A tank is filled with a fluid with density 54 kg/m^3The cross-section of the end of the tank is given by the shaded region shown below that is bounded by the y-axis, the function written as y = f(x) or x = g(y), and the line y = 12 as shown in the image below. (Note: Assume c = 5m and d = 12m.)(Image 1 see screenshots)The tank extends back a length of 9 meters as shown in the image below:(Image 2 see screenshots)Our goal in this problem is to compute the work required to pump the liquid 4 meters over the top edge of the tank:Report the volume, force and work for a slice:Volume of a slice: []Force for a slice: []Work for a slice: []Report the integral for the work. Note: a and b are the lower and upper limits of the integral:b = []work = integral [] dya = []Your answer will involve f(x) or g(y) (or possibly both). You will not be able to actually evaluate the integral.Inlude a drawing with the slice (dimensions labeled) in your work.Is there a tank of the same size and same basic shape that would require less work to empty? Explain/justify your answer. ## Educational Exercise: Calculating Work to Empty a Tank### Diagram Explanation#### Table:- **Rows and Columns**: - $ \Delta y $ - $ \Delta x $ - $ g(y) $ - $ f(x) $ - $ y $ - $ x $**Note**: The table suggests dimensions involved when calculating the volume, force, and work related to a physical slice of a substance in a tank.---### Calculation Prompts#### Volume of a Slice:- Calculate and input the volume of a slice in the given box.#### Force for a Slice:- Determine and input the force required to move the slice.#### Work for a Slice:- Calculate and input the work necessary to move the slice.#### Integral Setup for Total Work:- Report the integral for the work done. Use $ a $ and $ b $ as the lower and upper limits of the integral, respectively: - Enter $ b $ - Enter $ a $ - Input the integral expression $ \int_{a}^{b} $*Note*: Your answer should involve $ f(x) $ or $ g(y) $, or sometimes both. Direct evaluation of the integral is not required.#### Upload Section:- Include a handwritten rendition of your calculations in the designated upload box. Ensure you have a labeled drawing of the slice (with dimensions).---### Conceptual Question**Prompt**:Is there a tank of the same size and basic shape that would require less work to empty? Provide an explanation or justification for your answer in the provided text box. **Title: Work Required to Pump Liquid from a Tank****Context:**A tank is filled with a fluid with a density of $54 \, \text{kg/m}^3$.**Tank Specifications:**- The cross-section of the end of the tank is represented by the shaded region in the first diagram.- This region is bounded by the y-axis, the curve $y = f(x)$ or equivalently $x = g(y)$, and the horizontal line $y = 12$.- Assume $c = 5 \, \text{m}$ and $d = 12 \, \text{m}$.**First Diagram: Cross-Section View**- The diagram shows a coordinate system with a curve which marks the boundary of the shaded region.- The curve is labeled as $y = f(x)$ or $x = g(y)$, with boundaries at $c$ and $d$.**Tank Dimensions:**- The tank extends backward, maintaining a length of 9 meters.- The shape of this extension mirrors the cross-section shown.**Second Diagram: 3D Perspective**- Displays a 3D representation of the tank extending backward 9 meters.- The function annotations $y = f(x)$ or $x = g(y)$ highlight its shape.**Objective:**Calculate the work needed to pump the liquid to 4 meters above the tank's top.**Instructions:**To find the volume, force, and work for a slice of fluid:1. Use variables $\Delta x$ or $\Delta y$ to represent small changes in dimensions.2. Ensure capital 'Delta' is used ($\Delta$).3. Utilize the CalcPad navigation by selecting the ‘Vars’ tab to integrate these variables into your computations.**CalcPad Interface:**- Shows options like $\Delta y$, $\Delta x$, $g(y)$, and $f(x)$.This guide aids in computing the necessary physical work to elevate the liquid out of the tank, accounting for its dimensions and fluid properties.

Answered: A tank is filled with a fluid with…

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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Question

A tank is filled with a fluid with density 54 kg/m^3

The cross-section of the end of the tank is given by the shaded region shown below that is bounded by the y-axis, the function written as y = f(x) or x = g(y), and the line y = 12 as shown in the image below. (Note: Assume c = 5m and d = 12m.)

