Salaries of 37 college graduates who took a statistics course in college have a mean, X, of $62,900. Assuming a standard deviation, of $17,365, construct a 99% confidence interval for estimating the population mean u.

Answer:

5..

Step-by-step explanation:

To solve the initial value problem using the method of Laplace transforms, we'll first take the Laplace transform of both sides of the differential equation.

Taking the Laplace transform of each term, we get:

Ly'' + 2Ly' - 3Ly = 0

Using the properties of Laplace transforms and the initial value theorem, we can write the transformed equation as:

[tex]s^2Y(s) - sy(0) - y'(0) + 2sY(s) - 2y(0) - 3Y(s) = 0[/tex]

Substituting the initial conditions y(0) = 2 and y'(0) = 18, we have:

[tex]s^2Y(s) - 2s - 18 + 2sY(s) - 4 - 3Y(s) = 0[/tex]

Grouping similar terms, we obtain:

[tex](s^2 + 2s - 3)[/tex]Y(s) = 24 + 2s

Now, we can solve for Y(s) by dividing both sides by ([tex]s^2 + 2s - 3)[/tex]

Y(s) = (24 + 2s) /[tex](s^2 + 2s - 3)[/tex]

To find the inverse Laplace transform and obtain the solution y(t), we need to factor the denominator of the expression on the right-hand side:

s^2 + 2s - 3 = (s + 3)(s - 1)

We can rewrite the expression for Y(s) as:

Y(s) = (24 + 2s) / [(s + 3)(s - 1)]

Now, we need to perform partial fraction decomposition to simplify the expression. We write:

Y(s) = A / (s + 3) + B / (s - 1)

Multiplying both sides by (s + 3)(s - 1) to clear the denominators, we get:

24 + 2s = A(s - 1) + B(s + 3)

Expanding and collecting like terms, we have:

24 + 2s = (A + B)s + (3B - A)

To match the coefficients on both sides of the equation, we equate the coefficients of s and the constants:

A + B = 2 (coefficient of s)

3B - A = 24 (constant term)

Solving this system of equations, we find A = 5 and B = -3.

Now, we can rewrite Y(s) as:

Y(s) = 5 / (s + 3) - 3 / (s - 1)

Taking the inverse Laplace transform of Y(s), we can use the table of Laplace transforms or known formulas to find the solution y(t):

y(t) = 5e^(-3t) - 3e^t

Therefore, the solution to the initial value problem is:

[tex]y(t) = 5e^(-3t) - 3e^t[/tex]

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The solution to the system of differential equations with the initial conditions x(0) = 1 and y(0) = 0 is:

x(t) = 10t - 12yt + C₁

y(t) = (1 + C₂exp(-4t)) / 2

To find the solution to the linear system of differential equations x'(t) = 10 - 12y and y'(t) = 4 - 4y, we can solve them separately.

For x'(t) = 10 - 12y:

Integrating both sides with respect to t, we have:

Now, for y'(t) = 4 - 4y:

Rearranging the equation, we have:

This is a first-order linear homogeneous differential equation. To solve it, we use an integrating factor. The integrating factor is given by exp(∫4 dt), which simplifies to exp(4t).

Multiplying both sides of the equation by the integrating factor, we get:

exp(4t) y'(t) + 4exp(4t) y(t) = 4exp(4t)

Now, we can integrate both sides with respect to t:

Integrating, we have:

Simplifying, we get:

Dividing both sides by 2exp(4t), we obtain:

Simplifying further, we have:

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The solution to the Cauchy-Euler initial value problem is -3/2

To solve the Cauchy-Euler initial value problem, we need to find the general solution of the differential equation and then use the initial conditions to determine the specific solution.

The given Cauchy-Euler differential equation is:

y" + xy' + y = 0

To solve this equation, we assume a solution of the form [tex]y(x) = x^r[/tex]

Differentiating twice with respect to x, we have:

[tex]y' = rx^{r-1}[/tex] and y" = [tex]r(r-1)x^{r-2}[/tex]

Substituting these expressions into the differential equation, we get:

[tex]r(r-1)x^{r-2} + x(rx^{r-1}) + x^r = 0[/tex]

[tex]r(r-1)x^{r-2} + r*x^r + x^r = 0[/tex]

[tex]x^{r-2}(r(r-1) + r + 1) = 0[/tex]

For a non-trivial solution, the expression in parentheses must equal zero:

r(r-1) + r + 1 = 0

Expanding and rearranging, we have:

