Let's solve the differential equation to verify the solution you mentioned.

The differential equation we have is:
\[ \frac{dx}{dt} = -\frac{5x(t)}{50 - t} \]

This is a separable differential equation. To solve it, we separate the variables \(x\) and \(t\) and then integrate both sides.

### Step 1: Separate Variables
\[ \frac{dx}{x(t)} = -\frac{5}{50 - t} dt \]

### Step 2: Integrate Both Sides
Integrate the left side with respect to \(x\) and the right side with respect to \(t\):
\[ \int \frac{1}{x(t)} dx = -5 \int \frac{1}{50 - t} dt \]

The integrals are:
\[ \ln|x(t)| = 5 \ln|50 - t| + C \]

### Step 3: Simplify
We can exponentiate both sides to solve for \(x(t)\):
\[ x(t) = e^{5 \ln|50 - t| + C} \]
\[ x(t) = e^C \cdot e^{5 \ln|50 - t|} \]
\[ x(t) = A \cdot (50 - t)^5 \]

Here, \(A = e^C\) is a constant.

### Step 4: Apply Initial Condition
We know the initial condition \(x(0) = x_0\):
\[ x(0) = A \cdot (50 - 0)^5 \]
\[ x_0 = A \cdot 50^5 \]
\[ A = \frac{x_0}{50^5} \]

Thus, the solution to the differential equation is:
\[ x(t) = \frac{x_0}{50^5} \cdot (50 - t)^5 \]

This confirms the form you provided, where \(A = \frac{x_0}{50^5}\).

In summary:
\[ x(t) = \frac{x_0}{50^5} \cdot (50 - t)^5 \]

Do you want more details or have any questions?

Here are some questions for further exploration:
1. What is the behavior of \( x(t) \) as \( t \) approaches 50 minutes?
2. How does the initial concentration of the chemical affect the solution?
3. How does the concentration of the chemical change over time?
4. What would happen if the rates of inflow and outflow were different?
5. How can this model be applied to other similar scenarios?

**Tip:** When solving separable differential equations, carefully integrate each side and apply initial conditions to find the constant of integration. This ensures your solution is specific to the problem context.

Learn how to solve the differential equation x(t) = (x0 / 50^5) * (50 - t)^5 with a step-by-step solution. This problem involves separable differential equations and integration of natural logarithms. Suitable for college-level students and those interested in mathematical modeling.

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