The area of a rectangle is 216 inches squared. The ratio of the length to width is 3:2. Find the length and the width

Answer:

Step-by-step explanation:

Given that:    

The equation of the damped vibrating spring is y" + by' +2y = 0

(a) To convert this 2nd order equation to a system of two first-order equations;

let y₁ = y

y'₁ = y' = y₂

So;

y'₂ = y"₁ = -2y₁ -by₂

Thus; the system of the two first-order equation is:

y₁' = y₂

y₂' = -2y₁ - by₂

(b)

The eigenvalue of the system in terms of b is:

[tex]\left|\begin{array}{cc}- \lambda &1&-2\ & -b- \lambda \end{array}\right|=0[/tex]

[tex]-\lambda(-b - \lambda) + 2 = 0 \ \\ \\\lambda^2 +\lambda b + 2 = 0[/tex]

[tex]\lambda = \dfrac{-b \pm \sqrt{b^2 - 8}}{2}[/tex]

[tex]\lambda_1 = \dfrac{-b + \sqrt{b^2 -8}}{2} ; \ \lambda _2 = \dfrac{-b - \sqrt{b^2 -8}}{2}[/tex]

(c)

Suppose [tex]b > 2\sqrt{2}[/tex], then  λ₂ < 0 and λ₁ < 0. Thus, the node is stable at equilibrium.

(d)

From λ² + λb + 2 = 0

If b = 3; we get

[tex]\lambda^2 + 3\lambda + 2 = 0 \\ \\ (\lambda + 1) ( \lambda + 2 ) = 0\\ \\ \lambda = -1 \ or \ \lambda = -2 \\ \\[/tex]

Now, the eigenvector relating to λ = -1 be:

[tex]v = \left[\begin{array}{ccc}+1&1\\-2&-2\\\end{array}\right] \left[\begin{array}{c}v_1\\v_2\\\end{array}\right] = \left[\begin{array}{c}0\\0\\\end{array}\right][/tex]

[tex]\sim v = \left[\begin{array}{ccc}1&1\\0&0\\\end{array}\right] \left[\begin{array}{c}v_1\\v_2\\\end{array}\right] = \left[\begin{array}{c}0\\0\\\end{array}\right][/tex]

Let v₂ = 1, v₁ = -1

[tex]v = \left[\begin{array}{c}-1\\1\\\end{array}\right][/tex]

Let Eigenvector relating to  λ = -2 be:

[tex]m = \left[\begin{array}{ccc}2&1\\-2&-1\\\end{array}\right] \left[\begin{array}{c}m_1\\m_2\\\end{array}\right] = \left[\begin{array}{c}0\\0\\\end{array}\right][/tex]

[tex]\sim v = \left[\begin{array}{ccc}2&1\\0&0\\\end{array}\right] \left[\begin{array}{c}m_1\\m_2\\\end{array}\right] = \left[\begin{array}{c}0\\0\\\end{array}\right][/tex]

Let m₂ = 1, m₁ = -1/2

[tex]m = \left[\begin{array}{c}-1/2 \\1\\\end{array}\right][/tex]

∴

[tex]\left[\begin{array}{c}y_1\\y_2\\\end{array}\right]= C_1 e^{-t} \left[\begin{array}{c}-1\\1\\\end{array}\right] + C_2e^{-2t} \left[\begin{array}{c}-1/2\\1\\\end{array}\right][/tex]

So as t → ∞

[tex]\mathbf{ \left[\begin{array}{c}y_1\\y_2\\\end{array}\right]= \left[\begin{array}{c}0\\0\\\end{array}\right] \ \ so \ stable \ at \ node \ \infty }[/tex]

Answer:

1.07

Step-by-step explanation:

3.21/3=1.07

5.35/5=1.07

8.56/8=1.07

Answer:

if gray is what's left then it is C if white is what's left it is A

Step-by-step explanation:

Step-by-step explanation:

A because it shows the 2/3 of the pie colored meaning eaten

Answer:

10p + 12e + 10

General Formulas and Concepts:

Algebra I

Step-by-step explanation:

Step 1: Define

10p + 9 + 17e - 5e + 1

Step 2: Simplify

Answer:

10p + 10 + 12e would be your answer

I think 88 problems were on the test.

Answer:

Answer: The minimum number of full hours did Carson need to work at his father's business is 8 hours and 15 minutes. Happy to help!

Step-by-step explanation:

Answer:

Rita had a 28-mile head start.

Step-by-step explanation:

Answer:

the answer is A my lil' dudes

Step-by-step explanation:

Answer:

Answer:

-4

Step-by-step explanation:

if you switch it you get 12-16 and you have to get a negative so 16-12 is 4 so switch it to a nrgative to get -4

Answer: sarah has 36 cookies but she has 5 friends. she doesnt know how much cookies go to her friends. Write a equation to solve the problem

Step-by-step explanation:

Answer:

standard form is Ax + by + c = 0

Step-by-step explanation:

add 5x to each side to get 5x + y - 2 = 0

Answer:

Z is the standard normal random variable. The table value for Z is the value of the cumulative normal distribution. For example, the value for 1.96 is P(Z<1.96) = . 9750.

Step-by-step explanation:

Z is the standard normal random variable. The table value for Z is the value of the cumulative normal distribution. For example, the value for 1.96 is P(Z<1.96) = . 9750.

