Two pumps are being filled by two different pumps. The first vat has 16 gallons is being filled at a rate of 12 gallons per second. The second vat has 23 gallons and is being filled at a rate of 10 gallons per second. After how many seconds will both vats have the same amount of liquid?

Question

________seconds

Answer 1

Let \( t \) be the number of seconds after the filling begins.

For the first vat:

• Initial amount = 16 gallons
• Rate of filling = 12 gallons/second

The amount of liquid in the first vat after \( t \) seconds will be: \[ \text{Amount in first vat} = 16 + 12t \]

For the second vat:

• Initial amount = 23 gallons
• Rate of filling = 10 gallons/second

The amount of liquid in the second vat after \( t \) seconds will be: \[ \text{Amount in second vat} = 23 + 10t \]

We are looking for the time \( t \) when both vats have the same amount of liquid: \[ 16 + 12t = 23 + 10t \]

Now, let's solve for \( t \).

1. Subtract \( 10t \) from both sides: \[ 16 + 2t = 23 \]
2. Subtract 16 from both sides: \[ 2t = 23 - 16 \] \[ 2t = 7 \]
3. Divide by 2: \[ t = \frac{7}{2} = 3.5 \]

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