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 is continuing to fill at a rate of 4 gallons per minute. Write an equation to show when the two vats will have an equal amount of liquid.
Let's assume that after t minutes, the two vats will have an equal amount of liquid.
The amount of liquid in the first vat after t minutes will be 6t (since it is filling at a rate of 6 gallons per minute).
The amount of liquid in the second vat after t minutes will be 10 + 4t (since it already has 10 gallons and is filling at a rate of 4 gallons per minute).
To find when the two vats will have an equal amount of liquid, we can set these two expressions equal to each other:
6t = 10 + 4t
Simplifying this equation, we get:
2t = 10
t = 5
Therefore, the two vats will have an equal amount of liquid after 5 minutes.
To determine when the two vats will have an equal amount of liquid, we can use the equation:
EmptyVat + Pump1Rate * t = FilledVat + Pump2Rate * t
Where:
- EmptyVat represents the initial amount of liquid in the empty vat (0 gallons in this case).
- Pump1Rate represents the rate at which Pump 1 is filling the empty vat (6 gallons per minute).
- FilledVat represents the initial amount of liquid in the filled vat (10 gallons in this case).
- Pump2Rate represents the rate at which Pump 2 is filling the filled vat (4 gallons per minute).
- t represents the time in minutes.
Plugging in the values, the equation becomes:
0 + 6t = 10 + 4t
This equation represents the point in time when both vats will have an equal amount of liquid.
Let t represent the number of minutes elapsed. Since one vat starts empty and fills at a rate of 6 gallons per minute, the amount of liquid in the first vat after t minutes can be represented by the equation 6t.
Similarly, the second vat already has 10 gallons and fills at a rate of 4 gallons per minute. Therefore, the amount of liquid in the second vat after t minutes can be represented by the equation 10 + 4t.
To find when the two vats will have an equal amount of liquid, we can equate the two equations:
6t = 10 + 4t
Simplifying the equation:
6t - 4t = 10
2t = 10
Finally, to solve for t, we divide both sides of the equation by 2:
t = 10/2
t = 5
Therefore, the two vats will have an equal amount of liquid after 5 minutes.