Area Definition and 1000 Threads

I am not very good at proofs. The only thing I have come up with is the following regularity. However, I am not sure how this can be related to the above problem. Given a sphere ##S_a## with a center ##C## and a diameter of ##a##. I can now construct a line segment ##b## with the endpoints...

Find question here; Find solution here; I used the same approach as ms- The key points to me were; * making use of change of variables... $$A_{1}=\int_0^\frac{π}{4} {\frac {4\cos 2x}{3-\sin 2x}} dx=-2\int_3^{-2} {\frac {du}{u}}= 2\int_2^3 {\frac {du}{u}}=2\ln 3-2\ln2=\ln 9 - \ln 4=\ln...

I am still looking at this question. One thing that i know is that the distance ##AB=\dfrac {(λ+1)\sqrt {2λ^2-2λ+1}}{λ^2}## distance ##OA=\sqrt 2## distance ##OB=\dfrac{\sqrt{λ^2+1}}{λ^2}## Perpendicular distance from point ##B## to the line ##OA=\dfrac{\sqrt{2(λ^4+2λ^3+2)}}{2λ^2}## Therefore...

Below is an image to calculate the surface area of a sphere using dA. I can see how ##rcos\theta d\phi## works, but I don't understand how that side can't just be ##rd\phi## with a slanted circle representing the arc length. The second part I don't understand is why it is integrated from...

I’m wondering if there is a formula for calculating the coordinate points of a polygon given the following - Center point is known - area is known - Point A is known - Points B, C, and D are UNKNOWN I am NOT a math pro - this is for a puzzle I’m trying to solve and I can’t remember if this...

I have the solution for this problem using dydx as the area. Worse yet, I cannot find another solution for it. Everyone seems to just magically pick dydx without thinking and naturally this is frustrating as learning the correct choice is 99.9% of the battle... So, I was curious how one might...

Find the solution here; Find my approach below; In my working i have; ##A_{minor sector}##=##\frac {128.1^0}{360^0}×π×5×5=27.947cm^2## ##A_{triangle}##=##\frac {1}{2}####×5×5×sin 128.1^0=9.8366cm^2## ##A_3##=##\frac {90^0}{360^0}####×π×10×10##=##78.53cm^2## ##A_{major...

My interest is on part (c) only. Wow, this was a nice one! boggled me a little bit anyways; my last steps to solution, ##A= \dfrac{1}{2} ×16.8319^2 × 0.5707=80.84cm^2## bingo! Any other approach apart from using sine? Cheers guys

Sorry i don't understand English very well , if someone want to explain to me this problem?

My query in only on the highlighted part...c.ii. Find the question below; Find the markscheme here part c(ii) does not seem correct as i have; ##A_1=0.5 ×(0.65+0.84)0.3 ×2=0.447m^2## ##A_2 = 0.65 ×1.6=1.04m^2## ##A_3 = (0.3146 × 1.6)2=1.00672m^2## Total surface area =...

I used the parallel axis theorem to solve the question but my answer is still wrong. Any ideas where I slipped? I can't seem to figure out the problem?

This is the question, Now to my question, supposing the vectors were not given, can we let ##V=\vec {RQ}## and ##W=\vec {RP}##? i tried using this and i was not getting the required area. Thanks...

Normally when designing a retaining wall, you check for failure due to sliding, overturning, and insufficient bearing capacity. However, if I have a retaining wall which is around an enclosed pit, it doesn't seem reasonable to perform the same checks for sliding and overturning (the retaining...

I was looking at this problem today, and i was trying to figure out its area with the given dimensions shown. First, is this even possible?...i later looked at the problem in detail and realized that i had missed out on some dimension that was given on the text. Having said that, i would like...

Let ##\mathscr{H}## be a constant-##v## cross-section of the event horizon (area ##A##). The expansion is the fractional rate of change of the surface element, ##\theta = \frac{1}{\delta S} \frac{d(\delta S)}{dv}##. The problem asks to prove the formula ##\frac{dA}{dv} = \frac{8\pi}{\kappa}...

