h

=

and v

altitude, d

=

=

volume.

=

Write formulas for a cylinder of revolution, a cone of revolution, and the frustum of a cone of revolution.

503. Find s, S, and v by the formula above when d dm. and h = 10 dm.

504.

tion are r

8

The dimensions of the frustum of a cone of revolu= 4, and h = 9. Find s, S, and v.

=

6, r'

=

505. How many square yards of tin will be required to cover a conical tower whose base is 12 feet in diameter and whose altitude is 20 feet?

506. A ship's mast is 40 feet long, 12 inches in diameter at the lower base, and 5 inches at the upper base. Find its volume.

507. A cylindrical cistern is 5 m. deep and 23 m. in diameter. If 2 Hl. flow into it per minute, how long will the cistern be in filling?

508. Can you prove that the lateral surface of a pyramid circumscribed about a cone is tanget to the cone?

509. In a cylinder of revolution d=h, can you show that

510.

Two right circular cylinders have the diameter and 1 the altitude of each equal. If the volume of one is of that of

64

the other, what is the relation of their altitudes?

THE SPHERE.

DEFINITIONS.

603.

A sphere is a solid bounded by a surface all points of which are equally distant from a point within called the center.

604.

The radius of a sphere is the distance from the center to any point on the surface.

605.

The diameter of a sphere is any straight line passing through the center terminated by the surface of the sphere. It follows that all radii are equal, that any diameter is twice the radius, and that all diameters are equal.

606.

A line is tangent to a sphere when it has only one point in common with the sphere.

A plane is tangent to a sphere when it has only one point in common with the sphere.

Two spheres are tangent when they have only one point in common.

607.

Two spheres are concentric when they have a common

center.

608.

A polyedron is said to be inscribed in a sphere when its vertices lie in the surface of the sphere. The sphere may be said to be circumscribed about the polyedron.

A polyedron is circumscribed about a sphere when its faces are tangent to the surface of the sphere. In this case the sphere is inscribed within the polyedron.