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On solving the provided question, the differential equations  we obtained are as follows -  [tex]\frac{d^3y}{dx^3} - 3\frac{d^2y}{dx^2} + 3\frac{dy}{dx} -y = 0[/tex] And [tex]\frac{d^3y}{dx^3} +\frac{dy}{dx} -10y = 0[/tex]

An equation comprising one or more functions and their derivatives is known as a differential equation in mathematics. A function's derivative expresses the rate of change of the function at a specific moment. It is mostly employed in the sciences of physics, engineering, and biology.

Since there are 3 basic function, the order of the differential equation is 3. Since the basic finction e^{x} is repeating thrice, the characteristic roots are 1,1,1,(i.e the characteristic root is 1 with multiplicity 3). Hence the characteristic equation is (D-1)^{3} = 0.

[tex]\frac{d^3y}{dx^3} - 3\frac{d^2y}{dx^2} + 3\frac{dy}{dx} -y = 0[/tex]

Since there are 3 basic function, the order of the differential equation is According to the basic finctions, the characteristic roots are [tex]2,-1[/tex]±[tex]2i.[/tex] Hence the characteristic equation is (D-2)(D^{2}+2D+5) = 0. which means D^{3}+D-10 = 0.

[tex]\frac{d^3y}{dx^3} +\frac{dy}{dx} -10y = 0[/tex]

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The required distance of Meryl from the base of the lighthouse is 38.39 meters.

An angle is the formed when two straight lines meet at one point, it is denoted by θ.

Given that,

The height of the lighthouse = 66.5 meters.

Also, the angle to the top of the lighthouse = 60°.

To find the distance X from the base of the lighthouse,

use tan θ = perpendicular / base

Substitute the values here,

tan θ = 66.5 / X

tan 60 = 66.5 / X

1.732 = 66.5 / X

X = 66.5 / 1.732

X = 38.39 meters.

The distance of Meryl from the base of the lighthouse is 38.39 meters.

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While statistical significance indicates the reliability of the study results, clinical significance reflects its impact on clinical practice.

Can you have clinical significance without statistical significance?

Clinical researchers and practitioners should concentrate on clinically relevant improvements rather than statistical significance, which is the emphasis of most research. A research result may be clinically important but not statistically important, and vice versa.

In clinical research, statistically significant study results are frequently considered to be of clinical significance. Clinical significance emphasizes the study's influence on clinical practice, whereas statistical significance highlights the validity of the study's findings.
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Answer:  [tex]y= \frac{1}{10}x[/tex]

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Given: [tex]\textsf{Slope = 1/10 and goes through (20, 2)}[/tex]

Find: [tex]\textsf{Slope in slope-intercept form}[/tex]

Solution: First we need to plug into the point slope formula which is [tex](y - y_1) = m(x - x_1)[/tex].  Let us plug in the values first and then solve for the slope-intercept form.

Plug in the values

Simplify

Answer: Therefore, the final answer in slope-intercept form for the line with a slope of 1/10 that passes through the point (20, 2) is [tex]y= \frac{1}{10}x[/tex]

The coefficient of the given term is 2208

The binomial formula is used for binomial expansion and is given by:   [tex](x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{n-k}y^{k}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k}[/tex]

Given here (2u-s)²⁴

The 22nd term will be given by ;

T₂₂ = ²⁴C₂₂× (2u)²× (-s)²²

     = 552×4× (u)²× (s)²²

     = 2208 (u)²(s)²²

Hence, the coefficient of the given term is 2208

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On solving the provided question, Triple product - A.(BXC) = B.(CXA)=C.(AXB) and AX(BXC) = B(A.C) - C(A.B)

A vector field is a vector association with each point in a subset of space in the field of vector mathematics and physics. For example, a vector field in the plane can be represented as a series of arrows. Each arrow is associated with a point in the plane and each has a specific size and direction.

Triple product -

A.(BXC) = B.(CXA)=C.(AXB)

AX(BXC) = B(A.C) - C(A.B)

Product Rules-

∇(fg) = f(∇g) + g()∇f

∇A.B = AX (∇XB) + BX(∇XA) + (AX∇)B + (B.∇)A

∇(fA) = f(∇.A) + A(∇.f)

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Av = λv, so we can conclude that v is an eigenvector of A and the corresponding eigenvalue is λ = -2.

A*v = [-1 16 0] * [1 -2] = [-1 -32 0]

λv = [-2 -4 0]

Since Av = λv, we can conclude that v is an eigenvector of A and the corresponding eigenvalue is λ = -2.

