Question

A point is moving along the cubical parabola $12y=x^3$. The rate of ordinate is less than the rate of abscissa when

• A. $x<-2$ or $x>2$
• B. $x=\pm 2$
• C. $-2$
• D. $0$

Answer 1

The correct option is A $x<-2$ or $x>2$

Given parabola equation is $12y=x^3$

Given that the rate of the ordinate is less than the rate of the abscissa.

Therefore, $\frac{dy}{dt}<\frac{dx}{dt}$ ------(1)

Upon derivating the curve we get

$12\frac{dy}{dt}=3x^2\frac{dx}{dt}$

$⟹\frac{dy}{dt}=\frac{x^2}{4}\times \frac{dx}{dt}$ ------(2)

Substituting (2) in (1) we get

$\frac{x^2}{4}\times \frac{dx}{dt}<\frac{dx}{dt}$

$⟹(\frac{x^2}{4}-1)\times \frac{dx}{dt}<0$

$⟹(\frac{x^2}{4}-1)<0$

$⟹x^2-4<0$

$⟹(x-2)(x+2)<0$

$⟹x<-2$ or $x>2$