Question

Find the points on the curve $6y=x^3+2$ at which the y - coordinates is changing 8 times as fast the x - coordinates .

Answer 1

$Given$ $\frac{dy}{dt}=8\frac{dx}{dt}\therefore \frac{dy}{dx}=8$
but $6y=x^3+2$
$6\frac{dy}{dx}=3x^2\Rightarrow 6\times 8=3x^2$
$\Rightarrow x^2=16\Rightarrow x=\pm 4$
when $x=4,6y=66\Rightarrow y=11$
when $x=-4,6y=-62\Rightarrow y=\frac{-31}{3}$
hence the required point are $(4,11)$ and
$(-4,-31/3)$