Given two squares of sides x and y such that `y=x+x^(2)` what is the rate of change of area of the second square with respect to the area of the first square?

Options

A. `1+3x+2x^(2)`
B. `1+2x+3x^(2)`
C. `1-2x+3x^(2)`
D. `1-2x-3x^(2)`

Answer

A

Explanation

Let x be the side of first square and y be the side of second square
there Area of first square `A_(1)=x^(2)`
and area of second sqare `A_(2)=y^(2)`
`=(x+x^(2)^(2)) (before y=x+x^(2))`
`=x^(2)+x^(4)+2x^(3)`
Hence the rate of change of area of the second sqare with respect to the area of the first square
`=(dA_(2))/(dA_(1))=(2x+4x^(3)+6x^(2))/(2x)=1+2x^(2)+3x`