Let O and P be the centers of the lower and upper bases. Draw OE and PF perpendicular to BC and GH, respectively. Draw OA, PD, DML OA, and FN1 OE.

NE OE-PF = 7√3 - 3√3 = 4√3.

=

Ex. 1614. Let s = a side of the base, n the number of sides of the base, and H the altitude of a regular pyramid. Find the lateral edge, and the volume if

(a) n = 3, s =

12, H = 12.

: 4, 8 = = 6, H = 9.

(c) n = = 4, 8 = 12, H=7.

(d) n = 6, 8

Ex. 1615. Using the notations

lateral area if

Ex. 1616. Let s

base, n the number of

a side of the lower base, s' a side of the upper sides of the base, and H the altitude of a frustum.

Ex. 1617. Using the notations of the preceding exercise find the lateral area if

(a) s = 10, s'= 2, n =

(b) s 6, s'

(c) 8 =

4, H = 5.

4, n = = 4, H = 3.

8, s2, n = 3, H = 1.

(d) s = 8, s' = 6, n =

= 6, H = 3.

Ex. 1618. Find the volume of a regular quadrangular pyramid, if E 11, and s = 12.

Ex. 1619. The total surface of a cube is 336 sq. in. Find its volume. Ex. 1620. The lateral area of a right prism is 140, and its base is a triangle whose sides are 5, 7, and 8. Find its volume.

Ex. 1621. Find the volume of an oblique prism whose base is a regular hexagon inscribed in a circle of radius 5 in., if a lateral edge be 8 in. and its projection on the plane of the base be 3 in.

Ex. 1622. The base of a pyramid whose altitude is 24 is a square of side 10. Find the area of a section 4 in. from the vertex and parallel to the base.

Ex. 1623. How far from the vertex, on the lateral edge a, must a plane to the base be passed to divide a pyramid into two equivalent parts?

Ex. 1624. The altitude of a pyramid, having for its base a regular hexagon, each of whose sides is 8 in., is 28 in. Find the distance from 3√3 the vertex of a section whose area is 2

sq. in.

Ex. 1625. Find the surface and volume of a cube in which the diagonal of each face is 15 in.

Ex. 1626. Find the surface and volume of a cube whose diagonal is 30 in.

Ex. 1627. The base of a right prism is a quadrilateral ABCD, in which AB=7 in., BC= 20 in., CD = 15 in., DA = 24 in., and the angles at A and C are right angles. If the height of the prism is 12 in., find its entire surface and volume.

Ex. 1628. Find the lateral area of a right pyramid having the same base and height as a cube whose edge is 10 in.

Ex. 1629. The diagonals of a rectangular parallelopiped are equal.

Ex. 1630. Find the lateral area of the frustum of a regular pyramid the altitude of which is 15 in., and the bases of which are squares of sides 40 in. and 24 in., respectively.

Ex. 1631. A cone of revolution 12 in. in height and 16 in. in diameter at the base is cut by a plane parallel to the base and 9 in. from it. the lateral area and volume of the frustum so formed.

Find

Ex. 1632. The mid-points of two pairs of opposite edges of a tetrahedron determine a parallelogram.

Ex. 1633. The length of a right prism is 10 in., and the base is a regular hexagon whose sides are 8 in. Find the surface and volume of the greatest possible cylinder, of the same axis, that can be cut from the prism.

Ex. 1634. The base of a right pyramid is a regular hexagon whose sides are 20 in., and the lateral faces are inclined to the base at an angle of 60°. Find the volume.

Ex. 1635. Lines joining the mid-points of opposite edges of a tetrahedron meet in a point and bisect each other.

Ex. 1636. The altitude of a cone of revolution is 27 in., and its curved surface is 7 times the area of its base. Find the radius of the base.

Ex. 1637. Find the volume of a regular hexagonal pyramid, if the lateral edge equals 13, and the side of the base equals 11.

Ex. 1638. Find the volume and surface of a regular tetrahedron whose edge equals (a) 4 in., (b) n in.

Ex. 1639. Find the total surface of a regular tetrahedron, if the perpendicular from one vertex to the opposite face is 5 in.

Ex. 1640. The corners of a cube are cut off by planes which pass through the middle points of each set of edges meeting at a common vertex. If the edge of the cube is 2 ft., find the volume of the solid formed.

Ex. 1641. A cylindrical vessel 10 in. in diameter is required to hold 10 gal. Find the height to the nearest tenth of an inch, if one gallon equals 231 cu. in.

Ex. 1642. Find the weight of a hollow cylindrical column 5 ft. long, 1 ft. in outer diameter and 1 in. thick, if the material is 7 times as heavy as water (i.e. the specific gravity = 7.5), and 1 cu. ft. of water weighs 62.5 lb.

Ex. 1643. A cylindrical vessel, 6 in. in diameter, is partly filled with water. Upon immersing a body of irregular shape the surface of the water rises 2 in. Find the volume of the body.

Ex. 1644. How many yards of platinum wire in. thick can be made from a cubic inch of platinum?

Ex. 1645. By folding a semicircular piece of paper whose radius is 5 in. a conical surface may be formed. Find the volume and the total surface of the cone determined by this surface.

Ex. 1646. The area AA'B'B, bounded by two concentric arcs and parts of two radii is folded so as to form the lateral surface of a frustum of a cone whose lower base equals 36 π. Find ≤ 0, and the volume of the frustum if OA = 5 and OA' 10.

Ex. 1647. By folding a piece of paper similar to the figure of the preceding exercise we wish to obtain a frustum of a cone, whose slant height equals 5, and whose bases have the radii 9 and 6 respectively. Find OA, OB, and ≤ 0.

Ex. 1648. A cylindrical tin can closed at the top requires the least amount of tin if its height equals

its diameter. Find the diameter of such a can which holds one quart. (1 qt. 57.75 cu. in.)

=

Ex. 1649. How many square feet of ground are covered by a conical tent 8 ft. high, if the canvas contains 1884 sq. ft. ? (Assume = = 34.)

Ex. 1650. A conical vessel whose height equals the diameter of the base contains water to a depth of 6 in. Upon immersing a cube the water rises 2 in. Find the edge of the cube.

Ex. 1651. A cylindrical boiler which is partly filled with water has a length of 6 ft. and a diameter of 2 ft. Find the volume of the water if its greatest depth is 6 in. and the axis of the cylinder has a horizontal position.

Ex. 1652. A cylindrical log of wood, 10 ft. long and 1 ft. in diameter, floats on water. Find the volume and the weight of the displaced water if 9 in. of the vertical diameter are immersed in water. (1 cu. ft. of water weighs 62.5 lb.)

BOOK VIII

THE SPHERE

693. DEF. A sphere is a solid bounded by a surface, all the points of which are equally distant from a point. The fixed point is called the center of the sphere.

694. DEF. The radius of a sphere is a straight line drawn from the center to any point in the surface.

695. DEF. The diameter of a sphere is a straight line passing through the center and terminated at either end by the surface.

696. From the definitions it follows that

(1) All the radii of a sphere are equal, and all diameters are equal.

(2) A semicircle rotating about its diameter generates a sphere.

(3) Two spheres are equal if their radii or their diameters are equal, and conversely.

(4) A point is without a sphere if its distance from the center is greater than the radius.

Ex. 1653. The radii of two spheres are respectively 10 in. and 4 in., their line of centers (i.e. the line joining their centers) is 7 in. Does every point of the smaller sphere lie within the larger one?