A tank has a shape of a cone with a radius at the top of 3 m and a height of 5 m. The tank also has a 1 m spout at the top of the tank. The tank is filled with water up to a height of 2 m. Find the work needed to pump all the water out the top of the spout. (Use 9.8 m/s2 for g and the fact that the density of water is
1000 kg/m3.)


We are tasked with finding the work required to pump water out of a conical tank with a spout. To solve this, we'll need to use the following concepts:

- **Density of water**: \( \rho = 1000 \, \text{kg/m}^3 \)
- **Gravitational acceleration**: \( g = 9.8 \, \text{m/s}^2 \)
- The formula for work done in moving a volume of liquid against gravity is:
  \[
  W = \int F \, dy
  \]
  where \( F \) is the force, and \( dy \) represents an infinitesimal change in height.

### Step-by-step Plan:

1. **Volume Element**: We'll first calculate the volume of an infinitesimal slice of water at a height \( y \) above the bottom of the cone. Given the geometry of the cone, the radius of the water slice varies with height.

2. **Geometric Similarity**: Since the tank is a cone, the radius \( r \) at any height \( y \) is proportional to the height from the tip of the cone. For this cone:
   - At the top \( y = 5 \, \text{m} \), the radius is \( 3 \, \text{m} \).
   - Using the similarity of triangles:
     \[
     \frac{r}{y} = \frac{3}{5} \quad \text{or} \quad r = \frac{3}{5} y
     \]

3. **Infinitesimal Volume \( dV \)**: The infinitesimal volume \( dV \) of a slice of water at height \( y \) with thickness \( dy \) is the volume of a cylinder with radius \( r = \frac{3}{5} y \) and height \( dy \):
   \[
   dV = \pi r^2 \, dy = \pi \left( \frac{3}{5} y \right)^2 \, dy = \pi \frac{9}{25} y^2 \, dy
   \]

4. **Force to Lift Water**: The weight of this infinitesimal volume of water is \( \rho g dV \):
   \[
   dF = 1000 \times 9.8 \times \pi \frac{9}{25} y^2 \, dy
   \]

5. **Distance to Lift Water**: To pump water out of the spout, the water at height \( y \) needs to be lifted a distance of \( 6 - y \) meters (since the spout is 1 meter above the top of the cone).

6. **Work for Infinitesimal Slice**: The work to pump this infinitesimal slice is:
   \[
   dW = dF \times (6 - y) = 1000 \times 9.8 \times \pi \frac{9}{25} y^2 \, (6 - y) \, dy
   \]

7. **Total Work**: Integrate this expression from \( y = 0 \) to \( y = 2 \), as the tank is filled to a height of 2 meters:
   \[
   W = \int_0^2 1000 \times 9.8 \times \pi \frac{9}{25} y^2 \, (6 - y) \, dy
   \]
   Simplifying constants:
   \[
   W = 35280 \pi \int_0^2 y^2 (6 - y) \, dy
   \]

### Step-by-step Integration:
Now, we'll evaluate the integral:
\[
\int_0^2 y^2 (6 - y) \, dy = \int_0^2 6y^2 \, dy - \int_0^2 y^3 \, dy
\]

Let's calculate each part:

\[
\int_0^2 6y^2 \, dy = 6 \left[ \frac{y^3}{3} \right]_0^2 = 6 \times \frac{8}{3} = 16
\]
\[
\int_0^2 y^3 \, dy = \left[ \frac{y^4}{4} \right]_0^2 = \frac{16}{4} = 4
\]
Thus, the total integral is:
\[
16 - 4 = 12
\]

### Final Work Calculation:
Now substitute this result into the expression for \( W \):
\[
W = 35280 \pi \times 12 = 423360 \pi
\]
Approximating \( \pi \approx 3.1416 \), we get:
\[
W \approx 423360 \times 3.1416 = 1,329,305.6 \, \text{J}
\]

Thus, the total work required to pump all the water out of the tank is approximately \( \boxed{1,329,305.6} \, \text{J} \).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How would the work change if the height of the spout were increased to 2 meters?
2. What is the effect of changing the cone's base radius on the work required?
3. How would the work differ if the tank were only half-filled with water?
4. What if the water density were lower, like for oil (850 kg/m³)?
5. How does the shape of the tank (e.g., cylindrical vs. conical) affect the work required?

**Tip:** Always check the dimensions of your parameters to ensure consistency when solving physics-related problems!

A tank has a shape of a cone with a radius at the top of 3 m and a height of 5 m. The tank also has a 1 m spout at the top of the tank. The tank is filled with water up to a height of 2 m. Find the work needed to pump all the water out the top of the spout. (Use 9.8 m/s2 for g and the fact that the density of water is 1000 kg/m3.)

Learn how to calculate the work required to pump water from a conical tank with a 1 m spout. This problem uses calculus and the work-energy principle, applying key concepts such as geometric similarity and volume of revolution. Detailed step-by-step solution provided. Suitable for undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation