Volume Definition and 1000 Threads

I understand how to get the dimensions that equal 8436m^3. What I don't understand is how to find the range of all possible dimensions. I solved the inequality to get ##6w^{3}-13w^{2}-5w-8436## Using systematic guessing I found the root is x=4, so the factor is x-4. Dividing (x-4) into...

Hi, what I've done so far is solving equation 2) for ##U##, and replacing what I get in equation 1). Then, ##c_V## is equal to the partial derivative of ##S## with respect to T times T, so I've done that. The derivative is ##CNR/T##, so ##c_V=CNR## but those aren't the correct units for ##c_V##.

$\displaystyle\pi\int_{0}^{\infty} e^x \ dx = \pi$ Ok I looked at some of the template equations but came up with this.

First, I try to make a sketch and from that I take limit of integration from: 1. ##z_1 = 0## to ##z_2 = 4 - x -2y## 2. ##x_1 = 0## to## x_2 = 4- 2y ## 3. ##y_1 = 0## to ##y_2 = 2## Then, I define infinitesimal volume element in the first octant as ##dV = 1/8 dz dz dy##. Therefore, $$V=1/8...

Hi to everyone, do you know the "One World Trade Center"? Well, I've to calculate two things about it: -The volume, according to its particular shape -The surface of the glass plates which cover the whole structure Searching on internet i found two dimensions: 1) Total height without...

I started to understand how to apply Lagrange multiplier methods. But, for problem like this, I have difficulty to build the function to describe the volume that will be maximized. For the second question, I know from the example (in ML Boas) that ##V=8xyz## becase (x,y,z) is in the 1st octant...

Is this correct?

In the section 8-2 dealing with resolving the state vectors, we learn that |\phi \rangle =\sum_i C_i | i \rangle and the dual vector is defined as \langle \chi | =\sum_j D^*_j \langle j |Then, the an inner product is defined as \langle \chi | \phi \rangle =\sum_{ij} D^*_j C_i \langle j | i...

Purcell says that taking the surface integral of the magnetic field ##\textbf{B}## over the surfaces ##S_{1}, S_{2}, S_{3},...## below is a good way of finding the average of the volume integral of ##\textbf{B}## in the neighborhood of these surfaces. More specifically, he says in page...

In physics we often come across $$\rho=\dfrac{dq}{dV}$$ Does it mean: ##(i)## ##\displaystyle \lim_{\Delta V \to 0} \dfrac{\Delta q}{\Delta V}## OR ##(ii)## ##\dfrac{\partial}{\partial z} \left( \dfrac{\partial}{\partial y} \left( \dfrac{\partial q}{\partial x} \right) \right)## What does...

Elemental fixed streamtube control volume from Professor White’s textbook “Fuid Mechanics”: I was unable to develop the intermediate steps for the following approximations: (continuity equation according to the book ) Where and (Momentum equation according to the book) In...

Note this is in our Lagrangian Mechanics section of Classical Mechanics, so I assume he wants us to use Calculus of Variations to solve it. The surface area is fixed, so that'll be the constraint. Maximizing volume, we need a functional to represent Volume. This was tricky, but my best guess for...

Determine the volume of the shaded area around the Y-axis by using the theorem of Pappus Guldinus, where value of R = 143,3 cm. a) Determine the area of the shaded section. b) Determine the center of gravity of the shaded section. c) Detrmine the volume by using the theorem of Pappus Guldinus...

Will the available Volume of oxygen gas for use of patients increase when the pressure decreases from 12.4 MPa to 500 KPa? Is using boyle's law the right way to calculate the available volume?

A chocolate company produces triangular chocolate bars. The length of the chocolate bar is x cm, and its cross section is an isosceles triangle. The length of the base side of the cross section is 3 cm, the height is h cm, and the two base angles are 50 degrees. Moreover, the company uses a...

1. Area is the naming convention assigned to that which is within a closed diagram in the x-y dimensions. 2. Area is also the naming convention used in simplified Lorentzian diagrams in the x-t dimensions. 3. Volume is the naming convention used to that which is within a closed vessel in the...

