We pour water into a conical container of a radius of 10 cm with a height of 36cm at a rate of 70 cm^3/s. At what velocity does the water level increase when it reaches 12 cm?
dh/dt=?
To find the velocity at which the water level increases, we need to calculate the rate of change of the height of the water level with respect to time (dh/dt).
First, let's set up some given information:
- Radius of the conical container (r) = 10 cm
- Height of the conical container (H) = 36 cm
- Rate of pouring water (dV/dt) = 70 cm^3/s (This represents the rate at which the volume of water is increasing)
Now, we can use the formula for the volume of a cone to relate the height and radius of the water level in the conical container:
V = (1/3)πr^2h
Differentiating both sides with respect to time (t), we get:
dV/dt = (1/3)π * (2rh * dh/dt) ------------ Eq. 1
We have the value of dV/dt as 70 cm^3/s, but we are looking to find dh/dt.
Now, let's substitute the given values:
r = 10 cm
dh/dt = ?
dV/dt = 70 cm^3/s
Since it is given that the water level is at a height of 12 cm, we can substitute h = 12 cm in Eq. 1:
70 = (1/3)π * (2 * 10 * 12 * dh/dt)
Simplifying the equation further, we get:
70 = π * 240 * dh/dt
dh/dt = 70 / (240π)
dh/dt ≈ 0.092814 cm/s
Therefore, the velocity at which the water level increases when it reaches 12 cm is approximately 0.092814 cm/s.