Question:
If \[ 2x+\frac{y}{3}=12 \] and \[ xy=30, \] then \[ 8x^3+\frac{y^3}{27}= \]
(a) 1008
(b) 168
(c) 106
(d) none of these
Solution:
\[ \left(2x+\frac{y}{3}\right)^3 = (2x)^3+\left(\frac{y}{3}\right)^3 +3(2x)\left(\frac{y}{3}\right)\left(2x+\frac{y}{3}\right) \]
\[ 12^3 = 8x^3+\frac{y^3}{27} +3(2x)\left(\frac{y}{3}\right)(12) \]
\[ 1728 = 8x^3+\frac{y^3}{27} +6xy(12) \]
\[ 1728 = 8x^3+\frac{y^3}{27} +72(30) \]
\[ 1728 = 8x^3+\frac{y^3}{27} +2160 \]
\[ 8x^3+\frac{y^3}{27} = 1728-2160 \]
\[ =-432 \]
Hence, the correct answer is
\[ \boxed{-432} \]