Answer:

$2.85 per person

Step-by-step explanation:

$191.10 * 0.15 = $28.665 tip

total tip divide by 10 people

$28.665/10 = $2.85

Each person has to contribute $2.85 amount for the tip.

Given,

Total no. of person go out for dinner = carrie + her nine friends

                                                             = 1+9 = 10

Amount of the bill is = $191.10

Amount of tip is = 15% of the bill

                           = [tex]\frac{15 }{100}[/tex] × $  191.10

                           = 28.665

Total amount each person contribute an equal amount for tip is = [tex]\frac{28.665}{10}[/tex]

=$ 2.85

Each person contribute $2.85 amount for the tip.

For more information about ratio and proportion click the link given below.

https://brainly.com/question/25561616?referrer=searchResults

Answer:

10y²+24.5

Step-by-step explanation:

distribute.

10y²+24.5

Hope this helps plz hit the crown :D

The equivalent expression for the given expression is 10y²+24.5.

The given expression is 10(y² + 2.45).

Equivalent expressions are expressions that work the same even though they look different. If two algebraic expressions are equivalent, then the two expressions have the same value when we plug in the same value for the variable.

The equivalent expression is

10(y² + 2.45)

=10y²+24.5

Therefore, the equivalent expression for the given expression is 10y²+24.5.

To learn more about an equivalent expression visit:

https://brainly.com/question/28170201.

#SPJ2

Answer:

I think it is postive

Answer:

78

Step-by-step explanation:

x+(x+2)+(x+4)=228

x+x+2+x+4=228

3x+6=228

Subtract 6 from both sides

3x=228-6

3x=222

Divide both sides by 3

x=74.The first number is 74

The second number (x+2)=74+2=76

The third number (x+4)=74+4=78

                                                 Question # 14

Given the numbers

10     11     12     13     14     15     16     17     18     19     20

Let 'x' be the number

The condition breakdown:

I am less than 20.

I am more than 13.

I am less than 17.

Finally:

I am 4 more than 12

i.e. 12+4 = 16

Thus, the number is x = 16

                                               Question # 15

Part a)

Given the numbers

10     11     12     13     14     15     16     17     18     19     20

Let 'x' be the number

The condition breakdown:

I am more than 10.

I am less than 20.

I am more than 12.

I am less than 15.

As the numbers left after all the conditions are fulfilled are 13 and 14.

So, from the remaining number 13 and 14;

14 > 13

Thus, the number x = 14

Part b)

Drawing the number 14 in the place value:

          Chart

Tens                Ones

1                         4

Answer:

[tex]\dfrac{5}{6}[/tex]

Step-by-step explanation:

The problem says Two-thirds divided by StartFraction 4 Over 5 EndFraction.

Two thirds means = 2/3

It means,

[tex]x=\dfrac{2}{3}\ \text{divided by}\ \dfrac{4}{5}[/tex]

or

[tex]x=\dfrac{\dfrac{2}{3}}{\dfrac{4}{5}}\\\\=\dfrac{2}{3}\times \dfrac{5}{4}\\\\=\dfrac{5}{6}[/tex]

Hence, the required answer is [tex]\dfrac{5}{6}[/tex].

Answer:

1 3/4 gallon or 1.75

Step-by-step explanation:

1 (1/4) = 5/7

5/4 = 5/7

5/4 = 1.25

1.25/x = 5/7

7(1.25) = 5x

5x = 8.75

x = 1 3/4

Answer:

15, 30, 45, 60, and 75

Step-by-step explanation:

15x1

15x2

15x3

15x4

15x5

I hope this helps you

The answer is f(x) = g(x+2) + 1.

Answer:

Depth of water is increasing at 0.143 ft/min.

Step-by-step explanation:

9514 1404 393

Answer:

  4.75%

Step-by-step explanation:

When simple interest is charged, the payback amount is ...

  A = P(1 +rt) . . . . for annual rate r and a term of t years

Using the given values, we can solve for r.

  10280 = 8000(1 +6r)

  1.285 = 1 +6r . . . . . . . . divide by 8000

  0.285 = 6r . . . . . . . . . . subtract 1

  0.285/6 = r = 0.0475 = 4.75% . . . . . . divide by 6; express as %

Theo paid interest at the rate of 4.75%.

Answer:

Step-by-step explanation:

I don't really know what your question is asking for but I can tell you there's a quadratic formula that solves for x-ints. There's a quadratic equation which is Ax + bx + c = 0.

Answer:

603 hope this helps

Answer:

x = 2

y = 1

z = 4

Step-by-step explanation:

Step 1:

2x + y + z = 9

3x - y = z = 9

5x + 2z = 18 <-- add x and y

Step 2:

-x -y + z = 1

2x + y + z = 9

x + 2z = 10

Step 3:

-(x + 2z = 10) = -x - 2z = -10

5x + 2z = 18

-x - 2z = -10

4x = 8

x = 2

Step 4: solve for z

-x - 2z = -10 <-- Plug x

-(2) - 2z = -10

-2z = -8

z = 4

Step 5: solve for y

2x + y + z = 9

2(2) + y + 4 = 9

8 + y = 9

y = 1

Answer:

51

Step-by-step explanation:

just the denominator with the numerator