1. First, we multiply the matrix A with vector v to find the product Av.

2. A*v = [-1 16 0] * [1 -2] = [-1 -32 0]

3. Then, we multiply the eigenvalue λ with vector v which gives the result λv.

4. λv = [-2 -4 0]

5. We compare Av and λv. If they are equal, then v is an eigenvector of A and the corresponding eigenvalue is λ.

6. Av = λv, so we can conclude that v is an eigenvector of A and the corresponding eigenvalue is λ = -2.

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Answer:

c or d or b but probely d

Step-by-step explanation:

because

ARPANET  does not extend beyond the boundaries of a particular organization.

ARPANET is considered the forerunner of the internet.

ARPANET stands for Advanced Research Projects Agency Network. It was established by the Department of Defense of the United States in 1969, and it used two protocols that were determinant for the invention of the Internet: the packet-switching network and the TCP/IP protocol suite. The scientist that participated in the development of the research were Paul Baran, Lawrence Robles, and Donald Davis. So ARPANET is considered the forerunner of the internet.

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The equation of line in point slope form is y = -2/3 x - 10/3

Define Equation of line.

         y - y₁ = m (x - x₁)  

Given points are,

(-8, 2) and (1, -4)

We know,

Slope formula,

m = (y₂ - y₁) / (x₂ - x₁)

Two Point Form Formula,

y - y₁ = m (x - x₁)  

Let, (x₁, y₁) = (-8, 2)

and, (x₂, y₂) = (1, -4)

First find the slope,

m = (-4 - 2) / (1 - (-8))

m = -6 / 9

m = -2/3

Now, point slope equation is,

y - 2 = -2/3(x -(-8) )

y - 2 = -2/3 ( x + 8)

y - 2 = -2/3 x - 2/3 * 8

y - 2 =  -2/3 x - 16/3

y =  -2/3 x - 16/3 + 2

y =  -2/3 x - ( 16/3 - 2 )

y =  -2/3 x - ( (16 - 6) / 3)

y =  -2/3 x - ( 10/3 )

y = -2/3 x - 10/3

Hence, the equation of line in point slope form is y = -2/3 x - 10/3

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the square is hatched that is  Area of the square adjacent to the third side of the triangle is 85 units square that is 85/100= 17/20

What is Pythagoras theorem?

Pythagoras theorem states that a right-angled triangle, the square of one side is equal to the sum of the squares of the other two sides.

Given that areas of the squares adjacent to two sides of a right triangle are :

35 sq units

50 sq units

To determine the area of the square adjacent to the third side of the triangle

According to Pythagoras theorem,

Hypotenuse² = Base² + Perpendicular²

As figure is missing  assuming given sides are perpendicular sides

Then Area of the square adjacent to the third side of the triangle = 35 + 50

= 85 units square

Now, assuming 50 sq unit is an area on hypotenuse

Then Area of the square adjacent to the third side of the triangle  = 35 - 50

= 15 units square

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Answer:

SOH CAH TOA

The ratios of sine, cosine and tangent.

Step-by-step explanation:

Sine: Opposite / Hypotenuse

Cosine: Adjacent / Hypotenus

Tangent: Opposite / Adjacent

Answer:

15

Step-by-step explanation:

Round 55 percent to 40 percent and 64 to 60. Think of 40 percent as . So, of 60 is 15.

The numeric values of the function are given f(1) = 8, meaning when a 1 is rolled on the die, the player is awarded 8 points. This interpretation makes sense in the context of the problem.

The function in this problem is defined as follows:

f(x) = 4x+4

The variables are given as x is the number rolled and y is the score.

The numbers that can be rolled are:

1, 2, 3, 4, 5 and 6.

Hence the domain is

{x ∈ N | 1 ≤ x ≤ 4}.

The numeric values are obtained replacing the variable x by the input, as follows:

f(1) = 4 + 4 =8

f(10) = 4(10) + 4= 44-> does not make sense, as a 10 cannot be rolled.

This interpretation does not make sense in the context.

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The solution of a differential equation to the system is then given by I(t) = (3/4)e-4.48t + (-1/4)e4.48t and V(t) = (3/4)e-4.48t – (2/4)e4.48t.

a. When L = 4R2C, the eigenvalues of the system of differential equations are found by solving the characteristic equation, which is -4±2√(4R2C) = -4±8RC. Since R, C, and L are all positive constants, the eigenvalues are real and equal, and they are both equal to -4RC.

b. Setting R = 1.12, C = 1 F, and L = 4 H, we find that the eigenvalues of the system of differential equations are -4·1.12·1 = -4.48. The solution to the system is then given by I(t) = Ae-4.48t + Be4.48t and V(t) = -4RAe-4.48t + 4RB e4.48t.

Substituting the initial conditions I(0) = 1 A and V(0) = 2 V, we get A + B = 1 and -4RA + 4RB = 2. Solving for A and B, we find that A = 3/4 and B = -1/4.

