Question

$x$ and $y$ are the sides of two squares such that $y=x-x^2$. The rate of change of the area of the second square with respect to that of the first square is

• A. $2(1-x^2)x$
• B. $2x^2-3x+1$
• C. $2(2x^2-3x+1)$
• D. none of these

Answer 1

The correct option is B $2x^2-3x+1$

$\frac{dy}{dt}=1-2x\frac{dx}{dt}$
Or
$\frac{dy}{dx}=1-2x$
Now
$Are{a}_1=y^2$
and
$Are{a}_2=x^2$.
Hence
$\frac{d{A}_1}{d{A}_2}=\frac{y}{x}.\frac{dy}{dx}$
$=\frac{x-x^2}{x}.1-2x$
$=(1-x)(1-2x)$
$=1-3x+2x^2$.
$=2x^2-3x+1$.