the maths F3 ///

A right circular cone is divided into 3 portions A,B and C by planes parallel to the base as shown below. the hights of A,B And C ARE 1m, 2m,AND 3m respectively , And their base radii Are r1,r2 And r3 respectively .

http://hk.geocities.com/leemingki21/2m.bmp

find the ratios

(a)r,1 r2 ,r3,

Let [AB] be the cone formed by A+B,
and [ABC] be the cone formed by A+B+C.

Cone A, [AB] & [ABC] are similar cones, with height 1m, 3m & 6m resp.

Since they are similar cones, their ratio of radii = their ratio of height
I.E.　r1 : r2 : r3 = 1 : 3 : 6

(b)volume of A : volume of C

Volume of cone A : [AB] : [ABC] = 1³ : 3³ : 6³
　　Volume of A : A+B : A+B+C = 1 : 27 : 384

∴ By Volume :
　　A : A+B = 1 : 27
　　　27A = A+B
　　　　B = 26A

&　A : A+B+C = 1 : 384
　　　384A = A+B+C
　　　384A = A+26A+C
　　　　C = 357A

Volume of A : Volume of C = 1 : 357

(c) curved surface area of B : curved surface area of C

The curved surface area of cone A : [AB] : [ABC] = 1² : 3² : 6²
　　　　　　　　　　　area of A : [AB] : [ABC] = 1 : 9 : 64

∴ By Curved surface area :
　　　　A : A+B = 1 : 9
　　　　　　9A = A+B
　　　　　　B = 8A

&　A : A+B+C = 1 : 64
　　　　　64A = A+B+C
　　　　　　C = 55A

B : C = 8A : 55A = 8 : 55

The curved surface area of B : curved surface are of C = 8 : 55