DEFINITION. A straight line drawn from the vertex of a cone of revolution to a point in the circumference of the base is called the slant height of the cone. It is one of the positions of the hypotenuse of the generating triangle. 3. Find the volume of a cone of revolution if the slant height is 13 inches and the diameter of the base is 10 inches. DEFINITION. The slant height of a regular pyramid is the altitude of one of the equal isosceles triangles forming its lateral faces. 4. Find the lateral area and the total area of the pyramid in Exercise 1. 5. Find the lateral and total area of the cone in Exercise 2. Hint: The lateral surface of a cone of revolution may be spread out in the form of a sector of a circle. 6. Theorem. The lateral area of (a) a regular pyramid, (b) a cone of revolution, is half the product of the perimeter of the base and the slant height. 7. Find the volume generated by revolving about the x-axis the area bounded by the lines y 3x, y 0, and x = 3. Check the result by the theorem in the preceding section. = = 8. State and prove a theorem analogous to that on page 323 for the volume of a solid generated by revolving an area about the y-axis. 9. Find the volume generated by revolving about each axis the area bounded by the parabola y = 4 x2 and the positive parts of the axes. 10. Find the volume generated by revolving about the x-axis the area bounded by the parabola y x2 + 4x 3 and the x-axis. 11. Find the volume of the x2+ y2 = 16 about the x-axis. sphere generated by revolving the circle 12. Find the volume generated by revolving about the y-axis the area bounded by y = x2, x = 0, and y = 9. 13. Find the volume of the solid generated by revolving about the x-axis the area bounded by y x+2, the x-axis, and the ordinates at x and x = = = 2 A frustum of a cone is that part of a cone included between two planes perpendicular to the axis. It may be generated by revolving a trapezoid ABCD, such that AD and BC are perpendicular to AB, about AB. See, for example, the solid obtained in Exercise 13. 14. If that part of the line y = x + 2 which lies between x 1 and x = 4 is revolved about the x-axis, find the volume of the frustum of a cone which is generated. = 15. Find the volume of a frustum of a cone if the radii of the bases are 3 and 7 inches and the altitude is 19 inches. 16. The altitude of a conical receptacle is 10 inches, and the radius of the base is 6 inches. The axis is vertical with the vertex at the bottom. If water is poured into it at the rate of 20 cubic inches per second, how fast is the surface rising when the depth is 5 inches? 17. The water reservoir of a town is in the form of an inverted conical frustum with the sides inclined at an angle of 45°, the radius of the smaller base being 100 feet. If the depth is decreasing at the rate of 5 feet per day when the water is 20 feet deep, show that the town is being supplied with water at the rate of 72,000 π cubic feet per day. 18. The diameter of the base of a cone is found to be 6 inches and the altitude 15 inches. If the error in each measurement is not greater than one per cent, find the relative error in the computed volume, assuming first that the measurement of the diameter is exact, and then that the measurement of the altitude is exact. 19. The diameter of the base of a cone is found to be 8 inches and the slant height 10 inches. Assuming that the first measurement is exact, what is the admissible error in the second if the volume is to be determined within 0.1 of one per cent? 20. How many cubic feet of air will there be in the largest conical tent that can be made out of 200 square feet of canvas? 21. What is the least amount of canvas that can be used to make a conical tent of 1200 cubic feet capacity? 114. Volume of the Sphere. A sphere may be generated by revolving a circle about a diameter. Choosing the center of the circle as origin, a point P(x, y) will lie on the circle if and only if OP = r, where r is the Let V denote the volume from A to a section cutting the x-axis at a point whose abscissa is x. By the theorem on page 323, we have, from (1), Integrating D2V = πу2 = π(r2 — x2). V = π(r2x − x3/3) + C. Hence 0 = π(− p3 + p3/3) + C, whence С = 2πr3/3. V = π(r2x - x3/3) + 2πr3/3. When x = r, V is the volume of the sphere, hence the volume of the sphere is V Hence the = π(μ3 — r3/3) + 2πr3/3 = πr3. Theorem. The volume of a sphere of radius r is V = §πr3. 