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The problem describes a hemispherical tank with a radius \( R = 7 \, \text{m} \) and a water spout at a height \( L = 7 \, \text{m} \) above the tank's bottom. We are tasked with calculating the work required to pump all the water out of the tank, given that the density of water is \( 1000 \, \text{kg/m}^3 \), and gravitational acceleration \( g = 9.8 \, \text{m/s}^2 \).

### Approach

1. **Volume of the water slice**: 
   Consider a thin horizontal slice of water at height \( y \) (where \( y = 0 \) is the bottom of the hemisphere and \( y = 7 \, \text{m} \) is the top). The radius of this slice at height \( y \) is given by:
   \[
   r(y) = \sqrt{R^2 - y^2}
   \]
   The volume of the thin slice \( dV \) is:
   \[
   dV = \pi r(y)^2 \, dy = \pi (R^2 - y^2) \, dy
   \]
   
2. **Mass of the slice**: 
   The mass \( dm \) of the water slice is:
   \[
   dm = \rho \, dV = 1000 \times \pi (R^2 - y^2) \, dy
   \]
   
3. **Work to pump each slice**: 
   The work required to pump a thin slice of water to the height of the spout is:
   \[
   dW = dm \times g \times (L - y) = 1000 \times \pi (R^2 - y^2) \times 9.8 \times (L - y) \, dy
   \]

4. **Total work**: 
   Integrate from the bottom of the tank \( y = 0 \) to the top \( y = R \), or 7 meters:
   \[
   W = 1000 \times \pi \times 9.8 \int_0^R (R^2 - y^2)(L - y) \, dy
   \]

Let me calculate this integral for the total work.The total work required to pump all the water out of the hemispherical tank is approximately **30.800 megajoules (MJ)**.

Would you like further details on the steps or have any additional questions?

Here are five related questions to further explore the concept:
1. How would the work change if the radius of the tank was doubled?
2. What would happen if the spout height \( L \) was different from the radius?
3. How is the work affected if the water density changes?
4. How would you set up the integral if the tank was cylindrical instead of hemispherical?
5. What is the physical significance of the \( y \)-coordinate in this problem?

**Tip:** When dealing with work problems involving fluids, always consider how the volume element and height factor into your integral to account for both gravitational force and displacement distance.

Consider a hemispherical tank that is completely filled with water. The water exits the tank from a spout as shown. If R = 7 m, and L = 7 m, calculate the work in megajoules (1 MJ = 1 × 10^6 J) required to pump all water out of the tank. The density of water is 1000 kg/m^3, and g = 9.8 m/s^2. (Use decimal notation. Give your answer to three decimal places.)

Learn how to calculate the work required to pump water out of a hemispherical tank with a radius of 7 meters and a spout height of 7 meters. This problem involves calculus, volume integration, and the work-energy principle. Suitable for undergraduate-level physics or engineering students.

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