(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 dh (We will call this a related rate equation.) dt and
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
First image is the word problem. Please answer PART B. Please show all steps and explain with clear handwriting. No cursive if possible.
![### 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 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 \( h' \left( t \right) \) cannot be a constant function unless the filter is completely clogged (that is, unless \( k = 0 \)). [Hint: You may use \( h'' \left( t \right) \) to be able to answer this. There are other ways, too.]
1. Suppose the water is completely drained at a time \( t = t_0 \). Talk about the limits \( \lim_{t \to t_0^-} \left[ h(t) \right] \) and about \( \lim_{t \to t_0^-} \left[ h'(t) \right] \).
2. To answer these, make an assumption that \( h \left( t \right) \) is a monotonically decreasing continuous function (i.e., water doesn't climb back up!), and that \( h \left( t_0 \right) = 0 \). 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)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6653f5ea-9ae4-4965-89e5-4546e31fc625%2F2fe8eec7-eee4-42bd-96de-b86c580ed54f%2F4r9m9r.png&w=3840&q=75)
![### 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 the completion of 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.
#### Diagram Explanation
The diagram illustrates a semi-spherical bowl with radius \( R \) filled with water to height \( h \). Water is shown dripping from the bottom, indicating the draining process.
We will investigate the relationship between the water level \( h \) and the rate at which this level changes ( \( \frac{dh}{dt} \) ) 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
\[
\sqrt{2} \pi kh^2 = \pi h^2 \cdot \frac{dh}{dt} \implies \sqrt{2} k = \frac{dh}{dt}
\]
This tells us that the water level will decrease at a constant rate. In the case of a spherical funnel, will we arrive at a similar conclusion? We](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6653f5ea-9ae4-4965-89e5-4546e31fc625%2F2fe8eec7-eee4-42bd-96de-b86c580ed54f%2Fhh5cwgo.png&w=3840&q=75)
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