Questions (a,b), 1 (a,b)

Given the differential equation is y''+5y = e^(at) with initial conditions y(0) = 0 and y'(0) = 0,  the correct answer is: y(t) = (-1/5√(5)) cos (√(5)t)+(1/5) sin (√(5)t).

To solve the given initial value problem using Laplace transform, we need to apply Laplace transform on both sides of the differential equation.

y''+5y=e^(at) L{y''+5y} = L{e^(at)) s^2Y(s)-sy(0)-y'(0)+5Y(s)=1/(s-a) [by Laplace transform formula] s^2Y(s)+5Y(s)=1/(s-a) ... [i]

Applying Laplace transform on both sides, we get:

L{y''+5y}=L{e^(at))

Using the initial conditions, we get Y(s)=1/[(s-a)(s^2+5)] Y(s) = [A/(s-a)] + [(Bs+C)sin(t)+ (Ds+E)cos(t)]/√(5) (i)

To find the values of A, B, C, D, and E, we take the inverse Laplace transform of both sides of equation (i) using partial fraction expansion. Let's solve for A:

Y(s)=A/(s-a)+(Bs+C)sin(t)/√(5)+(Ds+E)cos(t)/√(5)

Multiplying by s-a on both sides: (s-a)Y(s)=A+Bssin(t)/√(5)+Csincos(t)/√(5)+Dscos(t)/√(5)+Esin(t)/√(5)

Taking the inverse Laplace transform: y(t)=Ae^(at)+(B/√(5))sin(√(5)t)+(C/√(5))cos(√(5)t)+(D/√(5))cos(√(5)t)+(E/√(5))sin(√(5)t)

Differentiating y(t) with respect to t, we get:

y'(t)=Aae^(at)+Bcos(√(5)t)-Csin(√(5)t)-Dsin(√(5)t)+Ecos(√(5)t)

Using the initial conditions, y(0)=0 and y'(0)=0 in equation (iii), we get:

0=A+E ...(iv)0=A+B/√(5)+D/√(5) ...(v)

Solving equations (iv) and (v) simultaneously, we get A=0, B=√(5)/5, D=-√(5)/5, and E=0

Substituting these values in equation (iii), we get: y(t)=(√(5)/5)sin(√(5)t)-(√(5)/5)cos(√(5)t)

Therefore, the correct answer is: y(t)= (-1/5√(5))cos(√(5)t)+(1/5)sin(√(5)t).

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The approximation of the integral I is   -5.1941 using the composite Trapezoidal rule with n = 4.

We need to divide the interval [0, 2] into subintervals and apply the Trapezoidal rule to each subinterval.

The formula for the composite Trapezoidal rule is given by:

I = (h/2) × [f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(xₙ₋₁) + f(xₙ)]

Where:

h = (b - a) / n is the subinterval width

f(xi) is the value of the function at each subinterval point

In this case, n = 4, a = 0, and b = 2. So, h = (2 - 0) / 4 = 0.5.

Now, let's calculate the approximation:

[tex]f\left(x_0\right)\:=\:f\left(0\right)\:=\:\left(0\:-\:3\right)e^{\left(0^2\right)}\:=\:-3[/tex]

[tex]f\left(x_1\right)\:=\:f\left(0.5\right)\:=\:\left(0.5\:-\:3\right)e^{\left(0.5^2\right)}\:=-2.535[/tex]

[tex]f\left(x_2\right)\:=\:f\left(1\right)\:=\:\left(1\:-\:3\right)e^{\left(1^2\right)}\:=\:-1.716[/tex]

[tex]f\left(x_3\right)\:=\:f\left(1.5\right)\:=\:\left(1.5\:-\:3\right)e^{\left(1.5^2\right)}\:=\:-1.051[/tex]

[tex]f\left(x_4\right)\:=\:f\left(2\right)\:=\:\left(2\:-\:3\right)e^{\left(2^2\right)}\:=\:-0.065[/tex]

Now we can plug these values into the composite Trapezoidal rule formula:

I = (0.5/2) × [-3 + 2(-2.535) + 2(-1.716) + 2(-1.051) + (-0.065)]

= (0.25)× [-3 - 5.07 - 3.432 - 2.102 - 0.065]

= -5.1941

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Answer:

Step-by-step explanation:

1) a + a

2) 2n+4 = 6n

3)   1

4n = 4

n = 1

Answer:

52

Step-by-step explanation:

Answer:

the second one is the answer

Step-by-step explanation:

hope that helps

Answer:

B; {x|–5 < x < 6}

Step-by-step explanation:

Answer:

3. 102

4. c

Step-by-step explanation:

For number 3:

(c^2 - a) + b becomes (10^2 - 3) + 5

10^2 = 100

100 - 3 + 5 = 102

For number 4:

Usually when you see distributive property, you're multiplying into parentheses. This time you're factoring out.

