Solve the following DE using Power series around x₁ = 0. Find the first eight nonzero terms of this DE. y" + xy' + 2y = 0.

Answer:

[tex] m\angle R = 50 \degree[/tex]

Step-by-step explanation:

By inscribed angle theorem:

[tex]m\angle R = \frac{1}{2} [360 \degree - (120 \degree + 140 \degree)] \\ \\ m\angle R = \frac{1}{2} [360 \degree -260 \degree] \\ \\ m\angle R = \frac{1}{2} \times 100 \degree \\ \\ m\angle R = 50 \degree \\ \\ [/tex]

1. The costs the firm spends on financing should be included in its capital budget cash flow analysis. This includes expenses related to obtaining funds for the project, such as interest payments on loans or fees for issuing stocks.

2. The source of a firm's financial leverage is the use of debt and preferred stock. By utilizing debt and preferred stock, a company can increase its financial leverage, which refers to the use of borrowed funds to finance its operations or investments.

3. Operating risk derives from the risk that comes from a firm's mix of fixed and variable costs. This refers to the uncertainty and potential negative impact on profitability due to the combination of fixed costs (such as rent, salaries) and variable costs (such as raw materials, utilities) in a company's cost structure.

4. The firm's liabilities exceeding its assets is not a legal constraint on the payment of dividends. This constraint is related to solvency and insolvency issues and not directly linked to the payment of dividends.

5. According to the information effect, investors view changes in a firm's dividend policy as a signal about the firm's financial condition.

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The simplified expression of integral 3 (x-2 ] 3 +4 dx is (A/3) + 12tanh[tex](sinh^{(-1)}[/tex](3(x-2))) + B

To find the integral ∫3(x-2)³+4 dx using the substitution sinh(y) = 3(x-2), we can start by differentiating both sides of the equation with respect to x to find the differential of y:

d(sinh(y))/dx = d(3(x-2))/dx

cosh(y) * dy/dx = 3

dy/dx = 3/cosh(y)

Now, let's solve for dx in terms of dy:

dx = (cosh(y)/3) dy

Substituting this value of dx in the integral:

∫3(x-2)³+4 dx = ∫(3/cosh(y)) * (3(x-2)³+4) dy

Now, we need to substitute the expression for x in terms of y using the given substitution:

3(x-2) = sinh(y)

x - 2 = sinh(y)/3

x = sinh(y)/3 + 2

Substituting this in the integral:

∫(3/cosh(y)) * (3((sinh(y)/3 + 2) - 2)³+4) dy

Simplifying:

∫(3/cosh(y)) * (sinh(y)³+4) dy

To integrate the expression ∫(3/cosh(y)) * (sinh(y)³+4) dy, we can simplify it first:

∫(3/cosh(y)) * (sinh(y)³+4) dy = 3∫(sinh(y)³/cosh(y)) dy + 12∫(1/cosh(y)) dy

To integrate the first term, we can use the substitution u = cosh(y), which implies du = sinh(y) dy:

3∫(sinh(y)³/cosh(y)) dy = 3∫(u³/u) du = 3∫(u²) du = u³/3 + C

For the second term, we can directly integrate 1/cosh(y) using the identity sech²(y) = 1/cosh²(y):

12∫(1/cosh(y)) dy = 12∫sech²(y) dy = 12tanh(y) + D

Now, substituting back y = [tex]sinh^{(-1)}(3(x-2))[/tex]:

u = cosh(y) = cosh[tex](sinh^{(-1)}(3(x-2))[/tex]) = √(3(x-2)² + 1)

Thus, the integral becomes:

∫(3/cosh(y)) * (sinh(y)³+4) dy = (u³/3 + C) + 12tanh(y) + D

Substituting back u = √(3(x-2)² + 1):

= (√(3(x-2)² + 1)³/3 + C) + 12tanh(y) + D

= (√(3(x-2)² + 1)³ + 3C)/3 + 12tanh(y) + D

= (√(3(x-2)² + 1)³ + 3C)/3 + 12tanh[tex](sinh^{(-1)}(3(x-2)))[/tex] + D

To simplify the expression and combine constants, let's assume (√(3(x-2)² + 1)³ + 3C)/3 = A, and 12D = B.

The simplified expression becomes:

(A/3) + 12tanh[tex](sinh^{(-1)}[/tex](3(x-2))) + B

Since [tex]sinh^{(-1)}(3(x-2))[/tex] is the inverse hyperbolic sine function, we can simplify it using the identity sinh[tex](sinh^{(-1)}(x))[/tex] = x:

(A/3) + 12tanh(3(x-2)) + B

This is the simplified form of the integral ∫(3/cosh(y)) * (sinh(y)³+4) dy after combining constants and simplifying the expression.

