Consider a hemispherical tank that is completely filled with water. The water exits the tank from the spout as shown.
If R = 8 m, and L = 3 m, calculate the work in megajoules (1 MU= 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.)

To calculate the work required to pump all the water out of a hemispherical tank, we can follow these steps:

1. **Geometry of the tank**: The radius of the hemispherical tank is \( R = 8 \) meters, and the spout (exit point) is \( L = 3 \) meters above the top of the tank.

2. **Work for an infinitesimal slice**: At a height \( y \) below the top of the hemisphere, the radius of the horizontal cross-section at that level is \( r(y) = \sqrt{R^2 - y^2} \).

3. **Volume of a thin slice of water**: Consider a thin slice of water of thickness \( dy \) at height \( y \). The volume \( dV \) of this slice is the area of the circular cross-section at height \( y \) times the thickness \( dy \):

   \[
   dV = \pi r(y)^2 dy = \pi (R^2 - y^2) dy
   \]

4. **Mass of the slice**: The mass \( dm \) of the water in this thin slice is the volume times the density of water \( \rho \):

   \[
   dm = \rho dV = 1000 \pi (R^2 - y^2) dy
   \]

5. **Distance the water needs to be lifted**: Each slice of water at height \( y \) must be lifted a distance of \( L + y \) meters to the spout.

6. **Work for a slice**: The work \( dW \) required to lift the slice is the force \( dm \cdot g \) times the distance \( (L + y) \):

   \[
   dW = g (L + y) dm = 9.8 \cdot (3 + y) \cdot 1000 \pi (64 - y^2) dy
   \]

7. **Total work**: Integrating this expression over the entire height of the tank from \( y = 0 \) (the top) to \( y = 8 \) (the bottom of the tank):

   \[
   W = 9.8 \cdot 1000 \pi \int_0^8 (3 + y)(64 - y^2) dy
   \]

We will now evaluate this integral to find the total work.The total work required to pump all the water out of the hemispherical tank is approximately **63.1 megajoules (MJ)**.

Do you have any questions or would you like further details? Here are 5 related questions for deeper exploration:

1. How would the work change if the radius of the tank were doubled?
2. What would be the work required if the spout were placed at ground level instead of 3 meters above the tank?
3. How does the shape of the tank influence the total work needed to pump the water out?
4. How would the work change if the fluid had a different density, such as oil?
5. What role does the acceleration due to gravity play in this calculation?

**Tip**: When calculating work, always consider the force required and the distance over which the force is applied.

Consider a hemispherical tank that is completely filled with water. The water exits the tank from the spout as shown. If R = 8 m, and L = 3 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 8 meters and a spout 3 meters above the tank. This problem involves integration, calculus, and the work-energy theorem. The water's density is 1000 kg/m^3, and g = 9.8 m/s^2. Find the result in megajoules (MJ), rounded to three decimal places.

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