115. Area of a Sphere. Imagine a very large number of planes drawn tangent to the sphere. The area and volume of the polyhedron bounded by these planes would be very nearly equal to the area and volume of the sphere. Planes passed through the center of the sphere and the lines of intersection of the tangent planes would divide the polyhedron into as many little pyramids as there are tangent planes. These pyramids would all have the radius for altitude, and the sum of their base areas would be the area of the polyhedron. Hence the volume of the polyhedron would be one-third its area multiplied by the radius. This would be true no matter how large the number of faces of the polyhedron. If the number of tangent planes is increased indefinitely in such a way that the area of the base of each little pyramid approaches zero, then the volume and area of the polyhedron approach respectively the volume and area of the sphere. Hence the volume of the sphere is also equal to one-third its area multiplied by the radius. Denoting the area of the sphere by S and applying the theorem in the preceding section, we have Hence the Sr Tr3, whence S 4πr2. = = Theorem. The area of a sphere of radius r is S = 4πr2. EXERCISES 1. Find the area and volume of a sphere whose radius is 5 inches. 2. Find the volume of the sphere inscribed in a cone of revolution whose altitude is 12 inches, the radius of whose base is 5 inches. 3. For what values of the radius is the number of units in the volume of a sphere less than the number of units in the area? For what values of the radius is DiV <DS? DiV>DS? In biology, we learn that a cell subdivides after a certain time. Why? 4. The diameter of a sphere is found by measurement to be 8.3 inches with a probable error of 0.1 inch. What is the error in the computed value of the area of the sphere? In the computed value of the volume? Is the percentage error in the area greater or less than the percentage error in the volume? 5. What is the relation between the rates of increase of the radius and volume of a soap bubble? When the radius is 3 inches and is increasing at the rate of 0.1 inch per second, how fast is the volume increasing? The surface? 6. A washbowl is in the form of a hemisphere of radius 8 inches. How much water is there in it when the water is 5 inches deep? 7. When the depth of the water in the bowl of Exercise 6 is 4 inches the surface is falling at the rate of of an inch per second. How rapidly does the water run off through the drain pipe? 8. Find the area and volume of the earth, assuming that it is a sphere whose radius is 3957 miles. Find the error and relative error in each case if the error in the radius is not more than 7 miles. How does the relative error in the area and in the volume compare with the relative error in the radius? 9. How many lead shot of an inch in diameter can be made from a piece of lead pipe 2 feet long whose outside and inside diameters are respectively 1.25 inches and 1 inch? MISCELLANEOUS EXERCISES 1. Find the volume generated by revolving the area bounded by the curves y x2 and x = y2 about the x-axis. = 2. A ball rolls down a smooth plane inclined at an angle of 18°. Find how far it will roll in t seconds. 3. The diameter of a cone is found to be 5 inches, and the altitude 8 inches. If the error in the diameter is 0.06 of an inch and the altitude is exact, find the error in the computed value of the lateral area. DEFINITION. At the center O of an equilateral triangle ABC erect a line perpendicular to the plane of the triangle. Take a point D on it such that AD AB, and join D to A, B, and C. The figure so obtained is called a regular tetrahedron [Fig. 193 (a)]. 4. Find the altitude DO and the volume of a regular tetrahedron (a) whose edge AB is 6 inches; (b) whose edge is e. DEFINITION. At the center O of a square ABCD erect a line perpendicular to the plane of the square, and extend it on both sides of the plane to points E and F such that EA = FA AB. Join E and F to the vertices of the square. The resulting figure is called a regular octahedron. = 5. Find the volume of a regular octahedron (a) whose edge AB is 6 inches; (b) whose edge is e. (Note that EF is a diagonal of the square AECF.) 6. Express the total area of the octahedron in terms of the edge. Do the same for the cube and the regular tetrahedron, and plot the graphs of these three functions of e on the same axes. For the same value of e, which solid has the greatest area? Which area increases the most rapidly as e increases? 7. The graph of x} + y} = a} is a parabola tangent to the coördinate axes whose axis of symmetry bisects the first quadrant. Find the area bounded by the curve and the axes. 8. Find the volume of the solid generated by revolving about the y-axis the area bounded by the parabola y = ax2, x = O, and y = h. Show that this volume is one-half the volume of a cylinder whose altitude is h and whose bases are equal to the circle forming the upper surface of the solid. 9. A point moves so that its coördinates at any time t are given by Find the components parallel to the axes of the velocity and acceleration. Show that the point moves in a circle (to eliminate t, square both equations |