Hi all I was wondering how I would find an expression for the rate of change of a spherical container which is being filled up with a liquid.
I am aware that I must use Derivatives as the liquid is being filled at a $\frac{dV_1}{dt}$.
$$ V_1 = \pi y^2 \left(\frac{r-y}{3}\right) $$ is the equation which defines the volume of the liquid in the container.
What I am told is to differentiate the equation with respect to $y$ and then rearrange for $\frac{dy}{dt}$
Any help would be greatly appreciated :)
$$V_1 = \pi y^2 \left(\frac{r-y}{3}\right)=\frac{\pi ry^2}{3}-\frac{\pi y^3}{3}$$ $$\frac{dV_1}{dt}=\frac{\pi r}{3}(2y)\frac{dy}{dt}-\frac{\pi}{3}(3y^2)\frac{dy}{dt}$$ $$\frac{dV_1}{dt}=\left(\frac{2\pi ry}{3}-\pi y^2\right)\frac{dy}{dt}$$ Now you can rearrange to get $\dfrac{dy}{dt}$ in terms of $\dfrac{dV_1}{dt}$.