To solve this problem, let’s first define the initial quantities and the rates at which water and sand are added.
Initial amounts:
- Water: 34 liters
- Sand: 22 liters
Rates of addition:
- Water is added at a rate of 5 liters per minute.
- Sand is added at a rate of 7 liters per minute.
Let \( t \) be the time in minutes after starting the addition of water and sand.
Amounts after \( t \) minutes:
- Amount of water after \( t \) minutes: \( W(t) = 34 + 5t \)
- Amount of sand after \( t \) minutes: \( S(t) = 22 + 7t \)
Part A: Finding when both amounts are equal
We want to find \( t \) when \( W(t) = S(t) \): \[ 34 + 5t = 22 + 7t \]
Now, let’s solve the equation:
- Rearrange the equation: \[ 34 - 22 = 7t - 5t \] \[ 12 = 2t \]
- Divide both sides by 2: \[ t = 6 \]
So, the tub will contain equal amounts of water and sand after 6 minutes.
Part B: Finding the amounts when they are equal
Now we need to find how much water and sand is in the tub after 6 minutes:
Substituting \( t = 6 \) into the equations for \( W(t) \) and \( S(t) \):
-
Amount of water: \[ W(6) = 34 + 5 \times 6 = 34 + 30 = 64 \text{ liters} \]
-
Amount of sand: \[ S(6) = 22 + 7 \times 6 = 22 + 42 = 64 \text{ liters} \]
When the amounts are equal, there will be 64 liters of both water and sand.
Final answers:
- Part A: 6 minutes
- Part B: 64 liters
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