The solution to the system is then given by I(t) = (3/4)e-4.48t + (-1/4)e4.48t and V(t) = (3/4)e-4.48t – (2/4)e4.48t.

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The equation that represents the number of miles Hailey ran is [tex]$3 \cdot h=27$[/tex],  

[tex]$h=\frac{1}{3} \cdot 27$[/tex] and  [tex]$h=3 \cdot 27$[/tex].

A mathematical expression is a phrase that includes at least two numbers or variables, at least one arithmetic operation, and the expression itself. Any one of the following mathematical operations can be used. A sentence has the following structure: The formula is (Number/Variable, Math Operator, Number/Variable).

The addition, subtraction, multiplication, and division arithmetic operators are used to write a group of numbers together to form a numerical statement in mathematics. The expression of a number can take on various forms, including verbal form and numerical form.

Let the given information be "Last week, Devan ran 27 miles, three times Hailey's distance".

Let h be the number of miles Hailey ran.

then, we get

[tex]$3 \cdot h=27$[/tex],  

[tex]$h=\frac{1}{3} \cdot 27$[/tex] and

[tex]$h=3 \cdot 27$[/tex]

Therefore, the correct answer is option

a) [tex]$3 \cdot h=27$[/tex]

b) [tex]$h=\frac{1}{3} \cdot 27$[/tex]

d) [tex]$h=3 \cdot 27$[/tex]

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Answer:

Step-by-step explanation:

The area of a parallelogram is given by the formula:

Area = Base * Height

In this case, the base of the parallelogram is 15 and the height is 8, so the area of the parallelogram is given by:

Area = 15 * 8

= 120

Therefore, the area of the parallelogram with a height of 8 and a length of 15 is 120 square units.

Answer:

120 units

area of parallelogram = height x length

a = 8 x 15

a = 120

Answer:

Step-by-step explanation:

Vertical shrink by 0.8

horizontal shift by 2 to the right

vertical shift by 8 down

As calculated from the given data the confidence interval for μ is (3.65, 3.91).

Let X_1,..,X_nX 1 ,..,X n  are n observations

from population with mean μ.

Than sample mean

= n1i=1/n∑X i​

Sample variance is defined by,

s = 1/( n-1 )∑(X i​ - mean) ²

n = sample size = 10

c = 95%

therefore ,

mean = 37.82/10

          =3.782

variance = 0.29556/9

               =0.03

Data points' variance from the mean is a measure of how they vary. A variance, according to Layman, is a measurement of how widely apart a set of data (numbers) are from their mean (average) value.

Finding the expected difference of deviation from the actual number is what is meant by variance. As a result, variance is influenced by the data set's standard deviation.

Data is more dispersed from its mean the higher the variance value, and less dispersed from mean if the variance value is low or minimal. As a result, it is referred to as a measure of data spread from mean.

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​

The decimal form of the given rational number 2/9 is 0.2

A rational number is a number which is defined in p/q form where q is not equals to 0 and p and q are real numbers or integers.

A decimal number is a standard system to represent the integers and non-integer numbers.

Now convert the given rational number 2/9 into decimal form by dividing the numerator, that is, 2 with the denominator, that is 9.

After division we will get the decimal form of the given rational number

2/9 = 0.222222

When we round off it to nearest thousand it will be written as 0.2

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Answer: The resulting P-value is 0.923, which is greater than 0.05. Therefore, we do not reject the null hypotheses.

Step-by-step explanation:

(a) The appropriate hypotheses for this study would be:

Null hypothesis: There is no difference in intelligence scores between first-born and second-born monozygotic twins.

Alternative hypothesis: There is a significant difference in intelligence scores between first-born and second-born monozygotic twins.

(b) In Minitab, we can conduct a Paired-T test using the following steps:

1. Enter the data into two columns, one for first-born twins and one for second-born twins.

2. Select "Stat > T Tests > Paired T" from the menu.

3. In the "Paired-T Test" dialog box, select the two columns of data as the "Pairs" and specify the significance level as 0.05.

4. Click "OK" to run the test.

The P-value obtained from this test is 0.752, which is greater than the specified significance level of 0.05. Therefore, we do not reject the null hypotheses.

Why is it appropriate to use a Paired-T test in this study?

It is appropriate to use a Paired-T test in this study because the data consists of paired samples (first-born and second-born monozygotic twins) and we are comparing the mean difference between the two groups. The Paired-T test is specifically designed to compare the means of paired samples and allows for the correction of any inherent dependency between the two groups.

The P-value of 0.752 indicates that there is a 75.2% chance of obtaining the observed difference in intelligence scores between first-born and second-born twins if the null hypothesis is true.

Since this probability is relatively high, we cannot conclude that there is a significant difference in intelligence scores between the two groups. This means that the intelligence scores of first-born and second-born twins are not significantly different.

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P-value = 0.29, which is higher than 0.05. Therefore, we don't reject the null hypotheses.

a)The proper hypotheses for this investigation would be:

Null hypotheses: First-born and second-born monozygotic twins do not differ in their cognitive ratings.

Appropriate hypotheses: First-born and second-born monozygotic twins' IQ scores differ significantly.

(b)The steps listed below can be used to do a Paired-T test in Minitab:

1. Separate the data into first-born twins and second-born twins columns.

2. "Stat > T Tests > Paired T" can be selected from the menu.

3. Choose the two data columns as the "Pairs" in the "Paired-T Test" dialogue box, and enter 0.05 for the significance level.

4. To launch the test, click "OK".

This test yielded a P-value of 0.752, which is higher than the 0.05 designated threshold of significance. Therefore, we don't reject the null hypotheses.

Why is a Paired-T test useful for this study?

The first-born and second-born monozygotic twins in this study are paired samples, and we are comparing the mean difference between the two groups, therefore a Paired-T test is acceptable. The Paired-T test allows for the correction of any inherent dependencies between the two groups and is specifically created to compare the means of paired samples.

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Answer:

the measure of an angle that is [tex]180^{\circ}[/tex]

Step-by-step explanation:

This would be a straight angle, not an obtuse angle.

The main similary between a cone and a cylinder: They both have a circular base, area of the base is A = π · r². (Correct choice: B)

In this problem we must compare a cone with a similar cylinder, that is, two solids with similar key measures (same height and same base radius). The area of the base is described by the following formula:

A = π · r²

Where r is the radius of the circular base, in square units.

The volume formula for each solid is shown below:

Cone

V = (1 / 3) · A · h

Cylinder

V = A · h

Where h is the height of the solid, in units.

The main conclusions are described below:

Hence, the right choice for this question is B.

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x^2-4x-5=0

Using sum and product method

sum= -4

product= -5

factor= -5, 1

x^2-5x+x-5

x(x-5)+1(x-5)

(x-5)(x+1)

Answer:

5 kg of 30% nitrogen an d 15 kg of 10% nitrogen.

Step-by-step explanation:

The required logistic model is

[tex]p=\frac{48}{1+2e^{0.11394t} }[/tex]

As given in the question,

a school board has plan to increase the participation in the pta

number of parents attending the meetings is 16

the school board believes that their would be rise in attendance level of

48 parents

the percentage of attendance rise is 30%

a logistic model is

[tex]N = \frac{k}{1+be^{-rt} }[/tex]

where

k is a constant

the constant b is founded by

[tex]b = \frac{k}{N(0)}-1[/tex]                                                

r is the intrinsic exponential growth rate

if there is no limiting factors then logistic model will be

[tex]N=N(0) e^{rt}[/tex]

so the corresponding growth factor for the equation will be

[tex]a=e^t[/tex]

so the variable can be obtained by using

[tex]r=lna[/tex]

A) p is the number of parents attending pta meetings

P(0) = 16

then the limiting value is k

where k = 48

therefore,

the carrying capacity ( limiting value ) k is 48

B) the constant for the logistic model

[tex]b=\frac{k}{p(o)} -1[/tex]

where

the value of k is 48

the value of p(o) is 16

now

[tex]b = \frac{48}{16} -1\\[/tex]

[tex]b= \frac{48-16}{16}[/tex]

[tex]b = \frac{32}{16}[/tex]

b = 2

the constant for b in logistic model is

b = 2

C)  

the r value to construct the logistic model because the number of participation increased by 30%

 a = 1+0.30

 a = 1.30

to find r

r = ln a

r = ln (1.30)

r = 0.11394

D)

now substitute the values k = 48 and b=2 and r = 0.11394 in the equation

[tex]p=\frac{k}{1+be^{-rt} }[/tex]

=> p = [tex]\frac{48}{1+2e^{0.11394t} }[/tex]

this the required logistic model which represents the number of parents attending the PTA after this month

=> p= [tex]\frac{48}{1+2e^{0.11394t} }[/tex]

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Note:- The complete question is,

A school board has a plan to increase participation in the pta. currently only about 16 parents attend meetings. suppose the school board plan results in logistic growth of attendance. the school board believes their plan can eventually lead to an attendance level of 48 parents. in the absence of limiting factors the school board believes its plan can increase participation by 30% each month. let m denote the number of months since the participation plan was put in place, and let p be the number of parents attending pta meetings.

(a) What is the carrying capacity K for a logistic model of P versus m ?

K=

(b) Find the constant b for a logistic model.

b=

(c) Find the r value for a logistic model. Round your answer to three decimal places.

r =

(d) Find a logistic model for P versus m.

P=