A partial variation has an equation of the form y=mx+b, where m represents the
slope and b represents the initial value.
True
False

Recent research has confirmed the Spillover Hypothesis by demonstrating a favourable association between life and job happiness.

According to the spillover hypothesis, behaviour or affect in a family system might flow immediately from one place or connection to another. Transfer takes place in the same valence, which means that one subsystem's negative affect is connected to another's negative affect, or stress at work spills over and intensifies stress at home. The compensatory hypothesis offers a different perspective by arguing that transfer between family subsystems occurs in the opposite valence, leading a person to seek fulfilment in one relationship or environment in order to make up for shortages in another.

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Answer:

\(\Huge \boxed{\£55}\)

________________________________________________________

A 17% reduction means that the dress cost 83% (100 - 17) of the original amount.

\(\large \fbox{\begin{minipage}{8.1 cm}83\% of the original price = \£45.65\\\\$\Rightarrow$1\% of the original price = $\frac{45.65}{83}$\\\\$\Rightarrow$1\% of the original price = 0.55\\\\$\Rightarrow$100\% of the original price = 0.55 \times 100\\\\$\Rightarrow$100\% \text{ of the original price = \£55}\end{minipage}}\)

To work out 83% of the original price, you multiply by 0.83. We can do the inverse, which is dividing by 0.83.

                                 ×0.83

Original Price →→→→→→→→→→→→→ Sale Price

        £?           ←←←←←←←←←←←←←     £45.65

                                ÷0.83

\(\large \boxed{\begin{minipage}{7 cm}Original Price = $\frac{\text{Sale Price}}{0.83}$\\\\$\Rightarrow$Original Price = $\frac{45.65}{0.83}$\\\\$\Rightarrow$Original Price = \£55\end{minipage}}\)

Therefore, the original price of the dress is £55.

________________________________________________________

Answer:

d. -5/2x + 5 (negative five-halves x + 5)

Step-by-step explanation:

In order to find the equation for the line that passes through these points, you must identify the slope and y-intercept. You can find the slope by finding the change in y divided by the change in x:

using (0, 5) and (2, 0),

change in y = 5

change in x = -2

(change in y)/(change in x) = -5/2

So, the slope is -5/2. You don't need to calculate the y-intercept since one of the points already is the y-intercept; when x is 0, y is 5, so the y-intercept is 5. Now, you can write your equation:

using y = mx + b, where m = slope and b = y-intercept,

y = -5/2x + 5

Answer:

d

Step-by-step explanation:

The probability that a single bulb will burn out within the next 10 minutes is approximately 0.6321. The expected amount of time until the first bulb goes out is 10 minutes. The probability density function (pdf) of the random variable T, representing the time when you leave the room after the last light bulb goes out, is given by \(g(t) = 20 * (1/10) * e^{(-(1/10)t)} * (1 - e^{(-(1/10)t))^(19)}\).

To find the probability that a single bulb will burn out within the next 10 minutes, we can use the exponential distribution. The exponential distribution with a mean of 10 minutes has a rate parameter λ = 1/10.

The probability density function (pdf) for an exponential distribution is given by \(f(x) = λ * e^{(-λx)}\)

In this case, we want to find the probability that a bulb burns out within the next 10 minutes, which corresponds to the cumulative distribution function (CDF) at x = 10. The CDF is given by \(F(x) = 1 - e^{(-λx)\)

So, substituting the values, we have:

\(F(10) = 1 - e^{(-(1/10)*10)\)

\(= 1 - e^{(-1)\)

= 1 - 0.3678794412

≈ 0.6321

Therefore, the probability that a single bulb will burn out within the next 10 minutes is approximately 0.6321.

The amount of time until the first bulb goes out follows an exponential distribution with a rate parameter of λ = 1/10 (since it has a mean of 10 minutes).

