To derive the expression relating the changing variable with time, let's consider the given information and apply some principles of fluid mechanics.
Given:
- Cross-sectional area of the tank: A
- Mass of water in the tank: m
- Density of water: ρ
- Area of the hole at the bottom: A₀
- Volumetric flow rate: q = C⋅h, where C is a constant and h is the water level in the tank.
We can start by relating the mass of water in the tank to its volume using the density:
m = ρ⋅V
The volume V can be calculated using the cross-sectional area A and the water level h:
V = A⋅h
Now, let's express the rate of change of mass with respect to time:
dm/dt = d(ρ⋅V)/dt
Using the product rule of differentiation, we can expand this expression:
dm/dt = ρ⋅dV/dt + V⋅dρ/dt
Next, let's consider how the volume V changes with time. Since water is draining out of the tank through the hole at the bottom, the volumetric flow rate q is equal to the cross-sectional area of the hole A₀ multiplied by the velocity v of the water draining out:
q = A₀⋅v
The velocity v can be related to the water level h by applying the principle of Torricelli's law for flow through an orifice:
v = √(2⋅g⋅h)
Where g is the acceleration due to gravity. Substituting this expression for v into the equation for q, we have:
q = A₀⋅√(2⋅g⋅h)
Now, let's differentiate the equation q = A₀⋅√(2⋅g⋅h) with respect to time t:
dq/dt = d(A₀⋅√(2⋅g⋅h))/dt
Using the chain rule of differentiation, we can calculate this:
dq/dt = A₀⋅(1/2)⋅(2⋅g/h)⋅(dh/dt)
Simplifying further, we have:
dq/dt = A₀⋅g/√h⋅(dh/dt)
Since we know that q = C⋅h, we can substitute this into the equation:
C⋅dh/dt = A₀⋅g/√h⋅(dh/dt)
Now, rearranging the equation to isolate the changing variable, we get:
C⋅dh/dt - A₀⋅g/√h⋅(dh/dt) = 0
Combining the terms on the left-hand side and factoring out the common factor of dh/dt, we have:
(dh/dt)⋅(C - A₀⋅g/√h) = 0
Since dh/dt cannot be zero (as the water level is changing), the expression in parentheses must be zero:
C - A₀⋅g/√h = 0
Solving for h, we get:
C = A₀⋅g/√h
Now, we can solve this equation to obtain an expression relating the changing variable (h) with time. By manipulating the equation further, we can isolate h:
√h = A₀⋅g/C
Squaring both sides:
h = (A₀⋅g/C)