For me is not to easy to understand volume element ##dV## in different coordinates. In Deckart coordinates ##dV=dxdydz##. In spherical coordinates it is ##dV=r^2drd\theta d\varphi##. If we have sphere ##V=\frac{4}{3}r^3 \pi## why then dV=4\pi r^2dr always?

Why is energy balance for a control volume dE/dt = dQ/dt-dW/dt-dm/dt(ΔH+ΔKE+ΔPE) 0 = dQ/dt-dW/dt-dm/dt(ΔH+ΔKE+ΔPE) whereas for other systems it is ΔE =Q-W-(ΔU+ΔKE+ΔPE) 0 = Q-W-(ΔU+ΔKE+ΔPE) with enthalpy, h = u +pv, replaced by only the internal energy? How is the pv term accounted for...

Afternoon all, Hopefully somebody can help me, I'm doing my final year project and it's looking at the effect of heat treatment on in17, when I run an XRD scan I found that I all the phases sort of hid behind the matrix and so can't really make them out. So I've been looking at using the SEM...

This is not homework. I have given myself two parameters; ##\theta##, and ##\alpha##. (see figure, it is a side view): The idea is to find an expression for the radius of the circles as ##x## varies on that line (figure), then sum up infinitely many cylinders of infinitesimal thickness. The...

1) I can't manage to find/justify the relation ##(1)## below, from the common relation ##(2)## of a volume. 2) It seems the variable ##r## is actually the comoving distance and not comoving coordinates (with scale factor ##R(t)## between both). The comoving volume of a region covering a solid...

Hi All, I am working on an engineering problem, where i have to calculate the total work needed to compress a volume of air (Locked in a cylindrical chamber similar to an IC chamber where the piston moves to compress the air mixture) I am defining the process with the below initial...

Q=heat capacity calorimeter*(-)change in T*moles =0.009089mol*-6.8C*4.38kj/C =-0.2707kj/mol This answer is wrong but it was the only one I could come up with right now. I just noticed units in the answer would be wrong too. Any suggestions?

We understand that the crucial thing about the problem is that the volume of water present in the three containers are not the same. Also, we note that in each case the weight of the container is the total weight of its contents. (A student might be confused as to why should be so - after all...

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My answer seems to be way-off/improbable, so I figured something is wrong with it. From the periodic table, Mr of tetraethyl orthosilicate = 208.33 Mr of ethanol = 46.069 Mr of water = 18.015 Mr of SiO2 = 60.084 Let the volume of tetraethyl orthosilicate, ethanol and water be x,y,z ml...

I want to express <m(x,y,z)> over a sphere of radius R in terms of $$<\rho(x,y,z)>$$ e.g $$<m>=\frac{\int_{sphere R}m(x,y,z)dv}{\int_{sphere}dv}$$ $$<m>=\frac{\int_{sphereR}(\int \rho(x,y,z)dv)dv}{\int_{sphere R}dv}$$

I'm studying fluid and propulsion mechanics by myself. I stumbled upon this website from MIT: http://web.mit.edu/16.unified/www/SPRING/propulsion/UnifiedPropulsion2/UnifiedPropulsion2.htm#fallingblock It states that "Newton’s second law for a control volume of fixed mass" is $$\sum...

https://www.wired.com/story/a-bizarre-form-of-water-may-exist-all-over-the-universe/ Black iceI knew the Black Ice Theories since around 1990 https://www.nature.com/articles/s41586-019-1114-6 -- Demontis, P., LeSar, R. & Klein, M. L. New high-pressure phases of ice. Phys. Rev. Lett. 60...

The answer given for (2) is " lower pressure" , isn't increase pressure, the reaction will proceed towards fewer moles of gas, therefore increase the product yield for this question.

Anyone who has an idea for how to calculate the irradition [W/m2] to the base of a cylinder with radius R, height H, absorption coefficient k, and temperature T? I've looked at the approach with mean beam length by Hottel but cannot figure out what to do when it is the base of the cylinder that...

