To find out how much water is in each pool after \( t \) minutes, we can create expressions for the amount of water in Pool A and Pool B as they lose water over time.
Pool A
Pool A starts with 400 gallons of water and loses water at a rate of 4 gallons per minute. Therefore, the expression for the amount of water in Pool A after \( t \) minutes is:
\[ \text{Water in Pool A} = 400 - 4t \]
Pool B
Pool B starts with 420 gallons of water and loses water at a rate of 8 gallons per minute. Thus, the expression for the amount of water in Pool B after \( t \) minutes is:
\[ \text{Water in Pool B} = 420 - 8t \]
Finding When Both Pools Have the Same Amount of Water
To find the time \( t \) when both pools have the same amount of water, we can set the two expressions equal to each other:
\[ 400 - 4t = 420 - 8t \]
Solving for \( t \)
- Rearrange the equation:
\[ 400 - 420 = -8t + 4t \]
\[ -20 = -4t \]
- Divide both sides by -4:
\[ t = \frac{20}{4} \]
\[ t = 5 \]
Finding the Amount of Water in the Pools at \( t = 5 \)
Now, we can substitute \( t = 5 \) back into either expression to find the amount of water in both pools at that time.
Using Pool A's expression:
\[ \text{Water in Pool A} = 400 - 4(5) = 400 - 20 = 380 \text{ gallons} \]
Using Pool B's expression:
\[ \text{Water in Pool B} = 420 - 8(5) = 420 - 40 = 380 \text{ gallons} \]
Conclusion
After 5 minutes, both Pool A and Pool B will have the same amount of water, which is 380 gallons.
- Time until equal amounts of water: \( 5 \) minutes
- Amount of water in both pools at that time: \( 380 \) gallons