A plane parallel to the base of the cone cuts off a cone with a base area of 4π from it. Find the radius of the base of the original cone if the plane divides the volume of the cone in a ratio of 1: 7, counting from the vertex.
Knowing the area of the base of the truncated cone, we determine its radius.
Ssection = π * OD1 ^ 2.
OD1 ^ 2 = Ssection / π = 4 * π / π = 4.
OD = 2 cm.
Since the plane divides the volume of the cone in a ratio of 1/7, the volume of the original cone is equal to: V1 = (7 + 1) * X cm3, and the volume of the cut off cone is V2 = 1 * X cm3.
Both cones are similar, and the volume ratio is: V2 / V1 = 1/8.
The ratio of the volumes of such cones is equal to the ratio of the cubes of their radii.
1/8 = O1D ^ 3 / AO ^ 3.
AO ^ 3 = 8 * O1D ^ 3 = 8 * 8 = 64.
AO = 8 cm.
Answer: The base radius of the original cone is 8 cm.
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