A winery has a vat with two pipes leading to it. The inlet pipe can fill the vat in 7 hours, while the outlet pipe can empty it in 10 hours. How long will it take to fill the vat if both pipes are left open?

Answer 1

To solve this problem, we’ll first determine the rates at which the inlet and outlet pipes fill and empty the vat, respectively.

1. Inlet Pipe:

   • The inlet pipe can fill the vat in 7 hours.
   • Therefore, the rate of the inlet pipe is \( \frac{1}{7} \) vat per hour.

2. Outlet Pipe:

   • The outlet pipe can empty the vat in 10 hours.
   • Therefore, the rate of the outlet pipe is \( \frac{1}{10} \) vat per hour.

Now, we need to find the net rate at which the vat is being filled when both pipes are open. This will be the rate of the inlet pipe minus the rate of the outlet pipe:

\[ \text{Net Rate} = \left(\frac{1}{7} - \frac{1}{10}\right) \text{ vat per hour} \]

To perform this calculation, we need a common denominator. The least common multiple of 7 and 10 is 70. We can rewrite the rates with a common denominator of 70:

\[ \frac{1}{7} = \frac{10}{70}, \quad \frac{1}{10} = \frac{7}{70} \]

Now substituting these values into the equation for the net rate:

\[ \text{Net Rate} = \frac{10}{70} - \frac{7}{70} = \frac{3}{70} \text{ vat per hour} \]

This net rate tells us that when both pipes are open, the vat is being filled at a rate of \( \frac{3}{70} \) vat per hour.

To find out how long it takes to fill the entire vat, we can set up the following equation where \( t \) is the time in hours it takes to fill the vat:

\[ \frac{3}{70}t = 1 \quad (\text{since 1 vat is being filled}) \]

Solving for \( t \):

\[ t = \frac{1}{\frac{3}{70}} = \frac{70}{3} \text{ hours} \]

Calculating \( \frac{70}{3} \):

\[ t \approx 23.33 \text{ hours} \]

Thus, it will take approximately \( 23 \) hours and \( 20 \) minutes to fill the vat if both pipes are left open.

Final Answer:

Approximately \( 23.33 \) hours or \( 23 \) hours and \( 20 \) minutes.