To solve the problem of how long it takes to fill the pond when both the inlet and discharge pipes are opened at the same time, we can follow these steps:
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Determine the rate of the inlet pipe: The inlet pipe can fill the pond in 2 hours, so its filling rate is: \[ \text{Rate of inlet} = \frac{1}{2} \text{ pond per hour} \]
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Determine the rate of the discharge pipe: The discharge pipe can empty the pond in 5 hours, so its emptying rate is: \[ \text{Rate of discharge} = -\frac{1}{5} \text{ pond per hour} \] (negative because it is removing water from the pond)
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Calculate the net rate when both pipes are open: \[ \text{Net rate} = \text{Rate of inlet} + \text{Rate of discharge} \] \[ = \frac{1}{2} - \frac{1}{5} \]
To perform this addition, we need a common denominator, which is 10: \[ = \frac{5}{10} - \frac{2}{10} = \frac{3}{10} \text{ pond per hour} \]
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Determine how long it takes to fill the pond at the net rate: Let \( t \) be the time taken to fill the pond. The equation relating the rate and time taken to fill the pond (1 pond) is: \[ \text{Net rate} \times t = 1 \text{ pond} \] \[ \frac{3}{10} t = 1 \]
Solving for \( t \): \[ t = \frac{1}{\frac{3}{10}} = \frac{10}{3} \text{ hours} \approx 3.33 \text{ hours} \]
Thus, it takes approximately \( 3.33 \) hours (or \( 3 \) hours and \( 20 \) minutes) to fill the pond when both the inlet and discharge pipes are opened at the same time.
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