(Image 1 see screenshots)

The tank extends back a length of 9 meters as shown in the image below:

(Image 2 see screenshots)

Our goal in this problem is to compute the work required to pump the liquid 4 meters over the top edge of the tank:

Report the volume, force and work for a slice:

Volume of a slice: []

Force for a slice: []

Work for a slice: []

Report the integral for the work. Note: a and b are the lower and upper limits of the integral:

b = []

work = integral [] dy

a = []

Your answer will involve f(x) or g(y) (or possibly both). You will not be able to actually evaluate the integral.

Inlude a drawing with the slice (dimensions labeled) in your work.

Is there a tank of the same size and same basic shape that would require less work to empty? Explain/justify your answer.

## Educational Exercise: Calculating Work to Empty a Tank

### Diagram Explanation

#### Table:
- **Rows and Columns**:
- $ \Delta y $
- $ \Delta x $
- $ g(y) $
- $ f(x) $
- $ y $
- $ x $

**Note**: The table suggests dimensions involved when calculating the volume, force, and work related to a physical slice of a substance in a tank.

---

### Calculation Prompts

#### Volume of a Slice:
- Calculate and input the volume of a slice in the given box.

#### Force for a Slice:
- Determine and input the force required to move the slice.

#### Work for a Slice:
- Calculate and input the work necessary to move the slice.

#### Integral Setup for Total Work:
- Report the integral for the work done. Use $ a $ and $ b $ as the lower and upper limits of the integral, respectively:
- Enter $ b $
- Enter $ a $
- Input the integral expression $ \int_{a}^{b} $

*Note*: Your answer should involve $ f(x) $ or $ g(y) $, or sometimes both. Direct evaluation of the integral is not required.

#### Upload Section:
- Include a handwritten rendition of your calculations in the designated upload box. Ensure you have a labeled drawing of the slice (with dimensions).

---

### Conceptual Question

**Prompt**:
Is there a tank of the same size and basic shape that would require less work to empty? Provide an explanation or justification for your answer in the provided text box.

**Title: Work Required to Pump Liquid from a Tank**

**Context:**
A tank is filled with a fluid with a density of $54 \, \text{kg/m}^3$.

**Tank Specifications:**
- The cross-section of the end of the tank is represented by the shaded region in the first diagram.
- This region is bounded by the y-axis, the curve $y = f(x)$ or equivalently $x = g(y)$, and the horizontal line $y = 12$.
- Assume $c = 5 \, \text{m}$ and $d = 12 \, \text{m}$.

**First Diagram: Cross-Section View**
- The diagram shows a coordinate system with a curve which marks the boundary of the shaded region.
- The curve is labeled as $y = f(x)$ or $x = g(y)$, with boundaries at $c$ and $d$.

**Tank Dimensions:**
- The tank extends backward, maintaining a length of 9 meters.
- The shape of this extension mirrors the cross-section shown.

**Second Diagram: 3D Perspective**
- Displays a 3D representation of the tank extending backward 9 meters.
- The function annotations $y = f(x)$ or $x = g(y)$ highlight its shape.

**Objective:**
Calculate the work needed to pump the liquid to 4 meters above the tank's top.

**Instructions:**
To find the volume, force, and work for a slice of fluid:

1. Use variables $\Delta x$ or $\Delta y$ to represent small changes in dimensions.
2. Ensure capital 'Delta' is used ($\Delta$).
3. Utilize the CalcPad navigation by selecting the ‘Vars’ tab to integrate these variables into your computations.

**CalcPad Interface:**
- Shows options like $\Delta y$, $\Delta x$, $g(y)$, and $f(x)$.

This guide aids in computing the necessary physical work to elevate the liquid out of the tank, accounting for its dimensions and fluid properties.

Expert Solution

Step 1: Evaluation of volume of slice

The slice is shown below:

The volume of slice is,

$increment v equals 9 x increment y bold italic d bold italic v bold equals bold 9 bold italic x bold italic d bold italic y$

$bold italic d bold italic v bold equals bold 9 bold italic g bold left parenthesis bold italic y bold right parenthesis bold italic d bold italic y$

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