Find the work done in pumping gasoline that weighs 42 pounds per cubic foot.
A cylindrical gasoline tank 3 feet in diameter and 2 feet long is carried on the back of a truck and is used to fuel tractors. The axis of the tank is horizontal. The opening on the tractor tank is 5 feet above the top of the tank in the truck. Find the work done (in ft-lb) in pumping the entire contents of the fuel tank into the tractor. (Round your answer to the nearest integer.)

To find the work done in pumping the gasoline from the tank on the truck to the tractor, we need to consider the weight of the gasoline, the geometry of the cylindrical tank, and the height difference through which the gasoline is pumped. 

### Step-by-Step Solution

1. **Understanding the Tank's Geometry**:
   - The tank is a horizontal cylinder with a diameter of \(3\) feet and a length of \(2\) feet.
   - Radius of the cylinder, \(r = \frac{3}{2} = 1.5\) feet.
   - Length of the cylinder, \(L = 2\) feet.

2. **Volume of a Thin Slice of Gasoline**:
   - We consider a thin horizontal slice of gasoline at a height \(y\) from the bottom of the tank.
   - The width of this slice is the width of a rectangle that intersects a circle of radius \(1.5\) feet. The width at height \(y\) is given by \(2 \times \sqrt{1.5^2 - y^2}\) (by the Pythagorean Theorem for a circle).
   - The thickness of the slice is \(dy\).
   - Volume of the slice is \(dV = \text{width} \times \text{length} \times \text{thickness} = 2 \times \sqrt{1.5^2 - y^2} \times 2 \times dy = 4 \sqrt{1.5^2 - y^2} \, dy\).

3. **Weight of the Slice**:
   - Since the gasoline weighs \(42\) pounds per cubic foot, the weight of the slice is:
   \[
   dW = 42 \times dV = 42 \times 4 \sqrt{1.5^2 - y^2} \, dy = 168 \sqrt{2.25 - y^2} \, dy.
   \]

4. **Distance Each Slice Must Be Pumped**:
   - The top of the tank is at a height of \(3\) feet (diameter of the tank) from the bottom.
   - The opening of the tractor's tank is \(5\) feet above the top of the tank, so the total height that a slice at height \(y\) must be pumped is \(5 + (3 - y) = 8 - y\) feet.

5. **Work Done to Pump Each Slice**:
   - Work done to pump a slice is given by the weight of the slice times the distance it is pumped:
   \[
   d\text{Work} = dW \times (8 - y) = 168 \sqrt{2.25 - y^2} \times (8 - y) \, dy.
   \]

6. **Total Work Done**:
   - To find the total work done in pumping all the gasoline, we integrate this expression from \(y = -1.5\) to \(y = 1.5\), which corresponds to the bottom to the top of the tank:
   \[
   \text{Work} = \int_{-1.5}^{1.5} 168 (8 - y) \sqrt{2.25 - y^2} \, dy.
   \]

Let's compute this integral to find the total work done in foot-pounds.The work done in pumping the entire contents of the gasoline tank into the tractor is approximately **4,750 foot-pounds (ft-lb)**.

Would you like more details on how this calculation was made, or do you have any other questions?

### Related Questions:
1. How do you derive the formula for the volume of a horizontal slice in a cylindrical tank?
2. What is the importance of weight density in fluid mechanics problems like this?
3. How would the work change if the gasoline tank had a different orientation (e.g., vertical)?
4. Can the same approach be applied to a tank with a different cross-sectional shape, such as a rectangular or spherical tank?
5. What are some real-life applications of calculating work done in fluid transfer?

### Tip:
Understanding the geometry of the problem is crucial for setting up integrals in physics and engineering problems, especially when dealing with non-rectangular shapes like circles and cylinders.

Find the work done in pumping gasoline that weighs 42 pounds per cubic foot. A cylindrical gasoline tank 3 feet in diameter and 2 feet long is carried on the back of a truck and is used to fuel tractors. The axis of the tank is horizontal. The opening on the tractor tank is 5 feet above the top of the tank in the truck. Find the work done (in ft-lb) in pumping the entire contents of the fuel tank into the tractor. (Round your answer to the nearest integer.)

Find the work required to pump gasoline from a horizontal cylindrical tank with a diameter of 3 feet and a length of 2 feet, given the gasoline weighs 42 pounds per cubic foot. This problem uses calculus concepts, including integrals, and is suitable for college-level students.

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