20c - 8d

It's not a, because that would be 20c + 8d

It's not b, because that would be 20c -32d

It is c, because 4 x 5c = 20 c and 4 x -2d = -8d

It's not d, because that would be 20c- 8cd

Answer:

0.25 is greater than 0.03

Step-by-step explanation:

You can solve this simply by looking for the position of the decimal point.

0.25  >  0.03

Answer:

Step-by-step explanation:

Answer:

$7969.24

Step-by-step explanation:

5000*1.06 to power of 8 = 7969.24037265

round to 2dp = $7969.24

Answer:

7400

Step-by-step explanation:

The correlation coefficient between the scores of students in courses Alb. X and Cae is approximately -0.333.

Correlation refers to the strength of the relationship between two variables while coefficient refers to the numerical value that measures the strength of the correlation.

To calculate the correlation coefficient between the scores of students in courses Alb. X and Cae, we need to first organize the data into two separate lists or arrays representing the scores in each course. Let's denote the scores in Alb. X as X_scores and the scores in Cae as C_scores:

X_scores: 7, 10, 3, 8, 3, 0, 9, 8

C_scores: 8, 5, 2, 9, 10, 1

Next, we need to calculate the mean (average) of both sets of scores.

X_mean = (7 + 10 + 3 + 8 + 3 + 0 + 9 + 8) / 8

X_mean = 48 / 8

X_mean = 6

C_mean = (8 + 5 + 2 + 9 + 10 + 1) / 6

C_mean = 35 / 6

C_mean ≈ 5.83

cov(X_scores, C_scores) = Σ((X_i - X_mean) * (C_i - C_mean)) / (n - 1)

where Σ denotes the sum, X_i and C_i are individual scores, X_mean and C_mean are the means calculated above, and n is the number of scores.

cov(X_scores, C_scores) = ((7-6)(8-5.83) + (10-6)(5-5.83) + (3-6)(2-5.83) + (8-6)(9-5.83) + (3-6)(10-5.83) + (0-6)(1-5.83) + (9-6)(8-5.83) + (8-6)(0-5.83)) / (8-1)

cov(X_scores, C_scores) ≈ -3.39

Next, we calculate the standard deviations of both sets of scores:

X_std = √(Σ(X_i - X_mean)² / (n - 1))

Let's calculate X_std:

X_std = √(((7-6)² + (10-6)² + (3-6)² + (8-6)² + (3-6)² + (0-6)² + (9-6)² + (8-6)²) / (8-1))

X_std ≈ 3.20

C_std = √(Σ(C_i - C_mean)² / (n - 1))

Let's calculate C_std:

C_std = √(((8-5.83)² + (5-5.83)² + (2-5.83)² + (9-5.83)² + (10-5.83)² + (1-5.83)²) / (6-1))

C_std ≈ 3.18

r = cov(X_scores, C_scores) / (X_std * C_std)

Let's calculate r:

r ≈ -3.39 / (3.20 * 3.18)

r ≈ -3.39 / 10.176

r ≈ -0.333

Therefore, the correlation coefficient between the scores of students in courses Alb. X and Cae is approximately -0.333.

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Answer:

The measure of Angle w in degrees is 50°

Step-by-step explanation:

We know that in a straight line, 180°

So,

90° + 40° + w = 180°

130° + w = 180°

w = 180° - 130°

w = 50°

Thus, The measure of Angle w in degrees is 50°

-TheUnknownScientist

The particular solution that does not involve any terms from the homogeneous solution is given by:[tex]Y(t) = C3 + C4te^(-t/2).[/tex]

To find a particular solution of the given differential equation using the method of variation of parameters, we follow these steps:

Solve the associated homogeneous equation: 4y" - 4y + y = 0.

