Water flows at the rate of 2000 cibic cm per second into a verical cylindrical tank 120 cm in diameter and 6 m high. How fast is the water level rising?

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To find the rate at which the water level is rising, we'll need to use the formula for the volume of a cylinder:

V = π * r^2 * h

where V is the volume, π is pi (approximately 3.14159), r is the radius, and h is the height.

Given that the diameter of the tank is 120 cm, the radius (r) can be found by dividing the diameter by 2:

r = 120 cm / 2 = 60 cm

The height of the tank is given as 6 m, which needs to be converted to centimeters:

height in cm = 6 m * 100 cm/m = 600 cm

Now let's substitute these values into the volume formula:

V = 3.14159 * (60 cm)^2 * 600 cm
V ≈ 678,578,947.3684 cubic cm (rounded to 4 decimal places)

Since the water is flowing into the tank at a rate of 2000 cubic cm per second, we can find the rate at which the water level is rising (dh/dt) by differentiating the volume equation with respect to time (t):

dV/dt = d(h * π * r^2)/dt
2000 = π * r^2 * dh/dt

Since we want to find dh/dt, we can rearrange the equation:

dh/dt = 2000 / (π * r^2)

Now let's substitute the known values and calculate the rate at which the water level is rising:

dh/dt = 2000 / (3.14159 * (60 cm)^2)
dh/dt ≈ 0.176 cm/sec (rounded to 3 decimal places)

Therefore, the water level in the tank is rising at a rate of approximately 0.176 cm/sec.

To find how fast the water level is rising, we need to determine the rate of change of the volume of water in the tank with respect to time.

The volume of a cylinder can be calculated using the formula V = πr^2h, where V is the volume, π is a constant (approximately 3.14), r is the radius of the cylinder's base, and h is the height of the cylinder.

Given that the tank has a diameter of 120 cm, the radius would be half of that, which is 60 cm or 0.6 m. The height is given as 6 m.

We are told that the water is flowing at a rate of 2000 cubic centimeters per second. However, we need to convert this to cubic meters since the other measurements are in meters.

To convert from cubic centimeters (cm^3) to cubic meters (m^3), divide by 1,000,000 (since there are 1,000,000 cubic centimeters in a cubic meter).

2000 cm^3/s ÷ 1,000,000 = 0.002 m^3/s

Now we can find the rate of change of the water level (dh/dt), which represents how fast the height is changing with respect to time.

Let's differentiate the formula for the volume of a cylinder with respect to time:

V = πr^2h

Differentiating both sides of the equation with respect to time (t), we get:

dV/dt = d(πr^2h)/dt

The derivative of πr^2 with respect to time is 0 since π and r^2 are constant.

dV/dt = πr^2(dh/dt)

Now we can substitute the given values into the equation:

0.002 m^3/s = π(0.6 m)^2 (dh/dt)

Simplifying the equation:

0.002 m^3/s = 0.36π (dh/dt)

Finally, to find dh/dt, we can rearrange the equation:

dh/dt = (0.002 m^3/s) / (0.36π)

Calculating this equation will give us the rate at which the water level is rising in meters per second.

the cross-section has area 3600π cm^2

2000 cm^3/s ÷ 3600π cm^2 = 5/(9π) cm/s

Or, if you want to use calculus
v = πr^2 h = 3600πh
dv/dt = 3600π dh/dt
2000 = 3600π dh/dt
dh/dt = 5/(9π) cm/s

Well, if I were the water level, I'd probably be rising pretty fast because the water is coming in at a rate of 2000 cubic cm per second! Talk about a splash!