Question

Two pumps are being filled by two different pumps. The first vat has 161616 gallons is being filled at a rate of 171717 gallons per second. The second vat has 232323 gallons and is being filled at a rate of 151515 gallons per second. After how many seconds will both vats have the same amount of liquid?

Answer 1

Let \( t \) represent the time in seconds after both vats start being filled.

1. For the first vat, we start with 161616 gallons and it is being filled at a rate of 171717 gallons per second. The amount of liquid in the first vat after \( t \) seconds can be expressed as:
\[
\text{Amount in first vat} = 161616 + 171717t
\]

2. For the second vat, we start with 232323 gallons and it is being filled at a rate of 151515 gallons per second. The amount of liquid in the second vat after \( t \) seconds can be expressed as:
\[
\text{Amount in second vat} = 232323 + 151515t
\]

We want to find the time \( t \) when the amounts in both vats are equal. Therefore, we set the two equations equal to each other:
\[
161616 + 171717t = 232323 + 151515t
\]

Now, we will simplify this equation. First, we can move all terms involving \( t \) to one side and constant terms to the other side:
\[
171717t - 151515t = 232323 - 161616
\]

Now simplify both sides:
\[
(171717 - 151515)t = 232323 - 161616
\]
\[
20102t = 70707
\]

Next, we solve for \( t \):
\[
t = \frac{70707}{20102}
\]

Calculating \( t \):
\[
t \approx 3.5
\]

Thus, \( t \) in seconds when both vats will have the same amount of liquid is approximately:
\[
\boxed{3.5} \text{ seconds}
\]