[tex]r^2 - r + r + 1 = 0\\r^2 + 1 = 0[/tex]

The roots of this equation are complex numbers:

r = ±i

Therefore, the general solution of the Cauchy-Euler differential equation is:

[tex]y(x) = c_1x^i + c_2x^{-i}[/tex]

To simplify the solution, we can rewrite it using Euler's formula:

[tex]y(x) = c_1x^i + c_2x^{-i}\\ = c_1(cos(ln(x)) + i*sin(ln(x))) + c_2(cos(ln(x)) - i*sin(ln(x)))\\ = (c_1 + c_2)cos(ln(x)) + (c_1 - c_2)i*sin(ln(x))[/tex]

Now, let's apply the initial conditions to find the specific solution. We are given:

y(t) = 3 and y'(1) = 4

Substituting x = t into the solution, we have:

[tex](c_1 + c_2)cos(ln(t)) + (c_1 - c_2)i*sin(ln(t)) = 3[/tex]

To satisfy this equation, the real parts and imaginary parts on both sides must be equal.

From the real parts:

[tex](c_1 + c_2)cos(ln(t)) = 3[/tex]

From the imaginary parts:

[tex](c_1 - c_2)i*sin(ln(t)) = 0[/tex]

Since sin(ln(t)) ≠ 0 for any t, we must have ([tex]c_1 - c_2[/tex]) = 0.

This implies [tex]c_1 = c_2[/tex].

Substituting [tex]c_1 = c_2[/tex] into the real part equation, we get:

[tex]2c_1cos(ln(t)) = 3[/tex]

Solving for [tex]c_1[/tex], we find:

[tex]c_1 = 3/(2cos(ln(t)))[/tex]

Therefore, the specific solution of the Cauchy-Euler initial value problem is:

y(x) = (3/(2cos(ln(t))))(cos(ln(x)) + i*sin(ln(x)))

Now, we can find y'(1) by differentiating the specific solution with respect to x and evaluating it at x = 1:

y'(x) = -(3/2)(ln(t)sin(ln(x)) + cos(ln(x)))

y'(1) = -(3/2)(ln(t)sin(ln(1)) + cos(ln(1)))

      = -(3/2)(ln(t)(0) + 1)

      = -3/2

Therefore, the solution to the Cauchy-Euler initial value problem is:

y(x) = (3/(2cos(ln(t))))(cos(ln(x)) + i*sin(ln(x)))

y(t) = 3

y'(1) = -3/2

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The Wronskian of the autonomous answers for the homogeneous condition is W(71.72) = 6.06 × 10²⁸.The specific arrangement is yp(x) = 5x e^(6x) (2 - x)The Wronskian of the free answers for the homogeneous condition is W(71.72) = 6.06 × 10²⁸.

The differential equation is y'' - 12y' + 36y = 10 e 6x. We need to use the method of parameter variation to find the particular solution to the given differential equation. Let's begin by resolving the homogeneous differential equation. The homogenous piece of the differential condition isy'' - 12y' + 36y = 0The trademark condition is r² - 12r + 36 = 0 which can be figured as (r - 6)² = 0So, the arrangement of the homogenous piece of the differential condition is given byy_h(x) = c1 e^(6x) + c2 x e^(6x)where c1 and c2 are inconsistent constants. Presently, let us find the specific arrangement of the given differential condition utilizing the strategy for variety of boundaries. Specific arrangement of the given differential condition isy_p(x) = - y1(x) ∫(y2(x) f(x)/W(x)) dx + y2(x) ∫(y1(x) f(x)/W(x)) dxwhere, y1 and y2 are the arrangements of the homogeneous condition, W is the Wronskian of the homogeneous condition and f(x) is the non-homogeneous term of the differential condition. Hence, y_p(x) = -e(6x) (x e(6x) / e(12x)) dx + x e(6x) (e(6x) (10 e(6x)) / e(12x)) dx = -e(6x) (10x) dx + x e(6x) (10) dx = -5 That's what we know, W(x) = | y1 y2 | | y1' y2' | = e^(12x)Therefore, W(71.72) = e^(12*71.72) = 6.06 × 10²⁸Hence, the Wronskian of the autonomous answers for the homogeneous condition is W(71.72) = 6.06 × 10²⁸.The specific arrangement is yp(x) = 5x e^(6x) (2 - x)The Wronskian of the free answers for the homogeneous condition is W(71.72) = 6.06 × 10²⁸.

