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.

Question

GPT-4o mini · Answer

Let $ t $ be the time in minutes. For the first vat, starting from empty and filling at 6 gallons per minute, the amount of liquid in the first vat after $ t $ minutes is: $$ 6t \text{ gallons} $$ For the second vat, which starts with 28 gallons and fills at 4 gallons per minute, the amount of liquid in the second vat after $ t $ minutes is: $$ 28 + 4t \text{ gallons} $$ We want to find the time $ t $ when the two vats have the same amount of liquid, so we set the two expressions equal to each other: $$ 6t = 28 + 4t $$ To solve for $ t $, we can first subtract $ 4t $ from both sides: $$ 6t - 4t = 28 $$ $$ 2t = 28 $$ Now, divide both sides by 2: $$ t = 14 $$ Thus, the two vats will have the same amount of liquid after **14 minutes**.