The gauge pressure of `3xx10^(5)N//m^(2)` must be maintained in the main water pipes of a city how much work must be done to pump 50,000 `m^(3)` of water at a pressure of `1.0xx10^(5)N//m^(2)`

Options

A. `10^(11)J`
B. `10^(10)J`
C. `10^(9)J`
D. `10^(8)J`

Answer

To solve the problem, we need to calculate the work done to pump water at a given pressure. Here's the step-by-step solution:

### Step 1: Identify the Given Values
- Gauge pressure to be maintained in the pipes, \( P_2 = 3 \times 10^5 \, \text{N/m}^2 \)
- Pressure at which water is pumped, \( P_1 = 1 \times 10^5 \, \text{N/m}^2 \)
- Volume of water to be pumped, \( V = 50,000 \, \text{m}^3 = 5 \times 10^4 \, \text{m}^3 \)

### Step 2: Calculate the Change in Pressure
The change in pressure (\( \Delta P \)) can be calculated as:
\[ \Delta P = P_2 - P_1 \]
Substituting the values:
\[ \Delta P = (3 \times 10^5) - (1 \times 10^5) = 2 \times 10^5 \, \text{N/m}^2 \]

### Step 3: Calculate the Work Done
The work done (\( W \)) to pump the water can be calculated using the formula:
\[ W = \Delta P \times V \]
Substituting the values:
\[ W = (2 \times 10^5) \times (5 \times 10^4) \]

### Step 4: Perform the Multiplication
Calculating the multiplication:
\[ W = 2 \times 5 \times 10^5 \times 10^4 = 10 \times 10^9 \, \text{J} \]
Thus, the work done is:
\[ W = 10^{10} \, \text{J} \]

### Step 5: Conclusion
The work done to pump 50,000 m³ of water at the specified pressure is \( 10^{10} \, \text{J} \).

### Final Answer
The work done is \( 10^{10} \, \text{J} \).

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