Answer:
dh/dt = 8/9π
Step-by-step explanation:
Given that
Volume of A cone = 1/3πr²h.
Assuming v is the volume of the tank, then dv/dt = 2 m³/min
Assuming h is the height of the water we're looking for dh/dt
To start, we need to find a relationship between dh/dt and dv/dt
We then write v as a function of h, instead of having it as h & r, and then by using similar triangles, we notice that
r/h = 2/4, essentially, r = h/2.
Now, we substitute for this r, in the formula for volume
V = 1/3πr²h
V = 1/3 * π * (h/2)² * h
V = 1/3 * π * h³/4
If we differentiate v, wrt t, we have
dv/dt = π/12 * 3h² * dh/dt
dv/dt = π/4 * h² * dh/dt
Remember that h = 3, so, substitute for h
2 = π/4 * 9 * dh/dt
8/π = 9 * dh/dt
dh/dt = 8/9π
