Question

Two pumps are filling large vats of liquid. One vat is empty, and the pump is filling it at a rate of 6 gallons per minute. The other vat already contains 28 gallons and is continuing to fill at a rate of 4 gallons per minute. When will the two vats have the same amount of liquid?(1 point) They will have the same amount of liquid after minutes.

Answer 1

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

For the first vat, which starts empty and fills at a rate of 6 gallons per minute, the amount of liquid in the vat after \( t \) minutes can be expressed as:

\[ \text{Amount in first vat} = 6t \]

For the second vat, which already contains 28 gallons and fills at a rate of 4 gallons per minute, the amount of liquid in the vat after \( t \) minutes can be represented as:

\[ \text{Amount in second vat} = 28 + 4t \]

To find out when both vats will contain the same amount of liquid, we set the two expressions equal to each other:

\[ 6t = 28 + 4t \]

Now, we will solve for \( t \):

1. Subtract \( 4t \) from both sides:

\[ 6t - 4t = 28 \]

\[ 2t = 28 \]

2. Divide both sides by 2:

\[ t = 14 \]

Thus, the two vats will have the same amount of liquid after \( \boxed{14} \) minutes.