A 50 L tank initailly contains 10 L of fresh water, At t=0, a brine solution containing 1 lb of salt per gallon is poured into the tank at the rate of 4 L/min, while the well-stirred mixture leaves the tank at the rate of 2 L/min. Then the amount of time required for overflow to occur in

Options

A. 30 min
B. 20 min
C. 10 min
D. 40 min

Explanation

To solve the problem, we need to determine the time it takes for the tank to overflow. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Initial Conditions
The tank has an initial volume of 10 liters of fresh water. We need to account for the inflow and outflow rates to determine when the tank will reach its maximum capacity of 50 liters.

### Step 2: Set Up the Inflow and Outflow Rates
- **Inflow Rate**: The brine solution is poured into the tank at a rate of 4 liters per minute.
- **Outflow Rate**: The well-stirred mixture leaves the tank at a rate of 2 liters per minute.

### Step 3: Calculate the Net Change in Volume
The net change in volume per minute can be calculated as follows:
\[ \text{Net Change} = \text{Inflow Rate} - \text{Outflow Rate} = 4 \, \text{L/min} - 2 \, \text{L/min} = 2 \, \text{L/min} \]

### Step 4: Write the Volume Equation
Let \( V(t) \) be the volume of the tank at time \( t \) in minutes. The volume at time \( t \) can be expressed as:
\[ V(t) = V_0 + \text{Net Change} \times t \]
Where \( V_0 = 10 \, \text{L} \) (initial volume). Therefore:
\[ V(t) = 10 + 2t \]

### Step 5: Determine When the Tank Overflows
The tank will overflow when \( V(t) = 50 \, \text{L} \). Setting up the equation:
\[ 10 + 2t = 50 \]

### Step 6: Solve for \( t \)
To find the time \( t \) when the tank reaches 50 liters, we rearrange the equation:
\[ 2t = 50 - 10 \]
\[ 2t = 40 \]
\[ t = \frac{40}{2} = 20 \, \text{minutes} \]

### Final Answer
The amount of time required for the tank to overflow is **20 minutes**.

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