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can u explain this solution if f(x)= 𝑥𝑙𝑜𝑔𝑒(𝑥) and can u explain why it is w+1?

Sure, let's break down the solution to this problem.

### Problem Explanation
The problem is about filling a tank to exactly half its capacity using water that flows through a hole drilled in the tank. We are given:
- The function \( y = f(x) = x \ln(x) \).
- The width of the tank \( a = 2 \).

We need to find the height \( h \) from the bottom of the tank to the hole such that the volume (area in this 2D case) of water in the tank is half of the total volume when filled to this height.

### Step-by-Step Solution

1. **Total Area Calculation:**
   The total area when the tank is full can be found by integrating the function \( f(x) = x \ln(x) \) from \( x = 0 \) to \( x = 2 \):
   \[
   \text{Total Area} = \int_0^2 x \ln(x) \, dx
   \]

2. **Half-Filled Area:**
   The area when the tank is half-filled is half of the total area:
   \[
   \text{Half-Filled Area} = \frac{1}{2} \int_0^2 x \ln(x) \, dx = \frac{1}{8} \, \text{units}^2
   \]

3. **Setting up the Equation:**
   To find the height \( h \) at which the tank is half-filled, we need to set up the integral from \( x = w \) to \( x = 2 \) and equate it to the half-filled area:
   \[
   \int_w^2 x \ln(x) \, dx = \frac{1}{8}
   \]
   Solving this integral involves the substitution and algebraic manipulation. However, to simplify, we solve for \( w \):
   \[
   \int_1^2 x \ln(x) \, dx - \int_1^{w+1} x \ln(x) \, dx = \frac{1}{8}
   \]

4. **Solving for \( w \):**
   From the equation above, we solve for \( w \):
   \[
   w \approx 0.598111
   \]

5. **Finding \( h \):**
   Given that \( h = f(w+1) \):
   \[
   h = f(0.598111 + 1) = (0.598111 + 1) \ln(0.598111 + 1)
   \]
   Simplifying further,
   \[
   h \approx 0.749
   \]

### Why \( w + 1 \)?
- The term \( w + 1 \) comes from the setup of the problem. We need the height \( h \) corresponding to the horizontal distance from 1 to \( w + 1 \) (i.e., from the point of the hole to the right edge of the tank).

### Summary
To fill the tank to exactly half its capacity, the height \( h \) from the ground to the hole should be approximately \( 0.749 \) meters.

Would you like more details or have any questions about the solution? Here are some additional questions to explore this concept further:

1. What is the integral of \( x \ln(x) \)?
2. How do we interpret the function \( x \ln(x) \) in a real-world scenario?
3. What other methods can be used to solve for \( w \) if we didn't use integration?
4. How does changing the value of \( a \) affect the solution?
5. Can you explain the importance of setting up the correct bounds in integration problems?
6. What are some common mistakes to avoid when solving integration problems?
7. How can we verify the correctness of the calculated height \( h \)?
8. What is the significance of the \( e \) (Euler's number) in the function \( x \ln(x) \)?

**Tip:** When solving complex integrals, always double-check your integration bounds and consider breaking the problem into smaller, more manageable parts.

Explore the function f(x) = x ln(x) and its application in filling a tank to half its capacity. Learn how to solve for the distance w + 1 to achieve this with a detailed step-by-step solution.

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