(e) Sketch a reasonable graph of h (t) and of h' (t) based on your answers in (c) and (d). Certain properties of graph (such as slope at t_0, concavity etc) must be shown clearly and explained.
(e) Sketch a reasonable graph of h (t) and of h' (t) based on your answers in (c) and (d). Certain properties of graph (such as slope at t_0, concavity etc) must be shown clearly and explained.
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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Topic Video
Question
First image is the word problem context. Please answer PART E from second image. Please show all steps and explain with clear handwriting. No cursive if possible.
![**Problem Choice 2.1**
In this problem, we will consider a filter that is shaped as a lower semi-sphere. The picture below shows a spherical filter with water filled to level \( h \). The following formulas are essential for completing this work:
- The volume of water filled to level \( h \) in a spherical bowl with radius \( R \) is given by
\[ V = \pi h^2 \left( R - \frac{h}{3} \right) .\]
- Another crucial formula is
\[ A = 2 \pi R h ,\]
for the surface area of the part of the sphere that is in contact with the body of water.
**Explanation of Diagram**
The diagram depicts a lower semi-spherical filter filled with water to a level \( h \). The radius of the sphere is marked as \( R \), and an arrow indicates the height \( h \) of the water level. Drops below the filter illustrate the draining water.
We will investigate the relationship between the water level \( h \) and the rate at which this level changes
\[ \left( \frac{dh}{dt} \right) \]
based on the assumption that the drain rate (the rate at which the volume of the water decreases) is proportional to the surface area of contact. (This is approximately what happens with an actual coffee filter!) Ultimately, we are interested in expressing \( h \) as a function of time \( t \).
Please read on for more detailed instructions below.
**CONTEXT**
To put your mind in a more concrete context, let's think about a cone-shaped filter. For the filter of any shape, water is drained at a rate that is proportional to the contact area. In the case of a right-angle cone filter, this area is \( \sqrt{2} \pi h^2 \). So,
\[ \frac{dV}{dt} = \sqrt{2} \pi kh^2 \]
for some negative constant \( k \). But, since
\[ V = \frac{1}{3} \pi h^3 \]
(for a right-angle cone) we have
\[ \sqrt{2} \pi kh^2 = \pi h^2 \cdot \frac{dh}{dt} \implies \sqrt{2} k = \frac{dh](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F28092663-10ce-474b-a101-95a46deb92bf%2F2efb0ace-207a-4051-a781-8c0da9db4563%2F9o4cmpb.png&w=3840&q=75)
Transcribed Image Text:**Problem Choice 2.1**
In this problem, we will consider a filter that is shaped as a lower semi-sphere. The picture below shows a spherical filter with water filled to level \( h \). The following formulas are essential for completing this work:
- The volume of water filled to level \( h \) in a spherical bowl with radius \( R \) is given by
\[ V = \pi h^2 \left( R - \frac{h}{3} \right) .\]
- Another crucial formula is
\[ A = 2 \pi R h ,\]
for the surface area of the part of the sphere that is in contact with the body of water.
**Explanation of Diagram**
The diagram depicts a lower semi-spherical filter filled with water to a level \( h \). The radius of the sphere is marked as \( R \), and an arrow indicates the height \( h \) of the water level. Drops below the filter illustrate the draining water.
We will investigate the relationship between the water level \( h \) and the rate at which this level changes
\[ \left( \frac{dh}{dt} \right) \]
based on the assumption that the drain rate (the rate at which the volume of the water decreases) is proportional to the surface area of contact. (This is approximately what happens with an actual coffee filter!) Ultimately, we are interested in expressing \( h \) as a function of time \( t \).
Please read on for more detailed instructions below.