Find area of square ABCD if OQ=OF=6.

Summary:: Does the surface area of a parachute affect its drag coefficient? If so, how? I have been trying to figure out the effect of surface area on the drag coefficient of a parachute. I have designed a lab in which parachutes of different surface areas are dropped and the terminal velocity...

Hello, My question relates to gamma spectroscopy. I understand how the net peak area is calculated for any photopeak. Fortunately, gamma-spec software (e.g., Genie-2000 from Canberra) provides Net peak area and associated uncertainty (for Cs-137 661.7 keV peak, as an example). My question: are...

This isn't homework, but I figured it's fine if I make it a HW problem and post here (if not, please let me know). Let ##z^*=0## be the vertex of the pyramid, and let ##z^*## run the altitude. It's easy to show the area of the base normal to the altitude is ##A = 4 \left.z^*\right.^2...

how to find volume and surface area of this without using the upper lid

In triangle ABC $AC=BD, CE=2, ED=1, AE=4$ and $\angle CAE=2 \angle DAB$. Find area ABC.

I was looking at the problem below in detail, attached find the problem and the mark scheme solution. Now this was my approach which is just similar to the Mark Scheme method ##2## above where they expressed ##x=f(y)##... I did it this way; ...There was some work involved particularly...

Find the length x if the shaded area is 1200 cm^2

For a streamlined and bluff bodies, why is it standard to have the projected area be a fixed reference area, but yet the angles of attack (AoA) vary? If one were to vary the AoA then the projected area would technically change. The following link discusses that it is a convention to avoid...

(I haven't been actively following this line of research... but I think it is possibly interesting reading. It's been in the science news today.) "Black Hole Area Law Tested" (synopsis) https://physics.aps.org/articles/v14/s87 "Testing the Black-Hole Area Law with GW150914" Maximiliano Isi...

A rather unique phenomenon is occurring in or near the Salton Sea region of California near Niland, California. It started in a farmer's field, but the puddle has migrated. There are emissions of CO2 and steam, and it appears some geothermal activity, which apparently is not unique. The...

Hello all, I am trying to search for different areas to do masters which would match my interests. I am broadly interested in , fluids (aligned to general aerodynamics) especially compressible fluids , turbomachinery, rockets. I am thinking to work in some sector related to gas turbines or jet...

Hello: Let's say you have a string and get data by changing the frequency a transverse wave in the string to get different standing modes. You measure the wavelength of each mode for each frequency. That is, the data you get are frequency and wavelength. Now, you are trying to find the...

Rectangle ABCD is inscribed in the circle shown. If the length of side $\overline{AB}$ is 5 and the length of side $\overline{BC}$ is 12 what is the area of the shaded region? $a.\ 40.8\quad b.\ 53.1\quad c\ 72.7\quad d \ 78.5\quad e\ 81.7$ well to start with the common triangle of 12 5...

To find ##\delta## for the 1st order, all I need to do is to square the diameter of the 2nd ring and subtract it to the square of the diameter of the first ring. $$\delta_{1st \; order} = {d^2}_{2nd \; ring} - {d^2}_{1st \; ring}$$ To find ##\Delta##, I can use the below equation...

Hello everyone, I am trying to do some calculations for the energy output of a solar farm that I am designing as my dissertation. However, when I trie to calculate the following formula: Wp = ηpvGBA from Equation (11) above, where: ηpv is module efficiency (18.4%) GB is solar irradiance (3.8)...

Figure shows six identical circles inside a rectangle. The radius of each circle is 24 cm. The radius of the circles is the greatest possible radius so that the circles fit inside the rectangle. The six circles form the pattern shown in Figure so that • each circle touches at least two other...