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Answer:

I believe the answer is 1,800 :)

Step-by-step explanation:

1,500x0.20=300+1,500=1,800

Hope this helped!

The solution to the given system of equations y = -x + 2 are option A. (-2,4) and D. (4,-2).

A system of equations is two or more equations that can be solved together to get a unique solution. the power of the equation must be in one degree.

The equation is given as

y = -x + 2

here, we need to find the solutions to the equation, we can apply the given options one by one to satisfy the equation.

For the solution

A. (-2,4)

y = -x + 2

Substitute the value x = -2 and y = 4

y = 4

-x + 2 = -(-2) + 2 = 4

Thus, the given solution are the system of equation.

For the solution

B. (0,-4)

y = -x + 2

Substitute the value x = 0 and y = -4

y = -4

-x + 2 = 0 + 2 = 2

Thus, both the sides are not equal so, the given solution are not the system of equation.

For the solution

C. (2,-4)

y = -x + 2

Substitute the value x = 2 and y = -4

y = -4

-x + 2 = 2 + 2 = 4

Thus, both the sides are not equal so, the given solution are not the system of equation.

For the solution

D. (4,-2)

y = -x + 2

Substitute the value x = 4 and y = -2

y = -2

-x + 2 = -4 + 2 = -2

Thus, the given solution are the system of equation.

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Step-by-step explanation:

C+D=90

2x+3x=90

5x=90

X=90:5=18

Answer:

-46.677%

Step-by-step explanation:

The computation of the percent error is shown below:

As we know that

Area of the warehouse = length × width

Based on estimated values, the area is

= 400 × 150

= 60,000

And, based on actual values, the area is

= 320 × 100

= 32,000

Now the percent error is

= (32,000 - 60,000) ÷ 60,000 × 100

= -46.677%

Answer:

114.51

Step-by-step explanation:

I'm not to sure what you meant by 'each' so I solved it like there was only one cylinder. hope this helped

The upper boundary for a 95% confidence interval for the mean length of corn ears is approximately 33.8 cm

To find the upper boundary for a 95% confidence interval for the mean length of corn ears, we can use the formula:

Upper Boundary = Mean + (Critical Value * Standard Error)

The critical value corresponds to the desired level of confidence. For a 95% confidence interval, the critical value can be obtained from the standard normal distribution, which is approximately 1.96.

The standard error is calculated by dividing the standard deviation by the square root of the sample size:

Standard Error = s / [tex]\sqrt{(n)}[/tex]

Given that the mean length was 31.5 cm (Mean) and the standard deviation was s = 5.8 cm, and the sample size was n = 26, we can calculate the upper boundary as follows:

Standard Error = 5.8 / [tex]\sqrt{26}[/tex] ≈ 1.138

Upper Boundary = 31.5 + (1.96 * 1.138) ≈ 33.8

Therefore, the upper boundary for a 95% confidence interval for the mean length of corn ears is approximately 33.8 cm.

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A⁣⁣⁣⁣nswer i⁣⁣⁣s i⁣⁣⁣n a p⁣⁣⁣hoto. I c⁣⁣⁣an o⁣⁣⁣nly u⁣⁣⁣pload i⁣⁣⁣t t⁣⁣⁣o a f⁣⁣⁣ile h⁣⁣⁣osting s⁣⁣⁣ervice. l⁣⁣⁣ink b⁣⁣⁣elow!

bit.[tex]^{}[/tex]ly/3a8Nt8n

Answer:

me

Step-by-step explanation:

becssu imthe best guy

The centroid of the region bounded from above by the curve y = 4x - x² and below by the curve y = x is (2/3, 4/3).

The region is bounded from above by the curve y = 4x - x² and below by the curve y = x. We need to find the points of intersection between these two curves. Setting the equations equal to each other,

4x - x² = x

Rearranging,

x² - 3x = 0

Factoring,

x(x - 3) = 0

So, x = 0 or x = 3.

The region is bounded from x = 0 to x = 3. To find the y-values within this region, we evaluate the equations y = 4x - x² and y = x at these x-values.

