The diameter of the liquid's surface is a function of the liquid's height. It will be
d = (20 cm)/(30 cm)*h = (2/3)h
Then, the volume will be
v = (1/3)*(π/4)d^2*h
= π/12*(2/3h)^2*h
= π/27*h^3
The derivative of this expression is
dv/dt = (π/27)*(3h^2)*dh/dt = π/9*h^2*dh/dt
Solving for dh/dt, we get
(dv/dt)*9/(π*h^2) = dh/dt
With your numbers, we have
(0.2 cm^3/s)*9/(π*(12 cm)^2) = dh/dt
(1.8/(144π)) cm/s = dh/dt
dh/dt ≈ 0.00398 cm/s
Perhaps the key is to realize that the diameter is a function of height. It is not constant. So, the volume is proportional to the cube of height. The cube introduces a factor of 3 to the derivative.
d = (20 cm)/(30 cm)*h = (2/3)h
Then, the volume will be
v = (1/3)*(π/4)d^2*h
= π/12*(2/3h)^2*h
= π/27*h^3
The derivative of this expression is
dv/dt = (π/27)*(3h^2)*dh/dt = π/9*h^2*dh/dt
Solving for dh/dt, we get
(dv/dt)*9/(π*h^2) = dh/dt
With your numbers, we have
(0.2 cm^3/s)*9/(π*(12 cm)^2) = dh/dt
(1.8/(144π)) cm/s = dh/dt
dh/dt ≈ 0.00398 cm/s
Perhaps the key is to realize that the diameter is a function of height. It is not constant. So, the volume is proportional to the cube of height. The cube introduces a factor of 3 to the derivative.
