Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Problem Explanation: Spherical Tanks Connected by a Tube**
Two spherical tanks are interconnected as illustrated in the provided diagram. Below are the key details about the system:
1. **First Tank**:
- **Radius**: 1 meter.
- **Position**: The tank is on the ground.
2. **Second Tank**:
- **Radius**: 0.5 meters.
- **Position**: The bottom of this tank is located 2 meters above the top of the first tank.
3. **Connecting Tube**:
- The tanks are connected by a tube of negligible width. This tube runs from the top of the first tank to the bottom of the second tank.
**Given:**
- \( \rho > 0 \) denotes the density (in kg/m³) of the liquid in both tanks.
**Objective:**
- Determine the work required to pump the liquid from the ground to fill both tanks.
**Diagram Details:**
- The diagram visually represents the two spherical tanks and the connecting tube.
- It features the dimensions for placement:
- The first tank (larger) is positioned on the line representing the ground.
- The second tank (smaller), is elevated, showing 2 meters distance above the top of the first tank.
**Steps to Solve:**
1. **Calculate the Volume of Both Tanks**:
- Volume of the first tank (V₁):
\[
V₁ = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi \, \text{m}^3
\]
- Volume of the second tank (V₂):
\[
V₂ = \frac{4}{3} \pi (0.5)^3 = \frac{4}{3} \pi \left(\frac{1}{8}\right) = \frac{1}{6} \pi \, \text{m}^3
\]
2. **Determine the Mass of Liquid Needed for Both Tanks**:
- Mass of liquid for the first tank (M₁):
\[
M₁ = \rho \times V₁ = \rho \times \frac{4}{3} \pi
\]
- Mass of liquid for the second tank (M₂):
\[
M₂ =](https://content.bartleby.com/qna-images/question/58f259fd-61e5-45b9-8d61-cdb877dc613f/f21c75cc-22ed-498a-bad4-fd99160b8840/g5jdl8_processed.jpeg)
Transcribed Image Text:**Problem Explanation: Spherical Tanks Connected by a Tube**
Two spherical tanks are interconnected as illustrated in the provided diagram. Below are the key details about the system:
1. **First Tank**:
- **Radius**: 1 meter.
- **Position**: The tank is on the ground.
2. **Second Tank**:
- **Radius**: 0.5 meters.
- **Position**: The bottom of this tank is located 2 meters above the top of the first tank.
3. **Connecting Tube**:
- The tanks are connected by a tube of negligible width. This tube runs from the top of the first tank to the bottom of the second tank.
**Given:**
- \( \rho > 0 \) denotes the density (in kg/m³) of the liquid in both tanks.
**Objective:**
- Determine the work required to pump the liquid from the ground to fill both tanks.
**Diagram Details:**
- The diagram visually represents the two spherical tanks and the connecting tube.
- It features the dimensions for placement:
- The first tank (larger) is positioned on the line representing the ground.
- The second tank (smaller), is elevated, showing 2 meters distance above the top of the first tank.
**Steps to Solve:**
1. **Calculate the Volume of Both Tanks**:
- Volume of the first tank (V₁):
\[
V₁ = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi \, \text{m}^3
\]
- Volume of the second tank (V₂):
\[
V₂ = \frac{4}{3} \pi (0.5)^3 = \frac{4}{3} \pi \left(\frac{1}{8}\right) = \frac{1}{6} \pi \, \text{m}^3
\]
2. **Determine the Mass of Liquid Needed for Both Tanks**:
- Mass of liquid for the first tank (M₁):
\[
M₁ = \rho \times V₁ = \rho \times \frac{4}{3} \pi
\]
- Mass of liquid for the second tank (M₂):
\[
M₂ =
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