Part A:
Let \( W(t) \) be the amount of water in the tub at time \( t \) minutes, and \( S(t) \) be the amount of sand in the tub at time \( t \) minutes.
Initially, we have:
- Water: \( 34 \) liters
- Sand: \( 22 \) liters
Water is added at a rate of \( 5 \) liters per minute, and sand is added at a rate of \( 7 \) liters per minute. Therefore, we can express \( W(t) \) and \( S(t) \) as:
\[ W(t) = 34 + 5t \]
\[ S(t) = 22 + 7t \]
We want to find \( t \) when \( W(t) = S(t) \):
\[ 34 + 5t = 22 + 7t \]
Rearranging gives:
\[ 34 - 22 = 7t - 5t \]
\[ 12 = 2t \]
\[ t = 6 \]
Thus, in 6 minutes, the tub will contain equal amounts of water and sand.
Answer to Part A: 6 minutes.
Part B:
Now, we need to find the amount of water and sand in the tub when they are equal. We substitute \( t = 6 \) into either \( W(t) \) or \( S(t) \):
Using the water equation:
\[ W(6) = 34 + 5(6) = 34 + 30 = 64 \text{ liters} \]
Using the sand equation:
\[ S(6) = 22 + 7(6) = 22 + 42 = 64 \text{ liters} \]
When the amounts are equal, there will be 64 liters of each.
Answer to Part B: 64 liters.
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