The problem involves the concept of related rates in calculus. Here's how you can solve it:

Step 1: Identify what you know and what you need to find out.
We know the rate at which the volume of water in the tank is increasing (dV/dt = 4 m³/min). We need to find out the rate at which the height of the water is increasing (dh/dt).

Step 2: Write down the formula for the volume of a cylinder, which is V = πr²h. In this case, the radius r is constant (3 m), but the height h is changing as the tank fills with water.

Step 3: Differentiate both sides of the volume formula with respect to time t. This gives us dV/dt = πr²dh/dt.

Step 4: Substitute the known values into the equation from step 3. This gives us 4 m³/min = π(3 m)²dh/dt.

Step 5: Solve the equation for dh/dt. This gives us dh/dt = 4 m³/min / (π(3 m)²) = 4/9π ≈ 0.141 m/min.

So, the height of the water in the tank is increasing at a rate of approximately 0.141 meters per minute.

Question

The problem involves the concept of related rates in calculus. Here's how you can solve it:

Step 1: Identify what you know and what you need to find out. 
We know the rate at which the volume of water in the tank is increasing (dV/dt = 4 m³/min). We need to find out the rate at which the height of the water is increasing (dh/dt).

Step 2: Write down the formula for the volume of a cylinder, which is V = πr²h. In this case, the radius r is constant (3 m), but the height h is changing as the tank fills with water.

Step 3: Differentiate both sides of the volume formula with respect to time t. This gives us dV/dt = πr²dh/dt.

Step 4: Substitute the known values into the equation from step 3. This gives us 4 m³/min = π(3 m)²dh/dt.

Step 5: Solve the equation for dh/dt. This gives us dh/dt = 4 m³/min / (π(3 m)²) = 4/9π ≈ 0.141 m/min.

So, the height of the water in the tank is increasing at a rate of approximately 0.141 meters per minute.

Knowee AI · Accepted Answer

The problem involves the concept of related rates in calculus. Here's how you can solve it:

Step 1: Identify what you know and what you need to find out. 
We know the rate at which the volume of water in the tank is increasing (dV/dt = 4 m³/min). We need to find out the rate at which the height of the water is increasing (dh/dt).

Step 2: Write down the formula for the volume of a cylinder, which is V = πr²h. In this case, the radius r is constant (3 m), but the height h is changing as the tank fills with water.

Step 3: Differentiate both sides of the volume formula with respect to time t. This gives us dV/dt = πr²dh/dt.

Step 4: Substitute the known values into the equation from step 3. This gives us 4 m³/min = π(3 m)²dh/dt.

Step 5: Solve the equation for dh/dt. This gives us dh/dt = 4 m³/min / (π(3 m)²) = 4/9π ≈ 0.141 m/min.

So, the height of the water in the tank is increasing at a rate of approximately 0.141 meters per minute.

A cylindrical tank with radius 3 m3 m is being filled with water at a rate of 4 m3/min4 m3/min . How fast is the height of the water increasing?

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