y = 2x or x = How much work does it take to pump all or part of the liquid from a container? Engi- neers often need to know the answer in order to design or choose the right pump, or to compute the cost to transport water or some other liquid from one place to another. To find out how much work is required to pump the liquid, we imagine lifting the liquid out one thin horizontal slab at a time and applying the equation W = Fd to each slab. We then evaluate the integral that this leads to as the slabs become thinner and more numerous. 10 10 - y 8. (5, 10) Ay EXAMPLE 5 The conical tank in Figure 6.39 is filled to within 2 ft of the top with olive oil weighing 57 lb/ft. How much work does it take to pump the oil to the rim of the tank? Solution We imagine the oil divided into thin slabs by planes perpendicular to the y-axis at the points of a partition of the interval [0, 8]. FIGURE 6.39 The olive oil and tank in Example 5.
y = 2x or x = How much work does it take to pump all or part of the liquid from a container? Engi- neers often need to know the answer in order to design or choose the right pump, or to compute the cost to transport water or some other liquid from one place to another. To find out how much work is required to pump the liquid, we imagine lifting the liquid out one thin horizontal slab at a time and applying the equation W = Fd to each slab. We then evaluate the integral that this leads to as the slabs become thinner and more numerous. 10 10 - y 8. (5, 10) Ay EXAMPLE 5 The conical tank in Figure 6.39 is filled to within 2 ft of the top with olive oil weighing 57 lb/ft. How much work does it take to pump the oil to the rim of the tank? Solution We imagine the oil divided into thin slabs by planes perpendicular to the y-axis at the points of a partition of the interval [0, 8]. FIGURE 6.39 The olive oil and tank in Example 5.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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### Work Required to Pump Liquid from a Container
**How much work does it take to pump all or part of the liquid from a container?** Engineers often need to know the answer in order to design or choose the right pump, or to compute the cost to transport water or some other liquid from one place to another. To find out how much work is required to pump the liquid, we imagine lifting the liquid out one thin horizontal slab at a time and applying the equation \( W = Fd \) to each slab. We then evaluate the integral that this leads to as the slabs become thinner and more numerous.
### Example 5
The conical tank in Figure 6.39 is filled to within 2 ft of the top with olive oil weighing 57 lb/ft³. _How much work does it take to pump the oil to the rim of the tank?_
**Solution**
We imagine the oil divided into thin slabs by planes perpendicular to the y-axis at the points of a partition of the interval \([0, 8]\).
**Details of Figure 6.39**
**Figure 6.39:**
The figure shows a vertical section of a conical tank filled with olive oil. Key elements in the diagram are labeled as follows:
- **y-axis:** Vertical axis pointing upwards.
- **x-axis:** Horizontal axis pointing to the right.
- **Conical Tank:** The tank has a height of 10 feet, tapering linearly from a radius of 5 feet at the top (point \( (5, 10) \)) to a point at the vertex (bottom of the cone).
- **Formula:** The equation of the line forming the side of the tank is given as \( y = 2x \) or equivalently \( x = \frac{1}{2}y \).
- **Current Oil Level:** The tank is filled up to 8 feet high, leaving a 2-foot gap at the top.
- **Differential Element \(\Delta y\):** A thin horizontal slab of the oil, whose thickness is \(\Delta y\), is illustrated.
- **Key Points:** The key points (5, 10) at the top boundary and varying points on the y-axis represent different heights of the oil within the tank.
By dividing the oil into these thin slabs and summing the work needed to lift each slab to the rim, we can determine the total work required.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf95109c-869d-4f91-a635-386ab8bbffdb%2Fb474a4ee-3efe-4a5f-9c4c-9b80d1d4841d%2Fiqcfgvp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Work Required to Pump Liquid from a Container
**How much work does it take to pump all or part of the liquid from a container?** Engineers often need to know the answer in order to design or choose the right pump, or to compute the cost to transport water or some other liquid from one place to another. To find out how much work is required to pump the liquid, we imagine lifting the liquid out one thin horizontal slab at a time and applying the equation \( W = Fd \) to each slab. We then evaluate the integral that this leads to as the slabs become thinner and more numerous.
### Example 5
The conical tank in Figure 6.39 is filled to within 2 ft of the top with olive oil weighing 57 lb/ft³. _How much work does it take to pump the oil to the rim of the tank?_
**Solution**
We imagine the oil divided into thin slabs by planes perpendicular to the y-axis at the points of a partition of the interval \([0, 8]\).
**Details of Figure 6.39**
**Figure 6.39:**
The figure shows a vertical section of a conical tank filled with olive oil. Key elements in the diagram are labeled as follows:
- **y-axis:** Vertical axis pointing upwards.
- **x-axis:** Horizontal axis pointing to the right.
- **Conical Tank:** The tank has a height of 10 feet, tapering linearly from a radius of 5 feet at the top (point \( (5, 10) \)) to a point at the vertex (bottom of the cone).
- **Formula:** The equation of the line forming the side of the tank is given as \( y = 2x \) or equivalently \( x = \frac{1}{2}y \).
- **Current Oil Level:** The tank is filled up to 8 feet high, leaving a 2-foot gap at the top.
- **Differential Element \(\Delta y\):** A thin horizontal slab of the oil, whose thickness is \(\Delta y\), is illustrated.
- **Key Points:** The key points (5, 10) at the top boundary and varying points on the y-axis represent different heights of the oil within the tank.
By dividing the oil into these thin slabs and summing the work needed to lift each slab to the rim, we can determine the total work required.
Transcribed Image Text:**Pumping Oil Problem**
**Problem 17:**
**Pumping Oil**
How much work would it take to pump oil from the tank in Example 5 to the level of the top of the tank if the tank were completely full?
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*Explanation:*
In this problem, you are tasked with determining the amount of work required to pump oil from the bottom of a tank to the top when the tank is completely full. This involves using principles from physics and calculus, specifically dealing with forces, distances, and the physical properties of the oil and tank.
Make sure to reference "Example 5" from your text for specific parameters and methods pertinent to solving this problem, which may include variables such as the shape and dimensions of the tank, density of the oil, and gravitational force.
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*Note:* There are no graphs or diagrams provided with this problem.
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