A rubber gasket has a circumference of 3.2 cm. When placed in service, it expands by a scale factor of 2. What is the circumference of the gasket when in service?
A.1.6 cm
B.3.2 cm
C.6.4 cm
D.13.2 cm

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 radian measure of angle A is (16/9)π.

To convert degrees to radians, we use the conversion factor:

[tex]1\ degree = \pi /180\ radians[/tex]

Given that angle A is 320 degrees, we can calculate its radian measure as follows:

Angle A in radians = (320 degrees) * (π/180 radians/degree)

                    = (320π)/180 radians

                    = (16/9)π radians

Therefore, the radian measure of angle A is (16/9)π.

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Answer:

Step-by-step explanation:

1) a + a

2) 2n+4 = 6n

3)   1

4n = 4

n = 1

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:

We cannot construct a confidence interval using the normal approximation in this case because the sample sizes of the control and treatment groups are relatively small (less than 30), and the data does not meet the assumptions required for the normal approximation.

The normal approximation assumes that the data follows a normal distribution, and for sample sizes less than 30, the distribution of the sample mean may not be well approximated by a normal distribution. In addition, the normal approximation assumes that the observations are independent, which may not hold true in this study due to the nature of the treatment and control groups.

If we constructed a confidence interval despite these problems, the interval may not accurately reflect the true difference in survival rates between the control and treatment groups. The interval could be biased or too wide/narrow, leading to incorrect conclusions about the effectiveness of the heart transplant program. Therefore, it is important to use appropriate statistical methods that account for the specific characteristics of the data and study design in order to obtain reliable estimates and valid inferences.

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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:

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:

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:

52

Step-by-step explanation:

Answer:

9!

Step-by-step explanation:

Answer:

3

Step-by-step explanation:

The closest angle to F in the triangle is 52 degrees.

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.

Given that Right triangle EFG it is known that the side opposite of acute angle F has a length of 18 inches and the leg adjacent to F has a length of 14 inches.

We can use the tangent function to find the measure of angle F:

tan(F) = opposite/adjacent = 18/14

we can find the inverse tangent of this value:

F = tan⁻¹(18/14)

= tan⁻¹(1.28)

F= 52.15°

Hence, the closest angle to F in the triangle is 52 degrees.

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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:

The speed is: 0.0142MHz

Step-by-step explanation:

Given

[tex]C(s) = 1234s^2 - 35s + 1900[/tex]

Required

The processor speed when the cost is at minimum

First, differentiate C(s) with respect to s

[tex]C(s) = 1234s^2 - 35s + 1900[/tex]

[tex]C'(s) = 2468s - 35[/tex]

The cost is at minimum when [tex]C'(s) = 0[/tex]

So, we have:

[tex]C' = 2468s - 35 = 0[/tex]

[tex]2468s - 35 = 0[/tex]

Solve for s

[tex]2468s = 35[/tex]

[tex]s=\frac{35}{2468}[/tex]

[tex]s\approx0.0142[/tex]

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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Answer:

the first one

Step-by-step explanation:

Answer:

45 meters

Step-by-step explanation:

I hope this helps.

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

Answer:

Choice D

Step-by-step explanation:

Choice D, 3072ft^2 is the correct answer.

Here's how! The length AND width are both 24, so we know that 24 * 24 is our start. Then the height of the pyramid is 16, which now makes our equation 24 * 24 * 16. When finding the volume of a pyramid, you must divide by 3, so now the answer is (24 * 24 * 16) / 3. Making the answer 3072ft^2.

Hope this helped!

Answer:

3072

Step-by-step explanation:

volume =(1/3) hight ×width×length

v=(1/3) × 16 × 24 × 24

v=3072

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.

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:

5/8m-3/8

Step-by-step explanation:

thats prolly it

Using the null-hypothesis to calculate the test-statistic at a significance level of 0.01, there is not enough evidence to reject the claim that 40% of the telephone company's customers have call-waiting service based on the sample data.

To test the hypothesis that 40% of the telephone company's customers have call-waiting service, we can conduct a hypothesis test.

Let's set up the null and alternative hypotheses:

Null hypothesis (H₀): The proportion of customers with call-waiting service is 40%.

Alternative hypothesis (H₁): The proportion of customers with call-waiting service is not 40%.

We can use the t-test for proportions to perform the hypothesis test. The test statistic is given by:

z = (p - p₀) / √((p₀ * (1 - p₀)) / n)

where p is the sample proportion, p₀ is the hypothesized proportion, and n is the sample size.

In this case, p₀ = 0.40, p = 0.37, and n = 100.

Calculating the test statistic:

z = (0.37 - 0.40) / √((0.40 * (1 - 0.40)) / 100)

z = -0.03 / √(0.24 / 100)

z = -0.03 / 0.049

z = -0.612

To determine if there is enough evidence to reject the null hypothesis, we compare the calculated z-value with the critical z-value at a significance level of α = 0.01.

The critical z-value for a two-tailed test at α = 0.01 is approximately ±2.576.

Since -0.612 falls within the range of -2.576 to 2.576, we fail to reject the null hypothesis.

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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:

the number of sponges per package bought be 32.5

Step-by-step explanation:

Let us assume the number of sponges per package be x

So the number of packages he bought be x - 14

And, the overall he bought 51 sponges

So

the number of packages of sponges did roger buy is

x + x - 14 = 51

2x = 51 + 14

2x = 65

x = 32.5

So the number of sponges per package bought be 32.5

Answer:

100.571429.

Step-by-step explanation:

2*22/7*(4)^2

=44/7*16

=100.571429

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:

325 grams is the weight.