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To diagonalize a matrix A, we need to find an invertible matrix P and a diagonal matrix D such that PAP^(-1) = D. Here's how to find the diagonalization of matrix A

1. Find the eigenvalues of A:

  - Calculate the characteristic polynomial by subtracting λI from A, where λ is a scalar variable and I is the identity matrix of the same size as A.

  - Set the characteristic polynomial equal to zero and solve for λ to find the eigenvalues.

2. Find the eigenvectors corresponding to each eigenvalue:

  - For each eigenvalue, substitute it back into the equation (A - λI)x = 0, where x is a vector, and solve for x.

  - Repeat this step for each eigenvalue to obtain a set of linearly independent eigenvectors.

3. Construct the matrix P:

  - Arrange the eigenvectors found in Step 2 as columns to form the matrix P.

4. Construct the diagonal matrix D:

  - Place the eigenvalues obtained in Step 1 on the diagonal of a matrix of the same size as A, with zeros elsewhere.

5. Verify the diagonalization:

  - Calculate PAP^(-1) and check if it equals D. If PAP^(-1) = D, then A is diagonalizable.

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Answer:

A)94.5 cm

Step-by-step explanation:

height = 9cm

Base 1 = 5cm

Area of a Trapezoid = 1/2 × (b1 + b2)h

h = 9cm

b1 = 5cm

b2 = (9 cm + 5cm + 2cm) = 16cm.

Area of Trapezoid

= 1/2 (5 + 16) × 9

= 1/2 × 21 × 9

= 94.5 cm

Option A is the correct answer

A. To save for her payout annuity with an ordinary annuity set up thirty years before her retirement, Holly Krech must make a monthly payment of $175.97.

B. If she sets up the ordinary annuity twenty years before her retirement, Holly Krech must make a monthly payment of $432.00.

To calculate the monthly payment for an ordinary annuity set up thirty years before retirement, we can use the formula for the present value of an ordinary annuity. Given the deposit amount of $218,437.048 and the total amount received from the annuity of $432,000, and solving for the monthly payment, we find that Holly must make a monthly payment of $175.97.

For an ordinary annuity set up twenty years before retirement, we use the same formula for present value. With the deposit amount and total amount received unchanged, we solve for the monthly payment, which comes out to be $432.00.

It's important to note that the monthly payment increases when the annuity is set up closer to the retirement date. This is due to the shorter time period available for saving, resulting in a higher required contribution to reach the desired payout amount.

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Total votes were casted in election are 4716

Given: The ratio of votes in favor to votes against in an election is 5 to 4. 2,620 votes are in favor.

To find: The total number of votes cast.

Let the number of votes against is 4x.

Given the ratio of votes in favor to votes against is 5 : 4

Then, the number of votes in favor is 5x.

According to the question, 2,620 votes are in favor.

So, 5x = 2,620x = 2,620/5x = 524

The number of votes against = 4x = 4 × 524 = 2096

The total number of votes cast = votes in favor + votes against= 2620 + 2096= 4716

Therefore, there were 4716 votes cast in the

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Answer: 60 mph

Step-by-step explanation:

Given

The red car left at 9 am and arrives at 1 pm

time taken by the red car [tex]t_a=4\ hr[/tex]

time taken by the blue car [tex]t_b=3\ hr[/tex]

Assume the speed of the red car is v

So, the speed of the blue car is v+15

distance traveled by them is the same

[tex]\Rightarrow v\times 4=(v+15)\times 3\\\Rightarrow 4v=3v+45\\\Rightarrow v=45\ mph[/tex]

Thus, the speed of the blue car is [tex]45+15=60\ mph[/tex]

Answer:

Well since the question ask for what in the green box, its 193

193 goes in the green box

If you solve everything it would be 13.89 (rounded to the nearest hundredths)

Step-by-step explanation:

So missing side of a right triangle, you can use the Pythagorean Theroum which is a^2+b^2=c^2

In this case we have the two legs which are a and b, we’re trying to find hypotnuse, “c”.

7^2+12^2=c^2

49+144=c^2

193= c^2

You basically do √193

which is the answer needed for this situation

Final answer: is 193 goes into the green box

Answer: $0.3

Step-by-step explanation:

Given

The owner buys 144 pens for $28.80 i.e. each pen costs

[tex]\dfrac{28.80}{144}=\$0.2[/tex]

owner sells the pen at $0.4 i.e. price marked up by

[tex]\Rightarrow \dfrac{0.4-0.2}{0.2}\times 100=100\%[/tex]

So, the claim of the owner is incorrect

The actual increase in the price to get 50% markup

[tex]\Rightarrow 0.2\times (1+0.5)=\$0.3[/tex]

Answer:

The answer is A.

Step-by-step explanation:

Answer:

Step-by-step explanation:

A is the correct answer on Edge

Hence, the required equation is `(24) Xlt)=12 Fo/m Sin (wo-w)t sin (wotw)t 7 Wo² - w² 2 2`.

Given damped forced oscillation equation is,`m d²x/dt² + c dx/dt + kx = Fo sin(wt)`Using trigonometric identities, we can write solution for the given damped forced oscillation equation as,X(t) = Acos(wt + Φ) + Xpwhere Xp = (Fo/k) sin(wt - δ)Let's substitute X(t) in the given equation to get the required equation.```
X(t) = Acos(wt + Φ) + Xp
=> dX(t)/dt = -Awsin(wt + Φ) + (Fo/k)wcos(wt - δ)
=> d²X(t)/dt² = -Aw²cos(wt + Φ) - (Fo/k)w²sin(wt - δ)

```Now, substitute these values in the given damped forced oscillation equation.`md²X(t)/dt² + cdX(t)/dt + kX(t) = Fo sin(wt)`⇒ `m(-Aw²cos(wt + Φ) - (Fo/k)w²sin(wt - δ)) + c(-Awsin(wt + Φ) + (Fo/k)wcos(wt - δ)) + k(Acos(wt + Φ) + (Fo/k)sin(wt - δ)) = Fo sin(wt)`Grouping the terms of sines and cosines, we get⇒ `{-Aw²mcos(wt + Φ) + Awcsin(wt + Φ) + (Fo/k)w²sin(δ) + kAcos(wt + Φ) + (Fo/k)wcos(δ)} = Fo sin(wt) - c(Fo/k)wcos(wt - δ)`Let's solve these equations for `δ` and `A`.```
-Aw²mcos(wt + Φ) + Awcsin(wt + Φ) + kAcos(wt + Φ) = 0      .....................(1)
(Fo/k)w²sin(δ) + (Fo/k)wcos(δ) = Fo sin(wt) - c(Fo/k)wcos(wt - δ)  .....(2)
```Squaring and adding both equations, we get,`(Aw)²m + kA² = (Fo/k)²`or `A = Fo/(k² - mω²)^(1/2)`From equation (1), we have,`(Aw)²m + kA² = 0`or `δ = tan⁻¹(Aw/k)`Substitute values of A and δ in equation (2), we get,`Xp = (Fo/k) sin(wt - δ) = Fo/(k² - mω²)^(1/2) sin(wt - tan⁻¹(Aw/k))`Therefore, solution for the given damped forced oscillation equation is,`X(t) = Acos(wt + Φ) + Xp`= `12 Fo/m Sin (wo-w)t sin (wotw)t / (wo² - w²)²`

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The probability that the 64 randomly selected TV sets will have a mean replacement time of 6 years or less is approximately 0.0048, or 0.48%.

To calculate this probability, we need to standardize the sample mean using the z-score formula and then find the corresponding probability from the standard normal distribution.

The formula for the z-score is:

z = (x - μ) / (σ / √n)

where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

In this case, x = 6, μ = 7.5, σ = 5, and n = 64. Substituting these values into the formula, we get:

z = (6 - 7.5) / (5 / √64)

Simplifying the expression:

z = -1.5 / (5 / 8)

z = -1.5 * 8 / 5

z = -2.4

From Table A (standard normal distribution table), the area to the left of z = -2.4 is approximately 0.0082.

However, since we are interested in the probability of obtaining a mean replacement time of 6 years or less, we need to find the area to the right of z = -2.4. This is given by:

1 - 0.0082 = 0.9918

Therefore, the probability that the 64 randomly selected TV sets will have a mean replacement time of 6 years or less is approximately 0.0048, or 0.48%.

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Absolute maximum of f (x, y) = 19 and Absolute minimum of f (x, y) = −3.

To find the absolute maximum and minimum of f (x, y) = x² + 2y² − 2x − 4y + 1 on D = {(x, y) 0 ≤ x ≤ 2, 0 ≤ y ≤ 3}, we need to follow these steps:Step 1: We need to find the critical points of f (x, y) in the interior of D. Step 2: We then need to evaluate f (x, y) at the critical points. Step 3: We need to find the maximum and minimum of f (x, y) on the boundary of D. Step 4: Compare the values obtained in steps 2 and 3 to get the absolute maximum and minimum values of f (x, y) on D.1. To find the critical points of f (x, y) in the interior of D, we need to find the partial derivatives of f (x, y) with respect to x and y respectively, and solve the resulting system of equations for x and y:fx = 2x − 2fy = 4y − 4Solving for x and y, we obtain (1, 1) as the only critical point in the interior of D.2. To evaluate f (x, y) at the critical point (1, 1), we substitute x = 1 and y = 1 into f (x, y) to get:f (1, 1) = (1)² + 2(1)² − 2(1) − 4(1) + 1 = −3.3. To find the maximum and minimum of f (x, y) on the boundary of D, we use the method of Lagrange multipliers. We set up the equations:g(x, y) = x² + 2y² − 2x − 4y + 1 = k1h1(x, y) = x − 0 = 0h2(x, y) = 2 − x = 0h3(x, y) = y − 0 = 0h4(x, y) = 3 − y = 0Solving for x and y, we obtain the critical points on the boundary of D: (0, 0), (0, 3), (2, 0), and (2, 3).4. Comparing the values obtained in steps 2 and 3, we have the following:f (1, 1) = −3f (0, 0) = 1f (0, 3) = 19f (2, 0) = −3f (2, 3) = 13The absolute maximum of f (x, y) on D is 19 at (0, 3), while the absolute minimum is −3 at (2, 0). Therefore, we have:Absolute maximum of f (x, y) = 19 and Absolute minimum of f (x, y) = −3.

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The center of the sphere is (-2, 1, 0).

The radius of the sphere is √65.

The given equation is x² + y² + 2 + 4x - 2y - 62 = 0.

We can rewrite the given equation as follows:

x² + 4x + y² - 2y = 60

Completing the square of x and y, we get:

(x + 2)² - 4 + (y - 1)² - 1 = 60

(x + 2)² + (y - 1)² = 65

Now, we know that the general equation of the sphere is :

(x - a)² + (y - b)² + (z - c)² = r² where (a, b, c) is the center of the sphere, and r is the radius.

To compare the given equation with the equation of the sphere, we will have to convert it into the standard form as follows:

(x + 2)² + (y - 1)² + (0 - 0)² = √65²

The center of the sphere is (-2, 1, 0), and its radius is √65.

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Answer:

6.4 because subtract 9.6 from 15

Step-by-step explanation:

so let's say that the width is x then the length is x+15

and the area of a square is length times width

x(x+15)=100

x^2+15x-100=0

(x+20)(x-5)=0

x=5 or x=-20 but a side length can't be negative so x would equal 5

Length=x+15 with 5 as x

Length=20

Width=5

Hope that helps :)

It takes approximately 14 minutes for the thermometer to reach a temperature of 58°F in the cold storage room. This calculation is based on Newton's Law of Cooling and the initial and final temperature readings.

To determine how long it takes for the thermometer to reach 58°F, we can use Newton's Law of Cooling. Let's plug in the given values into the equation and solve for the time (t):

88 = 37 + (96 - 37)e^(k * 5)

Simplifying the equation, we have:

51 = 59e^(5k)

Taking the natural logarithm of both sides:

ln(51/59) = 5k

Solving for k, we find:

k ≈ -0.0436

Now, we can use this value of k to find the time (t) when the thermometer reaches 58°F:

58 = 37 + (96 - 37)e^(-0.0436 * t)

Simplifying further, we have:

21 = 59e^(-0.0436 * t)

Taking the natural logarithm again:

ln(21/59) = -0.0436 * t

Solving for t, we find:

t ≈ 13.58

Rounding to the nearest whole minute, it takes approximately 14 minutes for the thermometer to reach 58°F.

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The final temperature of the cylinder is -148.68 °C .

To find the final temperature

Let the final temperature be T₂.

Let the final volume be V₂.

The mass of water inside the cylinder at initial conditions, m₁ = ρV₁

On opening the valve, the water enters the cylinder until the mass doubles. So the mass of water inside the cylinder after the valve is opened, m₂ = 2ρV₁

The pressure and mass are related by the equation, PV = mRT

On simplifying the equation we get,

P = (m/ρ) * RTSo Pρ = mRT ………… (1)

From equation (1),

P₁ρ₁ = m₁R T₁

Substituting the values in equation (1) for final conditions,

P₂ρ = m₂R T₂

We need to find T₂

So, T₂ = (P₂ρ/m₂) * R = (300000 N/m² * 1000 kg/m³)/[2 * 1000 kg] * 8.314 J/(mol K)

= 124.47 K or -148.68 °C

So, -148.68 °C is the final temperature approximately.

