Question

There are two different vats the first that has 12 gallons and is being filled at the rate of 3 gallons per minute the second that has 100 gallons and is leaking fluid at the rate of 2 gallons per minute. How many seconds both bats have the same amount of liquid.

Answer 1

To find the time when both vats have the same amount of liquid, we can set up equations for the amount of liquid in each vat over time.

1. **First vat**:
- Initial amount of liquid = 12 gallons
- Filling rate = 3 gallons/minute
- Amount of liquid after \( t \) minutes:
\[
V_1(t) = 12 + 3t
\]

2. **Second vat**:
- Initial amount of liquid = 100 gallons
- Leaking rate = 2 gallons/minute
- Amount of liquid after \( t \) minutes:
\[
V_2(t) = 100 - 2t
\]

To find when both vats have the same amount of liquid, we set \( V_1(t) \) equal to \( V_2(t) \):
\[
12 + 3t = 100 - 2t
\]

Now, we can solve for \( t \):
1. Combine like terms:
\[
3t + 2t = 100 - 12
\]
\[
5t = 88
\]
2. Solve for \( t \):
\[
t = \frac{88}{5} = 17.6 \text{ minutes}
\]

To convert minutes to seconds:
\[
t = 17.6 \text{ minutes} \times 60 \text{ seconds/minute} = 1056 \text{ seconds}
\]

Thus, both vats will have the same amount of liquid after **1056 seconds**.