Two spherical tanks are connected as in the figure below. The first tank has the radius 1m and is on the ground. The bottom of the second tank is located 2m above the top of the first tank. The second tank has the radius 0.5m. Two tanks are connected by a tube of negligible width that is going through the top of the first tank and the bottom of the second tank. Let p > 0 be the density (in kg/m^3) of the liquid in both tanks. Find the work required to deliver the liquid to fill both tanks from the ground.
Two spherical tanks are connected as in the figure below. The first tank has the radius 1m and is on the ground. The bottom of the second tank is located 2m above the top of the first tank. The second tank has the radius 0.5m. Two tanks are connected by a tube of negligible width that is going through the top of the first tank and the bottom of the second tank. Let p > 0 be the density (in kg/m^3) of the liquid in both tanks. Find the work required to deliver the liquid to fill both tanks from the ground.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F58f259fd-61e5-45b9-8d61-cdb877dc613f%2Ff21c75cc-22ed-498a-bad4-fd99160b8840%2Fg5jdl8_processed.jpeg&w=3840&q=75)
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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