Question

A rectangle with one side lying along the $x$-axis is to be inscribed in the closed region of the $xy$ plane bounded by the lines $y=0,y=3x$ and $y=30-2x$. The largest area of such a rectangle is

• A. $135/8$
• B. $45$
• C. $135/2$
• D. $90$

Answer 1

Answer: (C)

$A=\left(x_2-x_1\right)y$
$y=3x_1$ and $y=30-2x_2$
$A(y)=\left(\frac{30-y}{2}-\frac{y}{3}\right)y$
$6A(y)=(90-3y-2y)y=90y-5y^2$
$6A^{\prime }(y)=90-10y=0$
$\Rightarrow y=9;A^{\prime \prime }(y)=-10<0$
$x_1=3;x_2=21/2$
$\Rightarrow A_{\max }=\left(\frac{21}{2}-3\right)9=\frac{15\times 9}{2}=\frac{135}{2}$