The characteristic equation is:

[tex]4r^2 - 4r + 1 = 0.[/tex]

Solving the quadratic equation, we find two repeated roots: r = 1/2.

Therefore, the homogeneous solution is given by: y_h(t) = C1[tex]e^(t/2)[/tex] + C2t[tex]e^(t/2),[/tex] where C1 and C2 are constants.

Find the particular solution using the variation of parameters.

Let's assume the particular solution has the form:

[tex]y_p(t) = u1(t)e^(t/2) + u2(t)te^(t/2).[/tex]

To find u1(t) and u2(t), we differentiate this expression:

[tex]y_p'(t) = u1'(t)e^(t/2) + u1(t)(1/2)e^(t/2) + u2'(t)te^(t/2) + u2(t)e^(t/2) + u2(t)(1/2)te^(t/2).[/tex]

We equate the coefficients of e^(t/2) and te^(t/2) on both sides of the original equation:

[tex](1/2)(u1(t) + u2(t)t)e^(t/2) = 16e^(t/2).[/tex]

From this, we can deduce that u1(t) + u2(t)t = 32.

Differentiating again:

[tex]y_p''(t) = u1''(t)e^(t/2) + u1'(t)(1/2)e^(t/2) + u1'(t)(1/2)e^(t/2) + u1(t)(1/4)e^(t/2) + u2''(t)te^(t/2) + u2'(t)e^(t/2) + u2'(t)(1/2)te^(t/2) + u2(t)e^(t/2) + u2(t)(1/2)te^(t/2).[/tex]

Setting the coefficient of [tex]e^(t/2)[/tex]equal to zero:

[tex](u1''(t) + u1'(t) + (1/4)u1(t))e^(t/2) = 0.[/tex]

Similarly, setting the coefficient of [tex]te^(t/2)[/tex]equal to zero:

[tex](u2''(t) + u2'(t) + (1/2)u2(t))te^(t/2) = 0.[/tex]

These two equations give us a system of differential equations for u1(t) and u2(t):

u1''(t) + u1'(t) + (1/4)u1(t) = 0,

u2''(t) + u2'(t) + (1/2)u2(t) = 0.

Solving these equations, we obtain:

u1(t) = C3[tex]e^(-t/2)[/tex] + C4t[tex]e^(-t/2),[/tex]

u2(t) = -4C3[tex]e^(-t/2)[/tex] - 4C4t[tex]e^(-t/2).[/tex]

Substitute the values of u1(t) and u2(t) into the assumed particular solution:

[tex]y_p(t) = (C3e^(-t/2) + C4te^(-t/2))e^(t/2) - 4C3e^(-t/2) - 4C4te^(-t/2).[/tex]

Simplifying further:

[tex]y_p(t) = C3 + C4te^(-t/2) - 4C3e^(-t/2) - 4C4te^(-t/2).[/tex]

So, the particular solution that does not involve any terms from the homogeneous solution is given by:

[tex]Y(t) = C3 + C4te^(-t/2).[/tex]

Here, C3 and C4 are arbitrary constants that can be determined using initial conditions or boundary conditions if provided.

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Answer:

Quotient: p + 7

Remainder: 10

Step-by-step explanation:

To find the quotient of the expression (2p² + 7p - 39) ÷ (2p - 7), we can use long division or synthetic division. Let's use long division:

           ____________________

2p - 7  |   2p² + 7p - 39

We start by dividing the first term of the dividend by the first term of the divisor, which gives us 2p² ÷ 2p = p. We then multiply p by the divisor (2p - 7) and subtract it from the dividend:

         p

    ____________________

2p - 7  |   2p² + 7p - 39

        - (2p² - 7p)

       14p - 39

    ____________________

2p - 7  |   2p² + 7p - 39

        - (2p² - 7p)

        ___________

               14p - 39

We repeat the process by dividing the first term of the new dividend (14p - 39) by the first term of the divisor (2p - 7). This gives us (14p - 39) ÷ (2p - 7) = 7. We then multiply 7 by the divisor (2p - 7) and subtract it from the new dividend:

         p + 7

    ____________________

2p - 7  |   2p² + 7p - 39

        - (2p² - 7p)

        ___________

               14p - 39

        - (14p - 49)

        ___________

                10

We are left with a remainder of 10. Therefore, the quotient is p + 7 with a remainder of 10.

