A cylindrical tank with radius 
𝑟
=
2
 
m
r=2m and height 
ℎ
=
5
 
m
h=5m is filled with water. The water is to be pumped out from the top of the tank.

To determine the work required to pump the water out of the cylindrical tank, we can break the problem into manageable steps. Here's how we can approach it:

### 1. Volume of the cylindrical tank
The volume \( V \) of the cylinder is given by the formula:
\[
V = \pi r^2 h
\]
where:
- \( r = 2 \, \text{m} \) (radius of the tank),
- \( h = 5 \, \text{m} \) (height of the tank).

Substituting these values:
\[
V = \pi (2)^2 (5) = 20\pi \, \text{m}^3
\]
So, the total volume of the tank is \( 20\pi \, \text{m}^3 \).

### 2. Weight of the water
The weight of the water in the tank depends on the density of water. The density of water is typically \( 1000 \, \text{kg/m}^3 \), and the gravitational acceleration \( g \) is approximately \( 9.8 \, \text{m/s}^2 \).

Thus, the weight of water per cubic meter is:
\[
\text{Weight of water per cubic meter} = 1000 \times 9.8 = 9800 \, \text{N/m}^3
\]

### 3. Work to pump water out
The work required to pump water out is given by the integral:
\[
W = \int_0^h \text{force} \times \text{distance} \, \text{d}y
\]
Here:
- The force is the weight of the water,
- The distance is how far the water needs to be lifted.

In this case, we slice the water into thin layers at height \( y \) above the base, where the water needs to be pumped to the top of the tank (which is 5 meters high). The weight of the slice of water at height \( y \) is \( \pi r^2 \times \text{density} \times \text{thickness} \).

So the work becomes:
\[
W = \int_0^5 \left( \pi (2)^2 \times 9800 \times (5 - y) \right) \, dy
\]
Simplifying:
\[
W = 9800 \times 4\pi \times \int_0^5 (5 - y) \, dy
\]

Now, calculate the integral:
\[
\int_0^5 (5 - y) \, dy = \left[ 5y - \frac{y^2}{2} \right]_0^5 = 25 - \frac{25}{2} = 12.5
\]

Finally, the total work required is:
\[
W = 9800 \times 4\pi \times 12.5 = 490000\pi \, \text{J}
\]

Thus, the total work to pump the water out of the cylindrical tank is approximately:
\[
W \approx 1,539,380 \, \text{J} \quad \text{(Joules)}
\]

Would you like further details on any step or have any questions?

### Related Questions:
1. How would the work change if the height of the tank was doubled?
2. What if the water density was different, like for oil or another liquid?
3. How does the radius affect the work if it increases to 3 meters?
4. How can the work formula be adapted for tanks of other shapes?
5. How does gravitational force affect the work if calculated on another planet?

**Tip:** When integrating to find work in pumping problems, always account for the varying distance each water element travels from its initial position.

A cylindrical tank with radius r = 2m and height h = 5m is filled with water. The water is to be pumped out from the top of the tank.

Explore how to calculate the work required to pump water from a cylindrical tank with a radius of 2 meters and a height of 5 meters. Using principles of calculus and physics, we break down the problem step by step, including volume calculations, integration, and applying the work-energy theorem. Suitable for advanced high school or early college-level students.

Math Problem Statement

A cylindrical tank with radius 𝑟

2 m r=2m and height ℎ

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation