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The problem is asking for the work \( W \) required to pump all the water out of the tank. The tank is a trapezoidal prism filled with water, and the work required to pump water out is influenced by the weight of the water and the height it needs to be lifted to reach the spout.

### Step-by-Step Solution:

#### Given:
- Gravitational acceleration, \( g = 9.8 \, \text{m/s}^2 \)
- Density of water, \( \rho = 1000 \, \text{kg/m}^3 \)
- Dimensions of the tank:
  - Depth of water, \( h_{\text{max}} = 4 \, \text{m} \)
  - Length of tank, \( L = 15 \, \text{m} \)
  - Width at the bottom of the tank, \( b_{\text{bottom}} = 4 \, \text{m} \)
  - Width at the top of the tank, \( b_{\text{top}} = 4 + 4 = 8 \, \text{m} \)

#### Concept:
The work required to pump the water out of the tank is calculated by integrating the force (due to gravity) required to move each differential volume of water to the spout. The distance each layer of water must be lifted changes with its depth.

The work \( W \) is given by:

\[
W = \int_0^{h_{\text{max}}} \text{Force} \times \text{Distance} \, dy
\]

Where:
- \( \text{Force} = \rho g A(y) \Delta y \)
- \( \text{Distance} = h_{\text{max}} - y \)

The cross-sectional area of the water at a given height \( y \) depends on the shape of the trapezoid. The width of the tank at any height \( y \) can be found using the slope of the trapezoidal sides.

Let me calculate the total work required for this problem.The total work required to pump the water out of the spout is \( 6,272,000 \, \text{J} \) (joules).

Would you like a detailed breakdown of the steps or further explanations? 

Here are five related questions to explore the concept further:
1. How does the shape of the tank influence the work required to pump out the water?
2. What happens to the total work if the height of the water increases by 1 meter?
3. How would the work change if the tank were cylindrical instead of trapezoidal?
4. How would the work be affected if the spout were located halfway up the tank instead of at the top?
5. How does changing the gravitational constant \( g \) (e.g., on another planet) affect the work?

**Tip:** When calculating work in such problems, always consider the shape of the tank and how the distance each slice of water must be lifted varies with height.

A tank is full of water. Find the work W required to pump the water out of the spout. (Use 9.8 m/s² for g. Use 1000 kg/m³ as the weight density of water.)

Find the total work required to pump water out of a trapezoidal tank, using integration and hydrostatic principles. This problem covers the calculation of work, force, and the distance each layer of water must be lifted, with detailed formulas. Suitable for advanced high school students or college physics/calculus courses.

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