An aquarium 3 m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use 9.8 m/s2 for g and the fact that the density of water is 
1,000 kg/m3.)
Show how to approximate the required work (in J) by a Riemann sum. (Let x be the height in meters below the top of the tank. Enter xi* as xi.)

To solve this problem, we need to calculate the work required to pump half of the water out of the aquarium. The aquarium is a rectangular tank, so we can model the work as the integral of the force needed to pump the water over the distance it needs to be lifted.

### Step-by-step breakdown:

1. **Volume of the aquarium**:
   The aquarium has dimensions of 3 m (length), 1 m (width), and 1 m (depth). The total volume \( V \) of the tank is:
   \[
   V = 3 \times 1 \times 1 = 3 \text{ m}^3
   \]
   Since we're pumping half of the water out, the volume of water to pump is:
   \[
   V_{\text{half}} = \frac{3}{2} = 1.5 \text{ m}^3
   \]

2. **Weight of water**:
   The density of water is given as \( 1000 \, \text{kg/m}^3 \), and gravity is \( 9.8 \, \text{m/s}^2 \). Therefore, the weight density of water is:
   \[
   \text{Weight density} = 1000 \times 9.8 = 9800 \, \text{N/m}^3
   \]

3. **Riemann sum setup**:
   Let \( x \) be the height below the top of the tank, so that \( x = 0 \) corresponds to the top of the tank and \( x = 1 \) corresponds to the bottom. Each infinitesimal slice of water at height \( x \), of thickness \( \Delta x \), needs to be lifted a distance \( x \) to the top of the tank.

   - The area of each horizontal slice is \( 3 \times 1 = 3 \, \text{m}^2 \).
   - The volume of a thin slice of water at height \( x \) is approximately \( 3 \Delta x \, \text{m}^3 \).
   - The weight of that slice is:
     \[
     \text{Weight of slice} = 3 \times 9800 \times \Delta x = 29400 \Delta x \, \text{N}
     \]
   - The distance this slice needs to be lifted is \( x \).

4. **Work required to pump the slice**:
   The work to pump this slice of water is:
   \[
   \Delta W \approx \text{Weight of slice} \times \text{Distance lifted} = 29400 \Delta x \times x = 29400 x \Delta x \, \text{J}
   \]

5. **Riemann sum for the total work**:
   To approximate the total work needed to pump half the water, we sum the work for slices from \( x = 0.5 \) to \( x = 1 \), since we're pumping out the top half of the water. This gives us the Riemann sum:
   \[
   W \approx \sum_{i=1}^{n} 29400 x_i^* \Delta x
   \]
   where \( x_i^* \) is the height of the \( i \)-th slice, and \( \Delta x \) is the thickness of each slice.

6. **Integral for exact work**:
   In the limit as \( \Delta x \to 0 \), the Riemann sum becomes an integral. The total work \( W \) is given by:
   \[
   W = \int_{0.5}^{1} 29400 x \, dx
   \]

7. **Solve the integral**:
   The integral is straightforward:
   \[
   W = 29400 \int_{0.5}^{1} x \, dx = 29400 \left[ \frac{x^2}{2} \right]_{0.5}^{1}
   \]
   \[
   W = 29400 \left( \frac{1^2}{2} - \frac{0.5^2}{2} \right) = 29400 \left( \frac{1}{2} - \frac{0.25}{2} \right) = 29400 \times \frac{0.75}{2}
   \]
   \[
   W = 29400 \times 0.375 = 11025 \, \text{J}
   \]

Thus, the work required to pump half of the water out of the aquarium is **11,025 joules**.

---

Would you like further details or have any other questions?

### 5 Relative Questions:
1. How would the result change if the aquarium were taller but had the same base area?
2. How would you approach the problem if the water had a different density?
3. What would be the work required to pump all the water out of the tank?
4. Can you set up a similar Riemann sum for a tank of different shape, such as a cylinder?
5. What is the physical significance of integrating the height over which water is pumped?

### Tip:
Remember that when solving work problems involving lifting, the force and distance must be integrated, and the variable representing the changing distance should be carefully chosen based on the setup.

An aquarium 3 m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use 9.8 m/s2 for g and the fact that the density of water is 1,000 kg/m3.) Show how to approximate the required work (in J) by a Riemann sum. (Let x be the height in meters below the top of the tank. Enter xi* as xi.)

Learn how to calculate the work required to pump half the water out of a 3m x 1m x 1m aquarium using Riemann sums. This problem involves calculus concepts such as integration, work-energy principles, and volume calculations. Suitable for students in Grades 11-12.

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