So I basically took the integral and ended up with W=PVf-A(Vf^3)/3-PVi+A(Vf^3)/3 so 65.7=72*5.3-A(5.3)^3/3-72(2.4)+A(2.4)^3/3 But when I solve for A I get the wrong answer of 3.179 when the answer is suppose to be 5.05. I've checked my calculation with an algebra calculator too...

So first I found rate of heat change using the above equation, with T=883K, e=1, SA= 6*l^2=21.66 Now dQ/dt=746593.71 W Now I am not sure entirely what to do next. They give density so I likely have to get the mass from that, M=pV,=1.9^3*4037=27689.783 kg. My issue is that I don't know how to...

The surface area of the sphere is 4πr^2. dr/dt is given as 3cm^-1. dS/dt=dS/dr*dr/dt Differentiating 4πr^2 is dS/dr= 8πr dS/dt=8πr*3 dS/dt=24πr Given that r=5 dS/dt=24π*5=120 π The volume of the sphere is 4/3πr^3, differentiating which is dV/dr=4πr^2 dV/dt=dV/dr*dr/dt dV/dt= 4πr^2*3...

Regarding finding centers of mass of infinite figures, how one can show that $$ \int_{-\infty}^\infty \left(\frac1{x^2}-\cos \frac1x\right)dx=\pi $$ for instance, and other similar integrals, like the following? $$ \int_0^\infty (x^2-\frac6{x^4})dx=0 $$

this is another question that i saw on the internet...

R is the region bounded by the functions f(x)=3√x−4 and g(x)=3x/5−8/5. Find the area A of R. Enter answer using exact values.

I want to ask about the solution. The solution divides region R into two parts: curved part and triangle. The triangle is obtained by drawing line ##x=5##. Let say line ##x=5## cuts x-axis at point A so the triangle is PAQ For the curved part: $$\int_{-1}^{2} (3+3t) ~2t~ dt$$ My question: Why...

Find the perimeter and area of CD, if ABCE is a parallelogram and ADE is an equilateral triangle.

My Effort: Circumference = pi•d 10 •pi = pi•d 10•pi/pi = d 10 = d, where d is the diameter of the circle. Area = pi•r^2, where r is the radius of the circle. Diameter = 2 times the radius. 10pi = 2r 10pi/2 = r 5pi = r A = pi•r^2 A = pi(5pi)^2 A = 25•pi^3, which makes no sense. Only...

What is the minimal area of a right-angled triangle whose inradius is 1 unit?

Hello everyone. I am having trouble finding the area of the shaded region using the determinant area formula. I know where to plug in the numbers into the formula. My problem here is finding the needed points in the form (x, y) from the given picture for question 21.

I was reaching for some crackers across the table. My arm tipped over a full opened bottled water that splashed about 1/3 of its contents out of it. Of that 1/3, about 50% got on the table and the other 50% splashed across my laptop's keyboard area. I immediately reached for towels to soak...

Find the maximum area of a pentagon $ABCDE$ inscribed in a unit circle such that the diagonal $AC$ is perpendicular to the diagonal $BD$.

I am working on related rates problems involving figuring out how area of a square increases per second based on how much one side increases per second (or how the area of a circle increases based on increase of the radius, etc.). I was wondering about the practical significance of problems like...

In "An Introduction to Nuclear Physics by W. N. Cottingham, D. A. Greenwood" for the surface area of an oblate ellipsoid, the following equation is written for small values of ε : The book has said this without proof. I found the following formula for the desired shape: No matter how hard I...

The volume of a cuboid box with a square base is 2 litres. The production cost per unit of its top and its bottom is twice the production cost per unit of its lateral sides. Suppose the side length of its base is x and the height of the cuboid is h. The minimum production cost is reached when...

I'm looking for a source that fully derives the complete formula for the surface area of a general (triaxial) ellipsoid. I'd prefer a source that has more than just a full derivation, but also has a fair amount of prose discussion on this topic. Some historical context would be nice, as well...