Q. A rectangle with one side lying along the $x$-axis is to be inscribed in the closed region of the $x y$ plane bounded by the lines $y=0, y=3 x$ and $y=30-2 x$. The largest area of such a rectangle is
MHT CETMHT CET 2020
Solution:
$A=\left(x_{2}-x_{1}\right) y$
$y=3 x_{1}$ and $y=30-2 x_{2}$
$A(y)=\left(\frac{30-y}{2}-\frac{y}{3}\right) y$
$6 A(y)=(90-3 y-2 y) y=90 y-5 y^{2}$
$6 A^{\prime}(y)=90-10 y=0$
$\Rightarrow y=9 ; A^{\prime \prime}(y)=-10<0$
$x_{1}=3 ; x_{2}=21 / 2$
$\Rightarrow A_{\max }=\left(\frac{21}{2}-3\right) 9=\frac{15 \times 9}{2}=\frac{135}{2}$
