Correct option is (4) 162
\(y=3 x, 2 y=27-3 x\) and \(y=3 x-x \sqrt{x}\)
\(A=\int_{0}^{3} 3 x-(3 x-x \sqrt{x}) d x+\int_{3}^{9}\left(\frac{27-3 x}{2}-(3 x-x \sqrt{x})\right) d x\)
\(A=\int_{0}^{3} \mathrm{x}^{3 / 2} \mathrm{dx}+\int_{3}^{9} \frac{27}{2}-\frac{9 \mathrm{x}}{2}+\mathrm{x}^{3 / 2} \mathrm{dx}\)
\(\mathrm{A}=\left[\frac{2 \mathrm{x}^{5 / 2}}{5}\right]_{0}^{3}+\frac{27}{2}[\mathrm{x}]_{3}^{9}-\frac{9}{2}\left[\frac{\mathrm{x}^{2}}{2}\right]_{3}^{9}+\left[\frac{2 \mathrm{x}^{5 / 2}}{5}\right]_{3}^{9}\)
\(\mathrm{A}=\frac{2}{5}\left(3^{5 / 2}\right)+\frac{27}{2}(6)-\frac{9}{4}(72)+\frac{2}{5}\left(9^{5 / 2}-3^{5 / 2}\right)\)
\(\mathrm{A}=\frac{2}{5}\left(3^{5 / 2}\right)+81-162+\frac{2}{5} \times 3^{5}-\frac{2}{5} \times 3^{5 / 2}\)
\(\mathrm{A}=\frac{486}{5}-81=\frac{81}{5}\)
\(10 \mathrm{~A}=162\)