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To find the value of g(1), we need to integrate g'(x) and use the given initial condition g(4)=0.325. By integrating g'(x), we can determine the function g(x) and evaluate it at x=1 to find the desired value.

To find g(x), we integrate g'(x) with respect to x. The integral of 1/x * e^(-x) * cos(x^100) requires advanced techniques and cannot be expressed in elementary functions. Therefore, we rely on numerical methods or approximation techniques to evaluate the integral. Once we obtain the antiderivative of g'(x), denoted as G(x), we can use the initial condition g(4)=0.325 to determine the constant of integration.

Once we have the expression for g(x), we substitute x=1 to find g(1), which will provide the desired value.

Note that the process of evaluating the integral and determining g(x) can be computationally intensive and may require numerical approximation methods or specialized software tools.

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Answer:

Thanks for letting me know I might try that later today

Step-by-step explanation:

:)

Answer:

y=x+15

Step-by-step explanation:

Answer:

62

Step-by-step explanation:

find the area of each side

(3 * 5) + (2 * 3) + (2 * 5) + (2 * 5) + (2 * 3) + (3 * 5)

add them all

15 + 6 + 10 + 10 + 6 + 15 = 62

Answer: what do you need help with??

Step-by-step explanation:

Answer:

Oh Lol i didnt even see the pic

Step-by-step explanation:

Answer:

5*2=10

10*8=80

Step-by-step explanation:

Multiply all number!!!

The expression for dL/dA, which represents the rate of change of ladybugs (L) with respect to aphids (A), is 0.5L + 0.0001AL - 2A + 0.01AL.

To find the expression for dL/dA, we need to differentiate the equation dL/dt with respect to A. The given equations are:

dA/dt = 2A - 0.01AL

dL/dt = -0.5L + 0.0001AL

To find dL/dA, we differentiate dL/dt with respect to A:

dL/dA = (dL/dt) / (dA/dt)

Substituting the given equations into this expression, we have:

dL/dA = (-0.5L + 0.0001AL) / (2A - 0.01AL)

Simplifying further, we can rearrange the terms:

dL/dA = -0.5L / (2A - 0.01AL) + 0.0001AL / (2A - 0.01AL)

Combining the terms with a common denominator, we get:

dL/dA = (0.0001AL - 0.5L) / (2A - 0.01AL)

So, the expression for dL/dA is (0.0001AL - 0.5L) / (2A - 0.01AL), which represents the rate of change of ladybugs with respect to aphids in the given model.

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Trapezoid

---

hope it helps

sorry if i'm wrong

i used mental maths

Dude That's a square not a trapezoid

Answer:

taco bell

Step-by-step explanation:

per one taco at taco bell $0.53

per one taco at los comales $0.62

Japan's superconducting bullet that is being tested in Yamanashi Maglev Test Line will have an elevation of 2580 meters and actual distance of the track as 6.9 kilometers.

A. To calculate the elevation change in the steepest section of the track:

Grade = 40% (Given)

Horizontal distance = 6450 meters (Given)

Elevation change = Grade × Horizontal distance

= 40% × 6,450 meters

= 0.40 × 6,450 meters

= 2,580 meters

Therefore, the elevation change in this section of the track will be 2,580 meters.

B. To find the actual distance of the track in this section:

By using Pythagorean theorem, the horizontal distance represents the base of a right triangle, and the elevation change represents the height.

Actual distance of the track = √(Horizontal distance² + Elevation change²)

= √(6,450²  + 2,580² )

= √(41,602,500 + 6,656,400)

= √48,258,900

= 6,945 meters

= 6.9 kilometers

Therefore, the actual distance of the track in this section will be 6.9 kilometers.

C.  To determine which plane is closer to the tower:

Plane A: Altitude = 20,000 ft, Distance from tower = 5 km

Plane B: Altitude = 8,000 ft, Distance from tower = 7 km

1 ft is approximately equal to 0.0003048 km.

Altitude of Plane A in km = 20,000 ft × 0.0003048 km/ft ≈ 6.096 km

Altitude of Plane B in km = 8,000 ft × 0.0003048 km/ft ≈ 2.4384 km

On comparing the distances, we find that Plane A is closer to the tower than Plane B.

Therefore, Plane A is closer to the tower as compare to Plane B.

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Pictures listed in order... from A-C