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 is already filled with 10 gallons and it’s continuing to fill at a rate of 4 gallons per minute. Write an equation to show when the two vats will have equal amount of liquid. Use x as your variable

Answer 1

Let's define the amounts of liquid in each vat over time.

1. For the first vat that is empty and is filling at a rate of 6 gallons per minute, the amount of liquid in the vat after \( x \) minutes can be expressed as: \[ V_1(x) = 6x \]

2. For the second vat that starts with 10 gallons and is filling at a rate of 4 gallons per minute, the amount of liquid in that vat after \( x \) minutes can be expressed as: \[ V_2(x) = 10 + 4x \]

We want to find the time \( x \) when both vats have equal amounts of liquid. Therefore, we set the two equations equal to each other: \[ 6x = 10 + 4x \]

Now, we can solve for \( x \):

1. Subtract \( 4x \) from both sides: \[ 6x - 4x = 10 \] \[ 2x = 10 \]

2. Divide by 2: \[ x = 5 \]

So, the two vats will have equal amounts of liquid after 5 minutes.