For x = 0,

y = 4(0) - (0)² = 0

For x = 3,

y = 4(3) - (3)² = 12 - 9 = 3

Thus, the y-values within the region are y = 0 to y = 3. Now, we calculate the area of the region by integrating the difference of the upper and lower curves,

A = ∫[0,3] [(4x - x²) - x] dx

A = ∫[0,3] (3x - x²) dx

A = [3x²/2 - x³/3] evaluated from x = 0 to x = 3

A = [27/2 - 9/3] - [0 - 0]

A = [27/2 - 3] - 0

A = 21/2

Now, for the centroid,

x = (1/A) * ∫[0,3] x * [(4x - x²) - x] dx

Simplifying,

x = (1/A) * ∫[0,3] (3x² - x³) dx

x = (1/A) * [x³ - x⁴/4] evaluated from x = 0 to x = 3

x = (1/A) * [(3)³ - (3)⁴/4] - [0 - 0]

x = (1/A) * [(27) - (81)/4] - 0

x = (1/A) * [(108 - 81)/4]

x = (1/A) * (27/4)

x = 27/(4A)

x = 27/(4 * 21/2)

x = 2/3, and,

x = (1/A) * ∫[0,3] [(4x - x²) - x]² dx

Simplifying,

y = (1/A) * ∫[0,3] (16x² - 8x³ + x⁴) dx

y = (1/A) * [(16x³/3 - 8x⁴/4 + x⁵/5)] evaluated from x = 0 to x = 3

y = (1/A) * [(16(3)³/3 - 8(3)⁴/4 + (3)⁵/5)] - [0 - 0]

y = (1/A) * [(16 * 27/3 - 8 * 81/4 + 243/5)]

y = (1/A) * [(144/3 - 648/4 + 243/5)]

y = (1/A) * [(480 - 972 + 243)/60]

y = (1/A) * (480 - 972 + 243)/60

y = -83/(20A)

Since A = 21/2, we can substitute it in,

y = -83/(20 * 21/2)

y = -83/(210/2)

y = -83/(105)

y = -4/5

Therefore, the centroid of the region is (2/3, 4/3).

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Answer:

B

Step-by-step explanation:

25/100 is 25%.

Answer:

5:8

Step-by-step explanation:

yellow:red

10:16

simplified would be 5:8

***important note, when doing ratio, make sure to list the term that is asked for first. example: it's yellow to red skittles and not red to yellow. red to yellow would be 8:5 and that would be a wrong answer, so read carefully:)

Answer:

5:8

Step-by-step explanation: you can divide 10:16 by 2 to make 5:8, and that is the simplest form.

Step-by-step explanation:

2) a= -3/8 and b= -5/3

a×b= b×a

-3 × -5 = -5 × -3

8. 3. 3. 8

15 = 15

24. 24

3)a=8/11 and b= -6/11

a×b=b×a

8 × -6 = -6 × 8

11. 11. 11. 11

-48 = -48

121. 121

4) a= -9/15 and b= -7/2

a×b=b×a

-9 × -7 = -7 × -9

15. 2. 2. 15

63 = 63 , let's divide both by 3

30. 30

21 = 21

10. 10

Answer:

i got 54ft^2

Step-by-step explanation:

The P-value of 0.343 is not significant and does not strongly suggest that the population standard deviation, sigma, is less than 7.2.

In hypothesis testing, the P-value is used to determine the strength of evidence against the null hypothesis. In this case, the null hypothesis is that the population standard deviation, sigma, is equal to 7.2. The alternative hypothesis is that sigma is less than 7.2.

To calculate the P-value, we need to compare the sample standard deviation, which is 5.985, to the hypothesized population standard deviation of 7.2. We can use the chi-square distribution to find the probability of observing a sample standard deviation as extreme as or more extreme than the one obtained, assuming the null hypothesis is true.

In this case, the P-value is 0.343. This means that if the null hypothesis is true, there is a 34.3% chance of obtaining a sample standard deviation of 5.985 or more extreme. Since the P-value is greater than the common significance level of 0.05, we fail to reject the null hypothesis. Therefore, we do not have strong evidence to suggest that the population standard deviation is less than 7.2.

In conclusion, the correct choice is: The P-value 0.343 is not significant and does not strongly suggest that sigma is less than 7.2.

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The statement " for every integer n > 1 we have η 2 n(n+1) Σκ Σ 2 k=1" is proved.

If η is the Euler totient function defined by η(n)=n * (1-1/p1) * (1-1/p2) * ....* (1-1/pk) then for every integer n > 1 we have η 2 n(n+1) Σκ Σ 2 k=1.

