Water is being pumped into an inverted conical tank at a constant rate. The tank has height 13 meters and the diameter at the top is 5.5 meters. If the water level is rising at a rate of 29 centimeters per minute when the height of the water is 2 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute.
To find the rate at which water is being pumped into the tank in cubic centimeters per minute, we can use the concept of similar triangles and related rates.
First, let's calculate the radius of the water level in the tank when the height is 2 meters. Since the diameter at the top of the tank is 5.5 meters, the radius, r, can be determined by dividing the diameter by 2:
r = 5.5 m / 2 = 2.75 m
Now, we can find the rate at which the radius is changing using a similar triangle. Let's call the rate at which the radius of the water level is changing dr/dt (in meters per minute). Since we know that the height of the water level is rising at a rate of 29 centimeters per minute, we can convert it to meters per minute by dividing by 100:
dh/dt = 29 cm/min / 100 = 0.29 m/min
Using the concept of similar triangles, we know that the rate at which the radius is changing (dr/dt) is related to the rate at which the height is changing (dh/dt) by the formula:
dr/dt = (r / h) * dh/dt
Substituting the values we have:
dr/dt = (2.75 m / 2 m) * 0.29 m/min
dr/dt = 1.375 m * 0.29 m/min
dr/dt ≈ 0.39875 m²/min
Now, we can find the rate at which the volume of water is changing using the formula for the volume of a cone:
V = (1/3) * π * r^2 * h
Taking the derivative with respect to time, we get:
dV/dt = (1/3) * π * (2r * dr/dt * h + r^2 * dh/dt)
Substituting the values we have:
dV/dt = (1/3) * π * (2 * 2.75 m * 0.39875 m²/min * 2 m + (2.75 m)^2 * 0.29 m/min)
Calculating this expression, we can determine the rate at which water is being pumped into the tank in cubic meters per minute. Note that we have to convert the units from cubic meters to cubic centimeters:
dV/dt ≈ (3.759275 m³/min) * (1,000,000 cm³/m³)
Finally, we can convert the rate from cubic meters per minute to cubic centimeters per minute:
dV/dt ≈ 3,759,275,000 cm³/min
Therefore, the rate at which water is being pumped into the tank is approximately 3,759,275,000 cubic centimeters per minute.