RELATIVE POSITIONS OF SPHERES 796. The relative positions of two spheres are analogous to the relative positions of two circles. (See §§ 291-296.) 797. Line of Centers. The line joining the centers of two spheres is the line of centers. 798. Tangent Spheres. If two spheres are tangent to the same plane at the same point, they are tangent spheres. They may be tangent internally or externally. Spheres A and B are tangent internally; and A and C externally. 799. Concentric Spheres. Spheres that have a common center are concentric spheres. 800. Theorem. The intersection of two spheres is a circle, whose center is in a straight line joining the centers of the spheres and whose plane is perpendicular to that line. Ο'. Given two intersecting spheres O and To prove that the intersection of the spheres is a circle whose center is in OO' and whose plane is perpendicular to 00'. B Outline of proof. Through O, O', and any point A of the intersection, pass a plane. This plane intersects the two spheres in two great circles intersecting in two points. Why? If A and B are these points, then OO' is the perpendicular bisector of AB. § 295 Show that if the entire figure is revolved about 00', the great circles generate the spheres, and point A generates a circle with center C and radius CA, which is common to the spheres. Also show that the plane of the circle generated by A is perpendicular to 00'. Further, show that if any point of the intersection was outside of this circle, then the spheres would coincide. (§ 791). 801. Theorem. If two spheres are tangent to each other, the line of centers passes through the point of contact. Proof similar to that of § 296. EXERCISES 1. State and prove exercises concerning the sphere analogous to Exercises 1 to 5, page 115. 2. What is the locus of points at a distance r from a given point 0, and at a distance r' from a given point O'? 3. Find the center of a sphere which contains a given circle and also contains a given point not lying in the plane of the circle. 4. Two spheres with radii 34 and 50, respectively, intersect. If the distance between their centers is 56, find the radius of their circle of intersection, and the distance from the center of each sphere to the center of the circle of intersection. 5. The line of contact of a sphere inscribed in a circular cone with the conical surface of the cone, is a small circle of the sphere. 6. The line of contact of a sphere inscribed in a circular cylinder with the cylindrical surface, is a great circle of the sphere. MEASUREMENT OF THE SPHERE, AREA 802. Area of a Sphere. By the area of a sphere is meant the numerical measure of the surface, that is, the number of units of area in it. As with the cylinder and cone, a sphere is a curved surface, and a plane unit of area cannot be applied directly to it to find its numerical measure. Its area can be determined as a limit. It is based upon (7) of § 767, which states that a B sphere is generated by revolving a semicircle about its diameter, and the following statement which is accepted without proof: If in a semicircle generating a sphere, one-half of a regular polygon be inscribed so that a vertex lies at A each end of the diameter, then, as the number of the sides of the inscribed semipolygon is indefinitely doubled, the area generated by the semipolygon approaches the area of the sphere as a limit. 803. Theorem. The area of the surface generated by a segment of a straight line revolving about an axis, in its plane but not perpendicular to it and not intersecting it, is equal to the projection of the segment upon the axis multiplied by the circumference of the circle whose radius is the perpendicular erected at the midpoint of the segment and terminated by the axis. Given the segment AB revolving about the axis XY in its plane. To prove that S=00′×2× EF, where S denotes the area of the surface generated, 00' is the projection of AB upon XY, and FE is the perpendicular at the midpoint of AB. Proof. Three cases arise: CASE I, in which AB is oblique to and does not meet XY. The surface generated is the lateral surface of a frustum of a right circular cone. The proof of this is given in § 729. CASE II, in which AB is oblique to and meets XY. The surface generated is the lateral surface of a right circular cone. Show that S-TXOAXAB. ADEF AOAB. § 719 Then AB OB = EF: DF, and ABXDF=OBXEF. : Since OB 00' and OA=2DF, ABXOA=200'XEF. CASE III, in which AB is parallel to XY. The area generated is the lateral surface of a right circular cylinder. Give proof. 804. Theorem. The area of a sphere is equal to four times the area of a great circle of the sphere. S=4πr2. Given the area S of a sphere generated by the revolution of the semicircle O of radius r, about the diameter AE. To prove that S=4πr2. Proof. Inscribe in 'the semicircle one-half of a regular polygon of an even number of sides, as ABCDE. Let A denote the apothem of the polygon, and let F, O, and G be the projections of B, C, and D respectively upon the diameter AE. For each side, the apothem is the perpendicular bisector. Why? area generated by AB-AFX2πα. Then = Similarly area generated by BC= FOX2πа. Etc. § 803 Adding these equations and denoting by S' the area generated by the revolution of the semipolygon ABCDE, But S' = (AF+FO+···)2πα. Then S'=4πгα, which is always true as the number of the sides of the polygon is continuously doubled. Further S'-S and since a→r, 4πга-4πr2. §§ 802, 490, 485 (2) .. S=4πr2. § 485 (1) Show that other forms of the formula for the area of a sphere are S=2πrd, and S=πď2. 805. Theorem. The areas of two spheres are to each other as the squares of their radii; or, as the squares of their diameters. For, if S, and S' are the areas of two spheres, EXERCISES 1. Given the area of a sphere, to find its radius and its diameter. 2. Find the area of a sphere whose radius is 5. Of a sphere whose diameter is 16. Of a sphere whose circumference is 24 in. 3. Find the radius of a sphere whose area is 100 sq. in. 4. How many square feet of tin will it take to roof a hemispherical dome 40 ft. in diameter? 5. Find the diameter of a sphere that has the same area as a cube that is 8 in. on an edge. 6. Find the area of a sphere that is circumscribed about a cube that is 123 in. on an edge. Ans. 4071.5 sq. in. 7. The number of square inches in the area of a sphere is equal to the number of linear inches in the circumference of a great circle of the sphere. Find the radius of the sphere. 8. Show that, if a cylinder is circumscribed about a sphere, the lateral area of the cylinder is equal to the area of the sphere. Also show that the area of the sphere equals two-thirds the total area of cylinder. 9. The radius of the earth is 3960 miles and the radius of the moon is 1080 miles. Find the ratio of their areas. 10. Show how to determine the distance apart the points of a compass must be placed to draw a great circle upon a material sphere that has a diameter of 14 in. So as to draw a circle having a radius of 5 in. 11. What is the locus of the centers of all spheres tangent to a given plane at a given point? 12. What is the locus of the centers of all spheres passing through three given points? 13. What is the locus of points at a distance r1 from a point O1 and at a distance r2 from another point O2? Discuss. 14. Find the area of a sphere circumscribed about a tetraedron whose edge is 4 in. 15. The radii of two intersecting spheres are 10 in. and 12 in., respectively, and the distance between their centers is 16 in. Find the radius of the circle of intersection of the two spheres. 16. Three equal spheres each tangent to the other two rest on a plane. A fourth sphere rests on the first three spheres. Find the distance from the center of the fourth sphere to the plane, if each sphere has a radius of 8 in. Ans. 21.064 in. |