Question

a parallel to base divides a cone in two parts of equal volume , determine the ratio of the heights of the two parts that is the cone and the frustum..??

Answer 1

Volume of the cone = [$\pi$ * r² * h] / 3
                                = 1($\pi$rh)/3
 
Since the volume is divided equally, volume of the smaller cone = [1($\pi$r²h)/3 ] ÷ 2
= 1($\pi$r²h)/6

Both the cones formed are similiar so using the identity for similiar figures
V1/V2 = (L1/L2)³
Since the ratio of volumes is 1:2

1/2 = (L1/L2)³
1/(∛2) = (L1/L2)
This is the ratio of any of the length of the similiar shapes so the ratio of the smaller cone and the frustum is:
1 : (∛2 - 1)