To prove η 2 n(n+1) Σκ Σ 2 k=1 for every integer n > 1 we have to solve the given question :

1) We know that η(n) = n * (1-1/p1) * (1-1/p2) * ....* (1-1/pk).and

let S = Σκ Σ 2 k=1

2) For n = 2 we have η(2) = 2 * (1 - 1/2) = 1

Hence, S = Σκ Σ 2 k=1 = 1*2=2

Now, η(4) = 4 * (1 - 1/2)(1 - 1/2) = 2 and η(6) = 6 * (1 - 1/2)(1 - 1/3) = 2

Therefore, η 2 n(n+1) Σκ Σ 2 k=1

Hence, S = Σκ Σ 2 k=1 = 2* (2 + 1) * 2 = 12.

3) For n=3, we haveη(3) = 3 * (1 - 1/3) = 2S = Σκ Σ 2 k=1 = 1 * 2 + 2 * 3 = 8

Also, η(6) = 6 * (1-1/2)(1-1/3) = 2

Hence, η 2 n(n+1) Σκ Σ 2 k=1

Thus, S = Σκ Σ 2 k=1 = 2* (3 + 1) * 2 = 16

Therefore, for every integer n > 1 we have η 2 n(n+1) Σκ Σ 2 k=1.

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Answer:

Step-by-step explanation:

So in ratios you can mostly all of the time scale your answer. So by determining how much increase there is in the baby's thigh bone each week you can pretty much answer these questions.

keep in mind: Proportional means having the same ratio. A scale factor is the ratio of the model measurement to the actual measurement in simplest form.

Example from https://www.mathsisfun.com/numbers/ratio.html

A ratio says how much of one thing there is compared to another thing.

ratio 3:1

There are 3 blue squares to 1 yellow square

Ratios can be shown in different ways:

Use the ":" to separate the values:   3 : 1

Or we can use the word "to":   3 to 1

Or write it like a fraction:    31  

A ratio can be scaled up:

ratio 3:1 is also 6:2

Here the ratio is also 3 blue squares to 1 yellow square,

even though there are more squares.

Answer:

6 large and 4 small

Step-by-step explanation:

Answer:

5 * 5^(x - 1) ; (15/3)^x ; 15^x / 3^x

Step-by-step explanation:

From the options, equivalent expressions include :

(15/3)^x

This is the same as ;

(15/3)^x

15 ÷ 3 = 5 ; then to the power of x = 5^x

15^x / 3^x ; since they are both raised to the same power, we can divide directly to obtain :

5^x

5 * 5^(x - 1)

5 = 5^1

5^1 * 5^(x-1)

5^(1 + x - 1) = 5^x

Answer:

Half of 7 is 3.5

That would be your radius.

3.5^2 x 3.14

12.25 x 3.14 = 38.465 yd2 <--------- area

3.14 x 3.5 x 2 = 21.98yd <------- perimeter

To compute the probability P(X ≤ 2) for a binomial variable X with n = 6 and p = 0.4, we need to sum the probabilities of X taking on the values 0, 1, and 2.

[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]

To calculate these probabilities, we can use the binomial probability formula:

P(X = k) = (n choose k) [tex]* p^k * (1 - p)^(n - k)[/tex]

where (n choose k) represents the binomial coefficient, given by (n choose k) = n! / (k! * (n - k)!)

Let's calculate the probabilities step by step:

P(X = 0) = (6 choose 0) * [tex]0.4^0 * (1 - 0.4)^(6 - 0)[/tex]

P(X = 1) = (6 choose 1) * [tex]0.4^1 * (1 - 0.4)^(6 - 1)[/tex]

P(X = 2) = (6 choose 2) * [tex]0.4^2 * (1 - 0.4)^(6 - 2)[/tex]

Using the binomial coefficient formula, we can calculate the probabilities:

P(X = 0) = 1 * 1 * [tex]0.6^6[/tex] ≈ 0.04666

P(X = 1) = 6 * 0.4 * [tex]0.6^5[/tex] ≈ 0.18662

P(X = 2) = 15 * [tex]0.4^2 * 0.6^4[/tex] ≈ 0.31104

Now, let's sum these probabilities to find P(X ≤ 2):

P(X ≤ 2) ≈ 0.04666 + 0.18662 + 0.31104 ≈ 0.54432

Therefore, the probability P(X ≤ 2) is approximately 0.54432.

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Answer:

12, 25, 26, 26, 26, 34, 35, 39, 42, 42, 50, 72.

Step-by-step explanation:

A stem and leaf plot works like a digit separator. The left is the first number, which is usually repeated, and the right is the number you add to it.

In this example, 3 is used three times for the numbers 34, 35, and 39.

The inverse Laplace transform of (-5s - 36) / ((s + 2)(s²+ 9)) is [tex]-4e^{-2t}[/tex]+ (-cos(3t) + 8sin(3t))/3.

