Use the volume formula to find the volume of the prism.
help!!

20 points!!

This looks like a system of differential equations.

[tex]\begin{cases}x' + 2y' + x = 0 \\ x' - y' + y = 0 \\x(0)=0, y(0)=1\end{cases}[/tex]

Eliminating x' gives

[tex](x'+2y'+x)-(x'-y'+y) = 0 - 0 \implies 3y' - y + x = 0[/tex]

and eliminating y' gives

[tex](x'+2y'+x) + 2(x'-y'+y)=0+2\times0 \implies 3x' + x + 2y = 0[/tex]

so that we can rewrite the system as

[tex]\begin{cases}3x' = -x - 2y \\ 3y' = -x + y\end{cases}[/tex]

or equivalently in matrix form as

[tex]\begin{bmatrix}x\\y\end{bmatrix}' = \dfrac13 \begin{bmatrix}-1&-2\\-1&1\end{bmatrix} \begin{bmatrix}x\\y\end{bmatrix}[/tex]

Compute the eigenvalues for the coefficient matrix:

[tex]\det\begin{bmatrix}-1-\lambda&-2\\-1&1-\lambda\end{bmatrix} = (-1-\lambda)(1-\lambda) - 2 = \lambda^2 -3 = 0 \implies \lambda=\pm\sqrt3[/tex]

Compute the corresponding eigenvectors:

[tex]\lambda=\sqrt3 \implies \begin{bmatrix}-1-\sqrt3&-2\\-1&1-\sqrt3\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix} \\\\ \implies v_1=(1-\sqrt3)v_2 \implies \vec v = \begin{bmatrix}1-\sqrt3\\1\end{bmatrix}[/tex]

[tex]\lambda=-\sqrt3 \implies \begin{bmatrix}-1+\sqrt3&-2\\-1&1+\sqrt3\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix} \\\\ \implies v_1 = (1+\sqrt3)v_2 \implies \vec v = \begin{bmatrix}1+\sqrt3\\1\end{bmatrix}[/tex]

We end up multiplying the matrix by 1/3, so the eigenvalues also get scaled by 1/3 and λ = ±1/√3. The eigenvectors stay the same.

Then the characteristic solution to the system is

[tex]\begin{bmatrix}x\\y\end{bmatrix} = C_1 e^{t/\sqrt3} \begin{bmatrix}\frac{1-\sqrt3}3\\\frac13\end{bmatrix} + C_2 e^{-t/\sqrt3} \begin{bmatrix}\frac{1+\sqrt3}3\\\frac13\end{bmatrix}[/tex]

Use the initial conditions to solve for the constants.

[tex]\begin{bmatrix}0\\1\end{bmatrix} = C_1 \begin{bmatrix}\frac{1-\sqrt3}3\\\frac13\end{bmatrix} + C_2 \begin{bmatrix}\frac{1+\sqrt3}3\\\frac13\end{bmatrix}[/tex]

[tex]\implies \begin{cases}(1-\sqrt3) C_1 + (1+\sqrt3) C_2 = 0 \\ C_1 + C_2 = 3\end{cases}[/tex]

[tex]\implies C_1=\dfrac{3+\sqrt3}2, C_2=\dfrac{3-\sqrt3}2[/tex]

Then the particular solution is

[tex]\begin{bmatrix}x\\y\end{bmatrix} = \dfrac{3+\sqrt3}6 e^{t/\sqrt3} \begin{bmatrix}1-\sqrt3\\1\end{bmatrix} + \dfrac{3-\sqrt3}6 e^{-t/\sqrt3} \begin{bmatrix}1+\sqrt3\\1\end{bmatrix}[/tex]

or

[tex]\begin{cases}x(t) = -\dfrac1{\sqrt3} e^{\sqrt3 \, t} + \dfrac1{\sqrt3} e^{-\sqrt3\,t} \\\\ y(t) = \dfrac{3+\sqrt3}6 e^{\sqrt3\,t} + \dfrac{3-\sqrt3}6 e^{-\sqrt3\,t}\end{cases}[/tex]

3.375

hope it helps...!!

Answer: i think the answer is 116. if it is not i am terribly sorry.

