In the above question, the amount of salt in the tank at the moment of overflow is

Options

A. 20 lb
B. 50 lb
C. 30 lb
D. None of these

Explanation

To find the amount of salt in the tank at the moment of overflow, we will solve the given differential equation step by step.

### Step 1: Write the differential equation
The given differential equation is:
\[ \frac{dQ}{dt} + (10 + 2t)Q = 4 \]
This is a first-order linear differential equation.

**Hint:** Identify the standard form of a first-order linear differential equation, which is \(\frac{dy}{dx} + P(x)y = Q(x)\).

### Step 2: Identify \(P(t)\) and \(Q(t)\)
From the equation, we have:
- \(P(t) = 10 + 2t\)
- \(Q(t) = 4\)

**Hint:** Recognize that \(P(t)\) and \(Q(t)\) are functions of \(t\) that will help in finding the integrating factor.

### Step 3: Find the integrating factor
The integrating factor \(I(t)\) is given by:
\[ I(t) = e^{\int P(t) dt} = e^{\int (10 + 2t) dt} \]
Calculating the integral:
\[ \int (10 + 2t) dt = 10t + t^2 \]
Thus, the integrating factor is:
\[ I(t) = e^{10t + t^2} \]

**Hint:** Remember that the integrating factor is used to simplify the equation.

### Step 4: Multiply the entire equation by the integrating factor
Multiply the differential equation by \(I(t)\):
\[ e^{10t + t^2} \frac{dQ}{dt} + e^{10t + t^2}(10 + 2t)Q = 4e^{10t + t^2} \]

**Hint:** This step transforms the left side into the derivative of a product.

### Step 5: Rewrite the left side as a derivative
The left-hand side can be rewritten as:
\[ \frac{d}{dt}(e^{10t + t^2} Q) = 4e^{10t + t^2} \]

**Hint:** Recognize that this simplification allows us to integrate both sides easily.

### Step 6: Integrate both sides
Integrate both sides with respect to \(t\):
\[ \int \frac{d}{dt}(e^{10t + t^2} Q) dt = \int 4e^{10t + t^2} dt \]
This gives:
\[ e^{10t + t^2} Q = \int 4e^{10t + t^2} dt + C \]

**Hint:** The right-hand side may require integration techniques, possibly integration by parts.

### Step 7: Solve for \(Q\)
After integrating, we need to isolate \(Q\):
\[ Q = \frac{\int 4e^{10t + t^2} dt + C}{e^{10t + t^2}} \]

**Hint:** The constant \(C\) can be determined using initial conditions.

### Step 8: Apply initial conditions
Assuming at \(t = 0\), \(Q(0) = 0\):
Substituting \(t = 0\) into the equation will help us find \(C\).

### Step 9: Find \(Q\) at the moment of overflow
From the problem, we need to find \(Q\) when \(t = 20\):
\[ Q(20) = \frac{40 \cdot 20 + 4 \cdot 20^2}{10 \cdot 20 + 2 \cdot 20} \]
Calculating this gives:
\[ Q(20) = \frac{800 + 1600}{200 + 40} = \frac{2400}{240} = 10 \]

### Final Answer
The amount of salt in the tank at the moment of overflow is \(40\) (units of salt).

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