Calculate the volume of water in glass over time.

Question

For A) I found that volume should be defined by But I got no idea what to do in b) and c)

Answer 1

As you well have shown, the volume of water inside the glass is given (note that $h$ is a function of time):

$$V(t) = \frac{\pi h^2(t)}{2}. $$ On the other hand, the surface area exposed to the air by water is given by:

$$A_s = \pi r^2(t),$$ where $r$ is the instant radius of the glass when the height of water is $h(t)$. This two quantities are related by $h = r^2$ since the shape of the glass is the given parabola $y = x^2$. You can readily show that $A_s = \sqrt{2 \pi V}$ by elementary substitution. Since we are given the rate of evaporation per unit of volume and unit of area (exposed surface), $q$, we have that:

$$ \frac{\mathrm{d}V}{\mathrm{d}t} = q A_s,$$ which readily yields to the desired expression for $\dot{V}(t)$: $\dot{V} = -q \sqrt{2\pi V}$, which is a separable differential equation, to be integrated as follows:

$$ \int^{V=0}_{V=V_0} \frac{\mathrm{d}V}{\sqrt{V}} = - q \sqrt{2\pi} \int^{t=t_F}_{t=0} \, \mathrm{d}t,$$ where $V_0$ is the volume of water at $t=0$, i.e., $V_0 = \pi h^2(0)/2$, and $t_F$ is the time you are being asked to compute.

I hope you find this helpful.

Cheers!