A machinist creates a washer by drilling a hole through the center of a circular piece of metal. If the piece of metal has a radius of x + 10 and the hole has a radius of x + 6, what is the area of the washer?

Valentina Holland
2022-09-23
Answered

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gerasseltd9

Answered 2022-09-24
Author has **8** answers

Area of the Circle$=\pi {r}^{2}$

Area of the washer

$=\pi ({(x+10)}^{2}-{(x+6)}^{2})$

$=\pi (x+10+x+6)(x+10-x-6)$

$=\pi (2x+16)(10-6)$

$=8\pi (x+8)$

Area of the washer

$=\pi ({(x+10)}^{2}-{(x+6)}^{2})$

$=\pi (x+10+x+6)(x+10-x-6)$

$=\pi (2x+16)(10-6)$

$=8\pi (x+8)$

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Theorem:

Given a system of linear equations$Ax=b$ where

$A\in {M}_{m\times n}\left(\mathbb{R}\right),x\in {\mathbb{R}}_{\text{col}}^{n},b\in {\mathbb{R}}_{\text{col}}^{m}$ .

Deduce that a solution x exists if and only if$\text{rank}\left(A\mid b\right)=\text{rank}\left(A\right)$ where $A\mid b$ is the augmented coefficient matrix of this system

I am having trouble proving the above theorem from my Linear Algebra course, I understand that A|b must reduce under elementary row operations to a form which is consistent but I don't understand exactly why the matrix A|b need have the same rank as A for this to happen.

Please correct me if I am mistaken

Given a system of linear equations

Deduce that a solution x exists if and only if

I am having trouble proving the above theorem from my Linear Algebra course, I understand that A|b must reduce under elementary row operations to a form which is consistent but I don't understand exactly why the matrix A|b need have the same rank as A for this to happen.

Please correct me if I am mistaken

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Here is the example I encountered :

A matrix$M(5\times 5)$ is given and its minimal polynomial is determined to be $(x-2)}^{3$ . So considering the two possible sets of elementary divisors

$\{{(x-2)}^{3},{(x-2)}^{2}\}\text{}\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}\text{}\{{(x-2)}^{3},(x-2),(x-2)\}$

we get two possible Jordan Canonical forms of the matrix , namely$J}_{1$ and $J}_{2$ respectively. So $J}_{1$ has 2 and $J}_{2$ has 3 Jordan Blocks.

Now we are to determine the exact one from these two. From the original matrix M, we determined the Eigen vectors and 2 eigen vectors were linearly independent. So the result is that$J}_{1$ is the one .So, to determine the exact one out of all possibilities , we needed two information -

1) the minimal polynomial ,together with 2) the number of linearly independent eigen vectors.

Now this was a question-answer book so not much theoretical explanations are given . From the given result , I assume the number of linearly independent eigen vectors -which is 2 in this case - decided$J}_{1$ to be the exact one because it has 2 Jordan Blocks. So the equation

"Number of linearly independent eigen vectors=Number of Jordan Blocks"

must be true for this selection to be correct .

Now this equation is not proved in this book or the text book I have read says nothing of this sort

So, that is my question here : How to prove the equation "Number of linearly independent eigen vectors=Number of Jordan Blocks"?

A matrix

we get two possible Jordan Canonical forms of the matrix , namely

Now we are to determine the exact one from these two. From the original matrix M, we determined the Eigen vectors and 2 eigen vectors were linearly independent. So the result is that

1) the minimal polynomial ,together with 2) the number of linearly independent eigen vectors.

Now this was a question-answer book so not much theoretical explanations are given . From the given result , I assume the number of linearly independent eigen vectors -which is 2 in this case - decided

"Number of linearly independent eigen vectors=Number of Jordan Blocks"

must be true for this selection to be correct .

Now this equation is not proved in this book or the text book I have read says nothing of this sort

So, that is my question here : How to prove the equation "Number of linearly independent eigen vectors=Number of Jordan Blocks"?

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