Exercises (11). 1. Find the volume of a sphere whose radius is 1.) 5 inches. (2.) 2 feet 6 inches. (3.) 3 feet. 2. Find the radius of a sphere whose volume is (1.) 5000 cubic inches. | (2.) 3 cubic feet. 3. Find the volume of a sphere whose surface is (1.) 25 square feet. (2.) 100 square inches. 4. Find the diameter of a sphere which has the same volume as a cylinder whose height=h, and radius of base = r. 5. Find the radius of a sphere which has the same volume as a cone whose height = h, and radius of base = r. 6. Find the quantity of metal in a spherical shell whose outer diameter = d inches, and whose thickness t inches. = - 7. A cone circumscribes a given sphere, and the height of the cone is twice the diameter of the sphere; shew that the total surface of the cone is twice that of the sphere, and the volume of the cone twice that of the sphere. 8. Prove that when the side of the frustum of a cone is equal to the sum of the radii of the bases, the height of the frustum is twice the geometric mean of the radii, and the volume is found by multiplying the total surface by the sixth part of the height. 9. A triangle whose sides are 5, 4, and 3 revolves round the largest side as axis; compare the volumes generated by the other two sides. 10. Find the volume generated by an equilateral triangle revolving round one of its sides. II. The volumes generated by a parallelogram turning successively round two adjacent sides are inversely as the lengths of the sides. 12. Find the volume generated by a regular hexagon revolving round one of its sides. 13. In a sphere of which the radius is R, and centre O, to draw a plane AB cutting the sphere, such that the ratio of the surface of the zone BPA to the surface of the cone BOA, which has for vertex the centre O of the sphere, and for base the circle AB, may be 1. 14. Divide a sphere of given radius into two parts, so that the volume of the one part may be to that of the other in the ratio of m: n. 15. Express the volume of a sphere as a function of the circumference of a great circle. 16. Prove that the volume of a cylinder circumscribing a sphere is a mean proportional between the volume of the sphere, and that of the equilateral cone circumscribing. The cone is said to be equilateral when the diameter of the base is equal to the slant height. 17. The diameters of the earth, of the moon, and of the sun being proportional to the numbers 1, 1, 112; how are the volumes of the moon and of the sun expressed if that of the earth is taken for unity? 18. If we join the middle points of two sides of a triangle by a straight line, and make the triangle revolve round the remaining side; shew how to find the volumes generated by each of the two parts of the triangle. 19. The volumes generated by a rectangle turning successively round two adjacent sides are v and v'; find the diagonal of this rectangle. 20. Having given a sphere of which the radius is R, construct a cone which has the same volume as the sphere, and of which the height is half the radius of the sphere; what will the radius of the base be? 21. AB is the diameter of a sphere; shew how to draw a plane perpendicular to this diameter, dividing the sphere into two parts, whose 'surfaces are to each other in the ratio of 2 to 3. 22. An isosceles triangle of which the base is b, and other sides each equal to a, turns round its base; express the volume thus generated. 23. An equilateral triangle, ABC, revolves first round the side BC, and afterwards round a line DE, through A, parallel to BC; compare the volumes thus generated. 24. The difference of the radii of two spheres =d, and the difference of their volumes = V; find the radius of each. 25. The volumes of a right circular cone, of a sphere, and of a right cylinder of the same height, are proportional to the numbers 1, 2, 3, when the cone and the cylinder have for bases a great circle of the sphere. MISCELLANEOUS EXERCISES. 1. Given the total surface of a right circular cone, and also its volume, to determine the cone. from (4), we see that each of these roots must be less H (4); ... (5). than a√2, and consequently the values of y will both be positive and greater than those of x. We have therefore two solutions. and there is only one solution. 2. In a given sphere, it is required to inscribe a cylinder whose total surface is given. Let the radius of the sphere, x= the radius of the base of the cylinder, 2y= the height of the cylinder, 4a2 = total surface of cylinder. We will evidently have the following equations to determine x and y. ... (4); .. x = √2 (a2 + 1 (5). found from (2), we have 5x4-4(a2 + r2)x2 + 4a1=0 :(a2 + r2) ± 2 √(a2 + r2)2 − 5aa 5 As x is essentially positive, it is unnecessary to prefix the outside radical with the double sign ±. It follows from the above, that in order that the values of x2 in (4) may be real, we must have or, which is the same thing, we must have the condition that is, the total surface of the given cylinder must be less than xr2( √√5 + 1). If this condition is fulfilled, the two values of x2 are real and positive. Substituting each of these values of x2 in (3), we have corresponding values of y. But as the value of y ought also to be positive, we must have x2 L 2a2. 3. In a triangle ABC, draw AD perpendicular to BC; from D draw DE perpendicular to AB, and DF perpendicular to AC. Having given the lengths of these perpendiculars equal respectively to P, P1, and 2, find the sides of the triangle. 4. Given A, the area of a triangle, and the ratio of any side to the perpendicular on it from the opposite angle equal to mn; find this side. 5. Find the area of a square inscribed in a circle whose radius is r. 6. Find the radius of a circle whose area is equal to that of three circles whose radii are r1, 72, 73. 7. If d denote the difference between the radii of two circles, and S denote the difference between their areas, find the radius of each circle. 8. If A denote the area of a right-angled triangle, and h the hypothenuse, find the other two sides. 9. In a triangle whose sides are a, b, c, and whose area is S, we have given a, b, and S; it is required to find the remaining side c. (1.) Interpret geometrically the values of |