# The cylinder is entered in a cone So the lower basis lies on the basis of a cone, and the circle of the top basis belongs to a side surface of a cone. The volume of a cone is equal to 72. Search: 1) the volume of a cylinder which top basis halves cone height. 2) the largest volume of the described cylinder

Vasilkovskiy

327

There is drawing. Given: The volume of a cone is V (game) = 72; O1O2 cylinder Height = O2S = h = H/2 1) to Find the volume of a cylinder of V V (game) = 1/3*pi*R^2*H = 72 From here of pi*R^2*H = 72*3 = 216 (tsit) cone Volume As the cone with the basis of CO2D is similar to a cone with AO1B basis with coefficient of similarity of H/h = 2, r = to R/2. The volume of a cylinder of V (tsit) = pi*r^2*h = pi * (R/2) ^2 * (H/2) = 1/8*pi*R^2*H = 1/8*216 = 27 2) to Find the volume of the greatest such cylinder. In the 2nd drawing two cylinders in extreme provisions are designated in blue color. In both cases the volume of a cylinder is close to 0. Some average situation at which the volume of a cylinder is maximum is designated black. At a cone a tilt angle of the forming tg α = H/R. At the top cone too tg α = (H - h) / r = H/R. Means, at H cylinder - h = r*H/R; from here h = H - H*r/R = H * (R - r) / R Volume of a cylinder of V (tsit) = pi*r^2*h = pi*H/R*r^2 * (R - r) = pi*H*r^2 - pi*H/R*r^3 - gt; max Volume will be maximum when its derivative is equal 0 V (tsit) = 2pi*H*r - 3pi*H/R*r^2 = pi*H*r * (2 - 3*r/R) = 0 2 From here - 3*r/R = 0; r = 2/3*R; h = H * (R - 2/3*R) / R = H*1/3 = H/3 V = pi*r^2*h = pi*4/9*R^2*H/3 = 4/27*pi*R^2*H = 4/27*216 = 4*8 = 32

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