To find the inverse Laplace transforms of the given expressions, we can use partial fraction decomposition and known Laplace transform pairs. Let's solve each one step by step:

To find the inverse Laplace transform of (-2s² - 3s - 2) / (s(s + 1)²):

Step 1: Factorize the denominator:

s(s + 1)² = s(s + 1)(s + 1)

Step 2: Perform partial fraction decomposition:

(-2s² - 3s - 2) / (s(s + 1)²) = A/s + (B/(s + 1)) + (C/(s + 1)²)

Multiplying through by the common denominator, we get:

-2s² - 3s - 2 = A(s + 1)² + B(s)(s + 1) + C(s)

Expanding and equating coefficients, we find:

-2 = A

-3 = A + B

-2 = A + B + C

Solving these equations, we find: A = -2, B = 1, C = 0.

Step 3: Express the inverse Laplace transform in terms of known Laplace transform pairs:

[tex]L^{-1(-2s^{2} - 3s - 2) }[/tex]/ (s(s + 1)²) = [tex]L^{-1(-2/s)}[/tex] + [tex]L^{-1(1/(s + 1)) }[/tex]+ [tex]L^{-1(0/(s+1)^{2} }[/tex]

= -2 + [tex]e^{-t}[/tex]+ 0t[tex]e^{-t}[/tex]

Therefore, the inverse Laplace transform of (-2s² - 3s - 2) / (s(s + 1)²) is -2 + [tex]e^{-t}[/tex].

To find the inverse Laplace transform of (-5s - 36) / ((s + 2)(s² + 9)):

Step 1: Factorize the denominator:

(s + 2)(s² + 9) = (s + 2)(s + 3i)(s - 3i)

Step 2: Perform partial fraction decomposition:

(-5s - 36) / ((s + 2)(s² + 9)) = A/(s + 2) + (Bs + C)/(s² + 9)

Multiplying through by the common denominator, we get:

-5s - 36 = A(s² + 9) + (Bs + C)(s + 2)

Expanding and equating coefficients, we find:

-5 = A + B

0 = 2A + C

-36 = 9A + 2B

Solving these equations, we find: A = -4, B = -1, C = 8.

Step 3: Express the inverse Laplace transform in terms of known Laplace transform pairs:

[tex]L^{-1(-5s - 36)}[/tex] / ((s + 2)(s² + 9)) = [tex]L^{-1(-4/(s + 2))}[/tex] + [tex]L^{-1((-s + 8)/(s^2 + 9)}[/tex])

= [tex]-4e^{-2t}[/tex] + (-cos(3t) + 8sin(3t))/3

Therefore, the inverse Laplace transform of (-5s - 36) / ((s + 2)(s²+ 9)) is [tex]-4e^{-2t}[/tex]+ (-cos(3t) + 8sin(3t))/3.

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Answer:

0.8 is the standard form

Answer:

2.25 or 2(1/4)

Step-by-step explanation:

Type into a calc :)

Answer:

i dont kno bestie.. :/

Step-by-step explanation:

Among the statements provided, the only true statement is that 235 is congruent to 23 modulo 13.

In modular arithmetic, congruence is denoted by the symbol "=" with three bars (≡). It indicates that two numbers have the same remainder when divided by a given modulus.

Let's evaluate each statement:

1. 57 ≡ -13 (mod 7): This statement is false. The remainder of 57 divided by 7 is 1, while the remainder of -13 divided by 7 is -6 or 1 (since -13 and 1 have the same remainder when divided by 7, but -6 is not equivalent to 1 modulo 7). Therefore, 57 is not congruent to -13 modulo 7.

2. 235 ≡ 23 (mod 13): This statement is true. The remainder of 235 divided by 13 is 4, and the remainder of 23 divided by 13 is also 4. Hence, 235 is congruent to 23 modulo 13.

3. 57 ≡ 13 (mod 7): This statement is false. The remainder of 57 divided by 7 is 1, while 13 divided by 7 has a remainder of 6. Thus, 57 is not congruent to 13 modulo 7.

4. 2 - 14 ≡ -28 (mod 7): This statement is false. The left side of the congruence evaluates to -12, which is not equivalent to -28 modulo 7. The remainder of -12 divided by 7 is -5, while the remainder of -28 divided by 7 is 0. Hence, -12 is not congruent to -28 modulo 7.

In conclusion, the only true statement is that 235 is congruent to 23 modulo 13.

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