a solid right circular cone is cut into 4 pieces of equal height, with each cut being parallel to its base. If the semi vertical angle of the original cone is 30° what is the ratio of the total surface area of the original cone to the sum of the total surface areas of the 4 pieces?? ans= 12:19.. approach plzz

There are 4 cones - From the original (biggest cone) to the smallest cone. Let's call them 1, 2, 3 and 4 in that order.

Let's call the four pieces A, B, C and D from the bottom to the top.

Let the bottom radius of A = 4
So, top radius of A = Bottom radius of B = 3
Top radius of B = Bottom radius of C = 2
Top radius of C = Bottom radius of D = 1

TSA of A = TSA of cone 1 - TSA of cone 2 + 2*Top area of A
TSA of B = TSA of cone 2 - TSA of cone 3 + 2*Top area of B
TSA of C = TSA of cone 3 - TSA of cone 4 + 2*Top area of C
TSA of D = TSA of cone 4

So, total TSA of all the 4 pieces = TSA of cone 1 + 2*(Top area of A + Top area of B + Top area of C)
Bottom area of A = 16 pi
So, top area of A = $$(3/4)^2$$ * 16 pi = 9 pi
Top area of B = 4 pi
Top area of C = pi
Since radius of big cone = 4, slant height = 8
So, TSA of big cone = pi * 4 * 8 + pi * 16 = 48 pi
Required ratio = 48 pi / (48 pi + 2*(9 pi + 4 pi + pi)) = 48/(48 + 28) = 48/76 = 12/19