Find the work required to pump all of the water from an inverted conical tank with the flat side at the top, whose radius is 4 meters and height is 2 meters, to a height 2 meters above the top of the tank. Assume water weighs 9800 Newtons per cubic meter.
Find the work required to pump all of the water from an inverted conical tank with the flat side at the top, whose radius is 4 meters and height is 2 meters, to a height 2 meters above the top of the tank. Assume water weighs 9800 Newtons per cubic meter.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![### Problem Statement:
Find the work required to pump all of the water from an inverted conical tank with the flat side at the top, whose radius is 4 meters and height is 2 meters, to a height 2 meters above the top of the tank. Assume water weighs 9800 Newtons per cubic meter.
### Explanation:
This problem involves calculating the work done to pump water from a conical tank to a certain height above its top. Here are the specifics:
- **Shape of Tank**: Inverted conical tank, flat side at the top.
- **Dimensions**:
- Radius at the top: 4 meters.
- Height of the tank: 2 meters.
- **Pump Height**: 2 meters above the top of the tank.
- **Water Weight**: 9800 Newtons per cubic meter.
We need to integrate the force required to move each infinitesimal volume of water to the given height above the tank.
### Detailed Steps:
1. **Volume Element**:
Consider a thin slice of the conical tank at a height \( y \) from the vertex (the tip of the cone) and of thickness \( dy \). This slice will be a thin disk with radius \( r \).
2. **Relationship of Radius to Height**:
The radius \( r \) at height \( y \) can be found using similar triangles. Given the total height and radius of the cone:
\[
\frac{r}{y} = \frac{4}{2}
\]
\[
r = 2y
\]
3. **Volume of Thin Disk**:
The volume \( dV \) of the thin disk is:
\[
dV = \pi r^2 dy = \pi (2y)^2 dy = 4\pi y^2 dy
\]
4. **Force of Water Element**:
The weight \( dW \) of this water element is:
\[
dW = \text{Density of water} \times dV = 9800 \times 4\pi y^2 dy = 39200\pi y^2 dy
\]
5. **Distance to Lift Element**:
The distance to lift this thin slice of water to a height 2 meters above the tank is \( (2 + (2 - y)) = 4](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F199d58c4-213a-4e42-b645-d01318acf93a%2Ff396187c-319c-4f62-9ebe-d9a32795b7b0%2Fjy7grsg_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
Find the work required to pump all of the water from an inverted conical tank with the flat side at the top, whose radius is 4 meters and height is 2 meters, to a height 2 meters above the top of the tank. Assume water weighs 9800 Newtons per cubic meter.
### Explanation:
This problem involves calculating the work done to pump water from a conical tank to a certain height above its top. Here are the specifics:
- **Shape of Tank**: Inverted conical tank, flat side at the top.
- **Dimensions**:
- Radius at the top: 4 meters.
- Height of the tank: 2 meters.
- **Pump Height**: 2 meters above the top of the tank.
- **Water Weight**: 9800 Newtons per cubic meter.
We need to integrate the force required to move each infinitesimal volume of water to the given height above the tank.
### Detailed Steps:
1. **Volume Element**:
Consider a thin slice of the conical tank at a height \( y \) from the vertex (the tip of the cone) and of thickness \( dy \). This slice will be a thin disk with radius \( r \).
2. **Relationship of Radius to Height**:
The radius \( r \) at height \( y \) can be found using similar triangles. Given the total height and radius of the cone:
\[
\frac{r}{y} = \frac{4}{2}
\]
\[
r = 2y
\]
3. **Volume of Thin Disk**:
The volume \( dV \) of the thin disk is:
\[
dV = \pi r^2 dy = \pi (2y)^2 dy = 4\pi y^2 dy
\]
4. **Force of Water Element**:
The weight \( dW \) of this water element is:
\[
dW = \text{Density of water} \times dV = 9800 \times 4\pi y^2 dy = 39200\pi y^2 dy
\]
5. **Distance to Lift Element**:
The distance to lift this thin slice of water to a height 2 meters above the tank is \( (2 + (2 - y)) = 4
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