Given information: The following information has been given O is the center of the circle

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2020-12-13
Answered

To prove:The congruency of $\overrightarrow{SO}\stackrel{\sim}{=}\overrightarrow{TO}$ .

Given information: The following information has been given O is the center of the circle

$\mathrm{\angle}SOV=\mathrm{\angle}TOW$

$\mathrm{\angle}WSO=\mathrm{\angle}VTO$

Given information: The following information has been given O is the center of the circle

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okomgcae

Answered 2020-12-14
Author has **93** answers

Formula used : By the virtue of similarity, if any two angles of the triangles are congruent, the third one will also be congruent

Proof : From the given information we can say that by the virtue of similarity, triangles OWS and OVT are similar.

Now, we know that

$\overrightarrow{OW}=\overrightarrow{OV}=R$

Thus, now we can say that

$\mathrm{\u25b3}OWS\stackrel{\sim}{=}\mathrm{\u25b3}OVT$

Hence, for congruent triangles, we use virtue of congruency to tell us that

$\overrightarrow{SO}\stackrel{\sim}{=}\overrightarrow{TO}$

Hence, the given congruency is proved

Proof : From the given information we can say that by the virtue of similarity, triangles OWS and OVT are similar.

Now, we know that

Thus, now we can say that

Hence, for congruent triangles, we use virtue of congruency to tell us that

Hence, the given congruency is proved

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that i want to solve (to find the volume) with cylindrical coordinates. I am evaluating the integral to get:

$V=\iiint dV=\underset{R}{\iint}\phantom{\rule{-5pt}{0ex}}{\int}_{{x}^{2}+{y}^{2}}^{x+y}\phantom{\rule{thinmathspace}{0ex}}dz\phantom{\rule{thinmathspace}{0ex}}dA=\underset{R}{\iint}x+y-({x}^{2}+{y}^{2})\phantom{\rule{thinmathspace}{0ex}}dA.$

and from here i am trying to get the bounds for r by intersecting the two functions and i get that $r=0\text{}or\text{}r=1$ therefore i tought that $0\le r\le 1$ but it dosen't seems right. But i get the following result:

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and this is not the corrent answer, i should get $\frac{\pi}{8}.$. I know this is too specific to a given problem question i apologize for that but can someone tells me where i made a mistake

I have the following function:

$Z={x}^{2}+{y}^{2},\text{}Z=x+y$

that i want to solve (to find the volume) with cylindrical coordinates. I am evaluating the integral to get:

$V=\iiint dV=\underset{R}{\iint}\phantom{\rule{-5pt}{0ex}}{\int}_{{x}^{2}+{y}^{2}}^{x+y}\phantom{\rule{thinmathspace}{0ex}}dz\phantom{\rule{thinmathspace}{0ex}}dA=\underset{R}{\iint}x+y-({x}^{2}+{y}^{2})\phantom{\rule{thinmathspace}{0ex}}dA.$

and from here i am trying to get the bounds for r by intersecting the two functions and i get that $r=0\text{}or\text{}r=1$ therefore i tought that $0\le r\le 1$ but it dosen't seems right. But i get the following result:

$2\pi {\int}_{0}^{1}(r-{r}^{2})rdrd\theta =2\pi {\textstyle (}\frac{{r}^{3}}{3}-\frac{{r}^{4}}{4}{\textstyle )}{{\textstyle |}}_{0}^{1}=\frac{\pi}{6}$

and this is not the corrent answer, i should get $\frac{\pi}{8}.$. I know this is too specific to a given problem question i apologize for that but can someone tells me where i made a mistake

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Is there any way to obtain the closed-form of ${p}_{i}^{\ast}$ as the function of $w$, ${a}_{i}$, and ${d}_{i}$?

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where ${p}_{i}\in \{0,1\}$ is binary variable and $\sum _{i=1}^{I}{p}_{i}=1$. I need to find optimal ${p}_{i}^{\ast}$, where

${\mathbf{p}}^{\ast}=\mathrm{arg}\underset{\mathbf{p}}{max}w\text{ln}{\textstyle (}\sum _{i=1}^{I}{p}_{i}{a}_{i}{\textstyle )}-\sum _{i=1}^{I}{p}_{i}{d}_{i}$

Is there any way to obtain the closed-form of ${p}_{i}^{\ast}$ as the function of $w$, ${a}_{i}$, and ${d}_{i}$?

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