Since the assignment asks the work done by the gas, that should be equal to P1*(V2-V1) aka the area under the P1 line. Do I have to subtract the work done to the system or is this the solution already? If so, why do I need P2?

In this derivation,i am not sure why the second derivative of the vector ## S_j '' ## is equal to ## R^{u_j}{}_{xyz} s^y_j v^z y^x## could anyone explain this bit to me thank you it seems ## S_j '' ## is just the "ordinary derivative" part but it is not actually equal to ## R^{u_j}{}_{xyz}...

P1 = 2 bar V1 = 5.1L P2= 1bar V2 = V1P1/P2 = 10,2L, so the volume of gas would double? or should the absolute pressure be taken into account P1= 2bar (3bar absolute), V1=5.1L P2= 1 bar V2 = 15,3L?

EDIT: I thought I was in the math section for homework, sorry! My work is wrong, I don't see why though. Help much appreciated :)

I know that the formula for volume is equal to the definite integral ∫A(x)dx, where A(x) is the cross sectional. I found the definite integral where b=5 and a=0, for ∫4x2dx. I obtained the answer 500/3, however this was incorrect, and I'm unsure of where I went wrong? Thank you.

I don't know how to solve that integral, and to calculate the number of microstates first, then aply convolution and then integrate to find the volume of the phase space seems to be more complicated. Any clue on how to solve this? Thank you very much.

Hi, I find this... Please tell me your opinion on this. Thanks.

Sorry if i made any language errors, english is not my first language. Question: An area in the first quadrant (x=>0,y=>0) is limited by the axis and the graphs to the functions f(x)=x^2-2 and g(x)=2+x^2/4. When the area rotates around the y-axis a solid is created. Calculate the volume of...

Sorry if i made any language errors, English isn't my first language. Question: The limited area in the plane is created when the space between the line y=1 and the graph to the function f(x)=3*x/(x^2+1) rotates around the y-axis. Calculate the volume of the solid.I want to sum up all the...

I've come across discussions about the invariant properties of the 4 volume dV=dxdydzdt, but have yet to see its use in many equations. What is this object mostly used in and how is it or would it be used in quantum physics, cosmology, and relativity?

I attempt the solution on the attachment. The answer is 0.344 litre. Do I change 7L to m, so it is 0.007 cubic meters

I have attached the full answer in PDF file. I'm not sure about the answers. will really appreciate if they get checked

I am searching for the appropriate methodology/equation(s) to step beyond Boyle's Law to account for the phase change and solve this problem. All suggestions/guidance is greatly appreciated! Bruce

Using product rule, we have: [d/dx] (πr^2)(h) = (πr^2)(1 ) + (2πr)(h) Why is the two there? V = 2 πrh+2πr^2 The derivative of h is 1, not 2. Please help!

I am currently having trouble deriving the volume element for the first octant of an isotropic 3D harmonic oscillator. I know the answer I should get is $$dV=\frac{1}{2}k^{2}dk$$. What I currently have is $$dxdydz=dV$$ and $$k=x+y+z. But from that point on, I'm stuck. Any hints or reference...

In the chemical engineering text of Smith, VanNess, and Abbott, there is a section on partial molar volume. It states that Gibbs theorem applies to any partial molar property with the exception of volume. Why is volume different? In other words, when evaluating the partial molar volume of a...

For the probability of finding R out of N (indistinguishable) bosons in one half of a volume with a total of 2g states (g in each half) I get the following expression: PR = WR / WT where WT is the number of ways of distributing N particles in the total volume: WT = (N+2g-1)! / (N! (2g-1)!)...

According to following study 436 x 10^21 J of energy have been absorbed by the Earth's oceans since 1871. https://www.pnas.org/content/116/4/1126 What thickness of ice covering the globe would be needed to melt in order to absorb this amount of energy, assuming that all energy goes towards the...

Homework Statement: Base of solid is the region bounded by graphs ##y= \sqrt x## and ##y=x/2##. The cross sections perpendicular to the x-axis are squares whose sides run across the base of the solid. Find volume of solid. Homework Equations: - As stated above, I will want to calculate the...