Can I get help for this question please?
Suppose that a tank containing a liquid is vented to the air at the top and has an outlet at the bottom through which the liquid can drain. It follows from Torricelli’s law in physics that if the outlet is opened at time $t = 0$, then at each instant the depth of the liquid $h(t)$ and the area $A(h)$ of the liquid’s surface are related by
$$A(h){dh\over dt} = −k\sqrt h$$
where $k$ is a positive constant that depends on such factors as the viscosity of the liquid and the cross-sectional area of the outlet. Assume that $h$ is in metres, $A(h)$ is in square metres, and $t$ is in seconds. Now a conic tank in the accompanying figure is filled to a depth of 1 metre at time $t = 0$ and suppose that the constant in Torricelli’s law is $k = 0.025$.
(a) Find the depth $h(t)$ at time t while the liquid is draining.
(b) How many seconds will it take for the tank to drain completely?
This is what I have tried so far:
Since the base of the cone is just a circle, the area will then be:
$$\mathrm{Area} = {\pi}r^2$$
I thus got the equation:
$${\pi}r^2{dh \over dt}= -k {\sqrt h}$$
$${\pi}r^2{dh \over {\sqrt h}}= -k dt$$
$$\int{1 \over {\sqrt h}} dh = \int {-k \over {\pi}r^2} dt$$
I then got my final equation as:
$$h(t) = \left( {-2k \over {\pi}r^2}t + {c \over 2} \right)^2$$
In your attempt, you stated that the radius of the circle with area $A(h)$ is $r$. But note that this area is a function of $h$, which is why it says $A(h)$ rather than, say $A(r)$. So there must be some relation between the radius of this circle, and the height $h$.
To find this relation, we need to use some geometry. For the full cone, the diameter of the circle and the height of the circle are both equal to $1$m. If the water level falls to a height $h$, then the diameter of the new circle must become $h$. This fact is obtained by considering similar triangles. Diameter is twice the radius, so $r=h/2$.
So where you had written $${\pi}r^2{dh \over dt}= -k {\sqrt h}$$you should replace this $r$ with $h/2$, and solve the equation from there. This should yield the correct answer.