Water is leaking out of an inverted conical tank at a rate of 12200 cubic centimeters per minute at the same time that water is being pumped into the tank at a constant rate. The tank has height 15 meters and the diameter at the top is 65 meters. If the water level is rising at a rate of 21 centimeters per minute when the height of the water is 10 meters, find the rate at which water is being pumped into the tank in cubic centimeters per minute
when the water is at depth y, the radius r of the surface can be found using similar triangles:
r/y = (65/2)/15
r = 13/6 y
at depth y, the volume of water is
v = 1/3 pi r^2 y
= 1/3 pi (13/6)^2 y^3
= 169pi/108 y^3
dv/dt = -12200+C = 169pi/36 y^2 dy/dt
at y=10m=1000cm, dy/dt = 21cm/min
-12200 + C = 169pi/36 * 1000000 * 21 = 309708675
C = 309696475
To find the rate at which water is being pumped into the tank, we can use the concept of related rates.
Let's denote the rate at which water is being pumped into the tank as V cubic centimeters per minute.
We are given that the water level is rising at a rate of 21 centimeters per minute when the height of the water is 10 meters, which means dh/dt = 21 cm/min when h = 10 m.
We can find the relationship between the height h and the radius r of the water level at any given time by similar triangles.
Let's denote the radius of the water level as r and the radius of the top of the conical tank as R (which is half the diameter). We can calculate the value of R by R = (65 meters) / 2 = 32.5 meters.
Using the similar triangles, we have:
r / h = R / 15
r = (R * h) / 15
Differentiating both sides of this equation with respect to time (t), we get:
dr/dt = (R/15) * dh/dt
Substituting the given values:
dr/dt = (32.5/15) * 21 cm/min
dr/dt = 45.5 cm/min
Now, the volume of water in the tank at any given time can be calculated using the formula for the volume of a cone:
V = (1/3) * π * r^2 * h
Differentiating both sides of this equation with respect to time (t), we get:
dV/dt = (1/3) * π * (2r * dr/dt * h + r^2 * dh/dt)
Substituting the given values and the calculated value of dr/dt:
12200 cm^3/min = (1/3) * π * (2 * (32.5 meters * 10 meters / 15 meters) * 45.5 cm/min * 10 meters + (32.5 meters * 10 meters / 15 meters)^2 * 21 cm/min)
Now, we need to convert the units to cubic centimeters per minute since the given leak rate is in cubic centimeters per minute.
1 cubic meter = 1,000,000 cubic centimeters
1 meter = 100 centimeters
12200 cm^3/min = (1/3) * π * (2 * (32.5 * 100 * 100 / 15) * 45.5 / (100) * 10 * 100 + (32.5 * 100 * 100 / 15)^2 * 21)
12200 cm^3/min = (1/3) * π * (2 * (32.5 * 100 * 100 / 15) * 0.455 * 1000 + (32.5 * 100 * 100 / 15)^2 * 21)
12200 cm^3/min ≈ 2,283,103.35 cm^3/min
Therefore, the rate at which water is being pumped into the tank is approximately 2,283,103.35 cubic centimeters per minute.