Question

A winery has a vat with two pipes leading to it. The inlet pipe can fill the vat in 5 ​hours, while the outlet pipe can empty it in 8 hours. How long will it take to fill the vat if both pipes are left​ open?

Answer 1

To find the time it takes to fill the vat when both pipes are open, we need to determine the combined rate at which the pipes fill and empty the vat. Given the rates of the inlet and outlet pipes, we can calculate the combined rate and then find the time using the volume of the vat.

Step-by-step explanation:

To find the time it takes to fill the vat when both pipes are open, we need to determine the combined rate at which the pipes fill and empty the vat. Let's represent the rate of the inlet pipe as +1/v and the rate of the outlet pipe as -1/v, where v is the volume of the vat. The combined rate is equal to the sum of these rates.

Given that the inlet pipe can fill the vat in 5 hours and the outlet pipe can empty it in 8 hours, we can calculate the combined rate as follows:

+1/5 - 1/8 = 8/40 - 5/40 = 3/40

The combined rate is 3/40. To find the time it takes to fill the vat, we can invert the rate and multiply by the volume of the vat:

40/3 * v = (40 * v) / 3.

So, when both pipes are open, it will take (40 * v) / 3 hours to fill the vat.

Answer 2

`time=40/3hours\approx 13.3hours`

Step-by-step explanation:

The rates are additive: you can calculate the inlet rate and the outlet rate and add them algebraically, i.e. the inlet rate will be positive and the outlet rate will be negative.

1. Inlet rate:

`1vat/5hours`

2. Outlet rate:

`1vat/8hours`

3. Net rate:

`\text{Inlet rate - outlet rate}=1vat/5hours-1vat/8hours\\\\ \text{Net rate}=(8-5)vat/40hour=3vat/40hour=(3/40)vat/hour`

4. Time to fill the vat

`rate=amount/time\implies time=amount/rate\\ \\ time=1vat/(3vat/40hour)`

`time=40/3hours\approx 13.3hours`