Quotient: p + 7

Remainder: 10

Hope this helps!

p + 7 should be it.

I am not 100% sure?

To determine the local minimum and local maximum of the function A(x) = ∫₀ˣ f(t) dt on the interval (0, 6), we need to analyze the behavior of A(x) and its derivative.

Let's denote F(x) as the antiderivative of f(x), which means that F'(x) = f(x).

To find the local minimum and maximum, we need to look for points where the derivative of A(x) changes sign. In other words, we need to find the values of x where A'(x) = 0 or A'(x) is undefined.

Using the Fundamental Theorem of Calculus, we have:

A(x) = ∫₀ˣ f(t) dt = F(x) - F(0)

Taking the derivative of A(x) with respect to x, we get:

A'(x) = (F(x) - F(0))'

Since F(0) is a constant, its derivative is zero, and we are left with:

A'(x) = F'(x) = f(x)

Now, let's analyze the behavior of f(x) based on the given figure to determine the local minimum and maximum of A(x) on the interval (0, 6). Without the specific information about the shape of the graph, it is not possible to determine the exact values of x that correspond to local minimum or maximum points.

To find the local minimum, we need to locate a point where f(x) changes from decreasing to increasing. This point would correspond to x = x_min.

To find the local maximum, we need to locate a point where f(x) changes from increasing to decreasing. This point would correspond to x = x_max.

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Answer:

B. There is only one value for s

Step-by-step explanation:

I hope this works for u.. :3

Can I have a brainliest plz :))

Answer:

9!

Step-by-step explanation:

Answer:

20.00

Step-by-step explanation:

Answer:

There will be 13 gifts given in the party.

Step-by-step explanation:

Given that Ming invited 12 people to her chrismas party, with a total of 13 people at her party, and each person have each other person a present, to determine how many presents were given the following logical reasoning must be performed:

Given that there are 13 people at the party, and each of them will bring a gift, by mathematical logic there will be 13 gifts distributed in the party.

By using the method of variation of parameters to solve a nonhomogeneous DE with W = -3 W2 = e 112 and W = er, we have

Given: W1=-3, W2=e^t and W3=er.The general solution of the non-homogeneous differential equation, y" + p(t) y' + q(t) y = g(t) , where p(t) and q(t) are functions of t and g(t) is non-zero function is given by;{eq}y = y_c + y_p {/eq}Where {eq}y_c {/eq} is complementary function and {eq}y_p {/eq} is particular function obtained by using variation of parameters.The solution is as follows:The given differential equation is{eq}y''+3y'+2y=-3e^{-t}+e^{t}+re^t{/eq}Characteristic equation is{eq}m^2+3m+2=0{/eq}Solving above equation gives us, {eq}m=-1,-2{/eq}Therefore, complementary function {eq}y_c=c_1e^{-t}+c_2e^{-2t} {/eq}Now, we find the particular solution by using the method of variation of parameters.Let {eq}y_p=u_1e^{-t}+u_2e^{-2t}{/eq}be a particular solution where {eq}u_1{/eq} and {eq}u_2{/eq} are functions of {eq}t.{/eq}Here W is a Wronskian and is given as:{eq}W=\begin{vmatrix}W_1&W_2\\W_1'&W_2'\\\end{vmatrix}=\begin{vmatrix}-3&e^t\\-1&e^t\\\end{vmatrix}=2e^{2t}+3e^{t}{/eq}Now, we find {eq}u_1{/eq} and {eq}u_2{/eq} as follows:{eq}u_1=\frac{-\int W_2 g(t) dt}{W}=\frac{-\int e^t(-3e^{-t}+e^{t}+re^t)dt}{2e^{2t}+3e^{t}}=-\frac{r}{5}-\frac{7}{10}+\frac{3}{10}e^{t}{/eq}Similarly,{eq}u_2=\frac{\int W_1 g(t) dt}{W}=\frac{\int -3e^{-t}(-3e^{-t}+e^{t}+re^t)dt}{2e^{2t}+3e^{t}}=-\frac{r}{5}-\frac{1}{10}+\frac{3}{10}e^{-2t}{/eq}Hence, the general solution of the differential equation is {eq}y=y_c+y_p=c_1e^{-t}+c_2e^{-2t}-\frac{r}{5}-\frac{7}{10}+\frac{3}{10}e^{t}-\frac{r}{5}-\frac{1}{10}+\frac{3}{10}e^{-2t}{/eq}So, Option D, {eq}-4u_2=0,~u_1=-\frac{r}{5}-\frac{7}{10}+\frac{3}{10}e^{t},~u_2=-\frac{r}{5}-\frac{1}{10}+\frac{3}{10}e^{-2t}{/eq} is correct.