Step-by-step explanation:

Answer:

14.58

Step-by-step explanation:

x\27 = 54/100 cross multiply then divide by 100% x=Original price

so then you'll get 14.58

Answer:

72.5% out of 100%

Step-by-step explanation:

0.725 is 72.5% of 1. 100% technically counts as 1. so 72.5%

Answer:

vertex = (- 5, - 13 )

Step-by-step explanation:

given a quadratic in standard form : y = ax² + bx + c ( a ≠ 0 )

then the x- coordinate of the turning point ( vertex) is

x = - [tex]\frac{b}{2a}[/tex]

y = x² + 10x + 12 ← is in standard form

with a = 1, b = 10 , then

x = - [tex]\frac{10}{2}[/tex] = - 5

substitute x = - 5 into the equation and evaluate for y

y = (- 5)² + 10(- 5) + 12 = 25 - 50 + 12 = - 13

turning point = (- 5, - 13 )

Answer:

Now plot those points in graph and find intersection hope it helps you.......

Answer:

Side length ( perpendicular = 60 cm)

Hypotenuse = 61 cm

Base length

By Pythagoras theorem ,

Hypotenuse² = Perpendicular² + Base²

61² = 60² + base²

3721-3600 = Base²

√121 = base

11cm = base

Answer:

Step-by-step explanation:

In the question we are given ,

And we have asked to find the length of base that is denoted by x .

Solution : -

According to Pythagoras Theorem ,

[tex] \longrightarrow \purple{\boxed{H {}^{2} = P {}^{2} + B {}^{2} }} \longleftarrow[/tex]

Where ,

Now , substituting value of hypotenuse, perpendicular and base :

[tex] \hookrightarrow \: 61 {}^{2} = 60 {}^{2} + x {}^{2} [/tex]

Now by squaring 61 and 60 , we get :

[tex] \hookrightarrow \:3721 = 3600 + x {}^{2} [/tex]

Now transposing 3600 to left hand side :

[tex] \hookrightarrow \:3721 - 3600 = x {}^{2} [/tex]

Now subtracting 3600 from 3721 :

[tex] \hookrightarrow \:121 = x {}^{2} [/tex]

[tex] \hookrightarrow \: \sqrt{121} = x[/tex]

We know that 11 × 11 is equal to 121 that means square root of 121 is 11 . So :

[tex] \hookrightarrow \: \red{ \boxed{11 \: cm = x}}[/tex]

Answer:

2/3 × 2/5 = 4/15

Step-by-step explanation:

see image. If you just look at the horizontal lines the rectangle is divided into thirds. Two of the sections are highlighted indicating 2 out of 3, or 2/3.

If you then consider the vertical lines. There are 5 sections and 2 out of the 5 are highlighted indicating 2/5.

When you overlap both images (which is what the original image you posted was) you get a small section that is doubly highlighted. In your original image it is kind of purple (I marked them in yellow) In the double-layered image, there are 15 small sections. 4 are highlighted (double-highlighted) This is 4/15.

2/3 × 2/5 = 4/15

I supposed it could also be

2/5 × 2/3 = 4/15

Numerically, these are the same. They have the same result. I hope this helps!

Answer:D

Step-by-step explanation: Multiply the expression

The rational function shown in the graph is:

[tex]y = \frac{x - 6}{x - 5}[/tex]

First, we can see that we have an asymptote at x = 5, so the denominator becomes equal to zero when x = 5.

Then the rational function is something like:

[tex]y = \frac{g(x)}{x - 5}[/tex]

We also can see that when x = 6, the function intercepts the x-axis, then we must have that:

g(6) = 0

So we can define:

g(x) = x - 6

So the rational function is:

[tex]y = \frac{x - 6}{x - 5}[/tex]

If you want to learn more about rational functions, you can read:

https://brainly.com/question/1851758

Answer:

400

Step-by-step explanation:

Division cancels multiplication. Divide by 0.17 to isolate "x" on the right side of the equation.

68 = 0.17 * x

divide both sides by 0.17

400 = x

68 is 17 percentage of the number 400.

To find the number that 68 is 17% of, we can set up the equation:

17% of x = 68

To solve for x, we need to divide both sides of the equation by 17% (or 0.17 as a decimal):

x = 68 / 0.17

Evaluating the division:

x=400

Therefore, 68 is 17% of 400.

