To find the average rate at which the pump is working, we need to calculate the power output of the pump. Power is defined as the rate of doing work, which can be expressed mathematically as:
\[ P = \frac{W}{t} \]
where \( P \) is power, \( W \) is work done, and \( t \) is the time taken.
### Step 1: Calculate the Work Done by the Pump
The work done by the pump can be calculated using the formula:
\[ W = F \cdot d \]
where \( F \) is the force exerted by the pump and \( d \) is the displacement of the water.
### Step 2: Calculate the Force Exerted by the Pump
The force exerted by the pump can be calculated using the formula:
\[ F = m \cdot a \]
where \( m \) is the mass of the water ejected and \( a \) is the acceleration. However, since the water is being ejected at a constant speed, we can consider the force exerted to be equal to the weight of the water being pumped against gravity.
1. **Mass of water (m)** = 12000 kg
2. **Speed (v)** = 4 m/s
3. **Time (t)** = 40 s
Since the water is ejected at a constant speed, we can calculate the force as:
\[ F = m \cdot g \]
where \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)). However, in this case, we are more interested in the force required to maintain the speed of the water.
### Step 3: Calculate the Displacement of Water
The displacement \( d \) can be calculated using the formula:
\[ d = v \cdot t \]
Substituting the values:
\[ d = 4 \, \text{m/s} \cdot 40 \, \text{s} = 160 \, \text{m} \]
### Step 4: Calculate the Work Done
Now, we can calculate the work done by the pump:
\[ W = F \cdot d \]
Since the force is not directly calculated from acceleration, we can consider the effective force to maintain the speed, which is related to the momentum change. However, for our calculation, we can directly use the mass flow rate and speed to find the work done.
The work done can also be thought of as the kinetic energy imparted to the water:
\[ W = \frac{1}{2} mv^2 \]
But since we are ejecting water continuously, we can calculate the total work done over the time:
\[ W = m \cdot v^2 \]
### Step 5: Calculate the Power
Finally, we can calculate the power:
\[ P = \frac{W}{t} \]
Substituting the values we have:
1. **Work Done (W)** = \( 12000 \, \text{kg} \cdot (4 \, \text{m/s})^2 = 12000 \cdot 16 = 192000 \, \text{J} \)
2. **Time (t)** = 40 s
Now calculate the power:
\[ P = \frac{192000 \, \text{J}}{40 \, \text{s}} = 4800 \, \text{W} \]
To convert this to kilowatts:
\[ P = \frac{4800 \, \text{W}}{1000} = 4.8 \, \text{kW} \]
### Final Answer
The average rate at which the pump is working is **4.8 kW**.
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