**CONTEXT**
To put your mind in a more concrete context, let's think about a cone-shaped filter. For the filter of any shape, water is drained at a rate that is proportional to the contact area. In the case of a right-angle cone filter, this area is \( \sqrt{2} \pi h^2 \). So,
\[ \frac{dV}{dt} = \sqrt{2} \pi kh^2 \]
for some negative constant \( k \). But, since
\[ V = \frac{1}{3} \pi h^3 \]
(for a right-angle cone) we have
\[ \sqrt{2} \pi kh^2 = \pi h^2 \cdot \frac{dh}{dt} \implies \sqrt{2} k = \frac{dh
![### GUIDED INSTRUCTION
#### (a)
**Express the assumption that "the rate of change of the volume of water, \(V\), is proportional to the surface area \(A\)."**
Use a proportionality constant \(k\) (which may be named as "porosity"). Is it a positive or a negative constant? Since this is a crucial step, I will just write the answer for you.
\[
\frac{dV}{dt} = kA
\]
#### (b)
Use the given formulas for the volume and area (displayed above the diagram) to rewrite the equation in (a) as an equation for \(h\) and \(\frac{dh}{dt}\). (We will call this a **related rate equation**).
#### (c)
**Solve the related rate equation from (b) for \(\frac{dh}{dt}\) and simplify it.** Use this equation to explain why the drain rate \(h' (t)\) cannot be a constant function unless the filter is completely clogged (that is, unless \( k = 0\)). [Hint: You may use \( h'' (t) \) to be able to answer this. There are other ways, too.] Suppose the water is completely drained at a time \( t = t_0 \). Talk about the limits \(\lim_{t \to t_0^{-}} [h (t)]\) and \(\lim_{t \to t_0^{-}} [h' (t)]\). To answer these, make an assumption that \( h (t) \) is a monotonically decreasing continuous function (i.e., water doesn't climb back up!), and that \( h ( t_0 ) = 0 \). Verify your findings analytically by using the related rate equation.
#### (d)
**Study the concavity of the graph of the function \(h\).** We do this so we know whether the water level goes down faster and faster or slower and slower. Is the graph of \(h\) always concave up or always concave down or does it have an inflection point? Provide a physical explanation to your finding about the concavity.
#### (e)
**Sketch a reasonable graph of \(h (t)\) and of \(h' (t)\)** based on your answers in (c) and (d). Certain properties of the graph (such as slope at](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F28092663-10ce-474b-a101-95a46deb92bf%2F2efb0ace-207a-4051-a781-8c0da9db4563%2Fvmottr5.png&w=3840&q=75)
Transcribed Image Text:### GUIDED INSTRUCTION
#### (a)
**Express the assumption that "the rate of change of the volume of water, \(V\), is proportional to the surface area \(A\)."**
Use a proportionality constant \(k\) (which may be named as "porosity"). Is it a positive or a negative constant? Since this is a crucial step, I will just write the answer for you.
\[
\frac{dV}{dt} = kA
\]
#### (b)
Use the given formulas for the volume and area (displayed above the diagram) to rewrite the equation in (a) as an equation for \(h\) and \(\frac{dh}{dt}\). (We will call this a **related rate equation**).
#### (c)
**Solve the related rate equation from (b) for \(\frac{dh}{dt}\) and simplify it.** Use this equation to explain why the drain rate \(h' (t)\) cannot be a constant function unless the filter is completely clogged (that is, unless \( k = 0\)). [Hint: You may use \( h'' (t) \) to be able to answer this. There are other ways, too.] Suppose the water is completely drained at a time \( t = t_0 \). Talk about the limits \(\lim_{t \to t_0^{-}} [h (t)]\) and \(\lim_{t \to t_0^{-}} [h' (t)]\). To answer these, make an assumption that \( h (t) \) is a monotonically decreasing continuous function (i.e., water doesn't climb back up!), and that \( h ( t_0 ) = 0 \). Verify your findings analytically by using the related rate equation.
#### (d)
**Study the concavity of the graph of the function \(h\).** We do this so we know whether the water level goes down faster and faster or slower and slower. Is the graph of \(h\) always concave up or always concave down or does it have an inflection point? Provide a physical explanation to your finding about the concavity.
#### (e)
**Sketch a reasonable graph of \(h (t)\) and of \(h' (t)\)** based on your answers in (c) and (d). Certain properties of the graph (such as slope at
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