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Method 1 (not recommended)
The formula for the volume of a cone is V = (Pi* height * radius^2) * 1/3
Let take BO as "x",
we have OF = 2, thus AF = 10. Triangle AEF and ADO are in the same shape, then AF/AO = EF/BO = 10/12 =5/6, hence EF=5/6*x
Follow the same step, we get AD/AO = CD/BO = 6/12 = 1/2, hence CD = 1/2*x
For short, let's call G for grey, B for blue and O for orange
V (G+B+O) = Pi*12*x^2 *1/3 = Pi*4*x^2
V (G+B) = Pi*10*(5/6*x)^2 *1/3 = Pi* 125/54 * x^2
V (G) = Pi*6*(1/2*x)^2 *1/3 = Pi* 1/2 * x^2
They are asking us about the ratio among V (G), V (B) and V (O). Let's calculate them
V(G) = Pi*1/2*x^2 as above
V (B) = V(G+B) - V (G) = Pi*49/27*x^2
V(O) = V(G+B+O) - V(G+B) = Pi*91/54*x^2
V(G):V(B):V(O) = 1/2 : 49/27 : 91:54 = 27/54 : 98/54 : 91/54 = 27:98:91. Answer A wins.
Method 2
For those who hate equations, I have a more simple way to calculate this
Cone G, G+B, and G+B+O have the same shape and the ration among their height is 6 : 10: 12 = 1/2:5/6:1
The ratio among the volumes of those cone is the triple exponent of their height ratio, how can I get this?
The volume of a cone is calculated as V = Pi* r^2 * h * 1/3. Note that "r" is squared in this formula, so we must square the ratio among "r" multiply with ratio among heights, and the ratio among radiuses of those cones' bases is the same to that of their heights
V(G):V(G+B):V(G+B+O) = (1/2)^3 : (5/6)^3 : 1^3 = 1/8 : 125/216 : 1
V(B) = V(G+B) - V(G) = 125/216 - 1/8 = 49/108
V(O) = V(G+B+O) - V(G+B) = 1 - 125/216 = 91/216
V(G):V(B):V(O) = 1/8 : 49/108 :91/216 = 27/216 : 98/216: 91:216 = 27:98:91, Yeah, it's A.
Sorry for my bad English and expression, as well as small image ( dont know how to upload a bigger one)
Hope this could help.