To learn more on Percentage click:

https://brainly.com/question/24159063

#SPJ6

1. m∠5 = 75° (corresponding angles theorem)

2. Sum of angles in a triangle = 180°

3. m∠3 = 55°

4. Straight line angle = 180°

5. m∠2 = 50°

If two corresponding angles are formed when a transversal intersect parallel lines, they are both congruent based on the corresponding angles theorem.

1. m∠5 = m∠1 (corresponding angles theorem)

m∠5 = 75°

2. Sum of angles in a triangle = 180°

3. m∠3 + m∠4 + m∠5 = 180° (sum of triangle)

Substitute

m∠3 = 180 - 50 - 75 = 55°

4. Straight line angle = 180°

5. m∠2 = 180 - m∠3 + m∠1 (straight line angle)

m∠2 = 180 - 55 - 75

m∠2 = 50°

Learn more about corresponding angles theorem on:

https://brainly.com/question/5221905

Answer: I believe B and F

Step-by-step explanation:

To find the missing angle's measure

 ⇒ important to understand the sum of the angles of a triangle

    ⇒ the sum of every angle of any triangle in all cases equal to 180

         degrees

What is the information given:

 ⇒ one angle is of 94 degrees and the other of 46 degrees

Using the information in the first section and the information given in the diagram, let's set up an equation:

  [tex]94+46+x = 180\\140+x = 180\\x = 40[/tex]

The measure of the missing angle is 40 degrees

Hope that helps!

Answer: Heat pollution.

If you are to search up

heat pollution occurs because?

the answer that will pop up is:

The main reasons of thermal pollution are industries and power plants, withdrawing water (a powerful sink for heat waste) from streams for cooling their generators and machines, and then releasing back into the water bodies, which elevates the temperature of freshwater habitats (Miara et al., 2018).

Answer:

2*5*5

Step-by-step explanation:

50 ---> 25 * 2

25 = 5*5

50 = 2*5*5

Hope this helps :)

Have a nice day!

Answer:

the answer for this one it should be the probability or chances that the event b occurs when the event a has occured.

Step-by-step explanation: I hope this help you pal or folk and have a great day.

Answer:

Edna had $510 more amount of money than Toby in the end.

Step-by-step explanation:

LET :-

Edna's amount of money - x

Fatimah's amount of money - y

Toby's amount of money - z

Now,

By the question,

x = 4y

y : (y + z) = 5 : 8

y - 30 = z + 30

Now,

y : (y + z) = 5 :8

or, y/(y + z) = 5/8

or, 8y = 5y + 5z

or, 3y = 5z

so, y = 5z/3

Now,

y - 30 = z + 30

or, 5z/3 - 30 = z + 30

or, (5z - 90)/3 = z + 30

or, 5z - 90 = 3z + 90

or, 5z - 3z = 90 + 90

or, 2z = 180

so, z = 90

Toby's amount of money - $90

Now,

y = 5z/3

or, y = 5 * 90/3

or, y = 5 * 30

so, y = 150

Fatimah's amount of money - $150

Again,

x = 4y

or, x = 4*150

or, x = 600

Edna's amount of money - $600

Now,

$600 - $90

= $510

Therefore,Edna had $510 more amount of money than Toby in the end.

Answer:

sin (α )= 72 5sin (α) = 725

​

Step-by-step explanation:

Answer:

the answer your looking for would be 14, 48, 50.

Step-by-step explanation:

Hope this helps!

Answer:

14, 48, 50 have a good day

Step-by-step explanation:

Answer:

55°

Step-by-step explanation:

I hope that helps

Using proportions, it is found that the total purchase price of the bracelet is of $10,769.

A proportion is a fraction of a total amount.

In this problem, we consider the two taxes, of 6% and 11%, hence the price suffers an increase of 17%, that is, the price is multiplied by 117% = 1.17 to reach $12,600, hence:

1.17p = 12600

p = 12600/1.17

p = 10769.

The total purchase price of the bracelet is of $10,769.

More can be learned about proportions at https://brainly.com/question/24372153

Answer:

-7

Step-by-step explanation:

^

Answer:

-7

Step-by-step explanation:

Answer:

42

Step-by-step explanation:

40^2+13^2=x^2

1600+169=x^2

1769=x^2

1769−−−−√=x2−−√

±42≈x

Answer:

The answer is 3x.

Step-by-step explanation:

I used a calculator