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Answer:

y=2x+5

Step-by-step explanation:

u find the slope and since the y intercept is (0,5), that's the b value

hope this helps

Answer: 2x+5

Step-by-step explanation: calculate the slope of the points with the formula m= (y2-y1)/(x2-x1) then use the y-int to complete the equation

The standard form of the equation for the ellipse is ((x - 9)²/376) + ((y - 2)²/94) = 1.

The given equation represents an ellipse. The standard form of the equation is ((x-h)²/a²) + ((y-k)²/b²) = 1, where (h,k) is the center of the ellipse, 'a' is the semi-major axis, and 'b' is the semi-minor axis.

To determine the type of conic represented by the equation, we examine the coefficients of the x² and y² terms. In the given equation, both the x² and y² terms have positive coefficients, indicating that it represents an ellipse.

To write the equation in standard form, we need to complete the square for both the x and y terms. Let's rearrange the equation:

25x² - 450x + 100y² - 400y = 75

Now, we group the x and y terms and complete the square separately:

25(x² - 18x) + 100(y² - 4y) = 75

To complete the square for the x term, we take half of the coefficient of x (-18/2 = -9) and square it to get 81. Similarly, for the y term, we get 4.

25(x² - 18x + 81) + 100(y² - 4y + 4) = 75 + 25*81 + 100*4

Simplifying further:

25(x - 9)² + 100(y - 2)² = 9400

Dividing both sides by 9400, we get:

((x - 9)²/376) + ((y - 2)²/94) = 1

Therefore, the standard form of the equation for the ellipse is ((x - 9)²/376) + ((y - 2)²/94) = 1.

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The last one as we can see that as the wight increases, the miles per gallon decreases.

Answer:c?

Step-by-step explanation:

Answer:

B

Step-by-step explanation:

An isometry on R^n must be regular and the set of isometries forms a group under matrix multiplication.

An isometry is a linear transformation that preserves distances, meaning the norm of the transformed vector is equal to the norm of the original vector. To show that an isometry must be regular (i.e., invertible), we can assume there exists a non-invertible isometry matrix A. In this case, there exists a nonzero vector x such that Ax = 0. However, this contradicts the property of an isometry since ||Ax|| = ||0|| = 0, but ||x|| ≠ 0. Thus, an isometry must be regular.

The set of isometries forms a group under matrix multiplication because it satisfies the group axioms: closure (the product of two isometries is an isometry), associativity (matrix multiplication is associative), identity (identity matrix is an isometry), and inverses (the inverse of an isometry is also an isometry).

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Answer:

A. Shading above a solid line.

Step-by-step explanation:

Let [tex]f(x) = 2\sqrt{3}\cdot x + 2[/tex], whose domain is all real numbers, since it is a first order polynomial (linear function) and meaning that for all element of [tex]x[/tex] exists one and only one value for [tex]f(x)[/tex], meaning a solid line. If [tex]g(x) > f(x)[/tex], then the range of all possible results is a shade area above the solid line.

Hence, the correct answer is A.

Answer:

true

Step-by-step explanation:

Answer:

Step-by-step explanation:

17x + 14 + 4x -2 = 21x + 12

21x + 12 = 180

-12         -12

21x = 168

x = 8 degrees

[tex]\boxed{\large{\bold{\green{ANSWER~:) }}}}[/tex]

Here,

The two lines l and m are parallel,

cut by a transversat line t

So The two angles are supplementary each other.

The sum of two supplementary angles are 180°

(17x+14)°+(4x-2)°=180°

17x+14°+4x-2°=180°

17x+4x+14°-2°=180°

21x+12°=180°

21x=180°-12°

21x=168°

x=168°/21

x=8°

The value of x is 8°

Answer:

5/8m-3/8

Step-by-step explanation:

thats prolly it

Answer:

100.571429.

Step-by-step explanation:

2*22/7*(4)^2

=44/7*16

=100.571429