The probability density function (pdf) for the time until the first bulb goes out is given by\(f(t) = λ * e^{(-λt).\)

To find the expected amount of time until the first bulb goes out, we need to calculate the mean (or expected value) of this distribution.

The expected value of an exponential distribution with rate parameter λ is equal to 1/λ. In this case, the expected value is 1/(1/10) = 10 minutes.

Therefore, the expected amount of time until the first bulb goes out is 10 minutes.

To find the probability density function (pdf) of the random variable T, which represents the time when you leave the room (after the last light bulb goes out), we need to consider the distribution of the maximum of the exponential random variables.

Since there are 20 light bulbs in the room, and each follows an exponential distribution with a rate parameter λ = 1/10, the time until the last bulb goes out can be modeled as the maximum of 20 exponential random variables.

The pdf of the maximum of independent exponential random variables with the same rate parameter λ is given by \(g(t) = n * λ * e^{(-λt)} * (1 - e^{(-λt))^(n-1)}\), where n is the number of random variables.

In this case, n = 20, and λ = 1/10. Thus, the pdf of T is \(g(t) = 20 * (1/10) * e^{(-(1/10)t)} * (1 - e^{(-(1/10)t))^(19)}\)

This expression represents the pdf of the random variable T, which denotes the time when you leave the room after the last light bulb goes out.

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A sample size of 76 would be required to estimate the population mean with a margin of error of 4 and a 98% level of confidence.

To determine the sample size required to estimate the population mean with a specific margin of error and level of confidence, we can use the formula:

n = (Z * σ / E)²

where

n = required sample size

Z = Z-score corresponding to the desired level of confidence

σ = standard deviation of the population

E = desired margin of error

here, we have that

the standard deviation (σ) is given as 15,

the margin of error (E) is 4 and

the level of confidence is 98% and For a 98% confidence level, the Z-score is approximately 2.33.

Plugging the values into the formula:

n = (2.33 * 15 / 4)²

n = (8.7375)²

n ≈ 76.34

n = 76

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Hi there! :)

\(\huge\boxed{x = -2}\)

m(x) = 4x + 15

Find the value of "x" that would satisfy m(x) = 7:

4x + 15 = 7

Subtract 15 from both sides:

4x = -8

Divide both sides by 4:

x = -2.

The equilibrium point for the supply and demand equations p² + 4q = 253 and 183p² + 6q = 0 is (q, p) = (3, 10).

To find the equilibrium point, we need to solve the system of equations formed by the supply and demand equations. By substituting the value of q = 3 into the first equation, we get p² + 4(3) = 253, which simplifies to p² + 12 = 253.

Solving this equation gives us p = 10. Substituting the values of q = 3 and p = 10 into the second equation, we get 183(10)² + 6(3) = 0, which simplifies to 18300 + 18 = 0.

Since this equation holds true, we have found the equilibrium point to be (q, p) = (3, 10), where the equilibrium quantity is q = 3 and the equilibrium price is p = 10.

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For a sample of born babies' weight in a large hospital, probability that the mean weight of the sample babies would differ from the population mean by greater than 43 grams is equals to 0.5390.

We have a sample of born babies' weight in a large hospital.

Mean of weight = 4090 grams

Variance of weight = 31300

Standard deviations = 560

Sample size or included babies = 64

standard error = \(\frac{560}{\sqrt(64)}\) = 70

We have to determine probability that the mean weight of the sample babies would differ from the population mean by greater than 43 grams. Let the population mean would be \(\mu\). Now, the probability, \(P ( \bar X - \mu > 43 ) \). Using Z-score formula, \(Z = \frac{X -\mu}{\frac{\sigma}{\sqrt(n)}} \)

Substitute all known values then, z= ± 43/70, z= ± 0.614

P(z<-0.614 or z>0.614) = P(z<-0.614)+ P(z>0.614)

=2 × 0.2695

=0.5390

Hence, required value is 0.5390.

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Answer: 21

Step-by-step explanation: 5        7

                                             10       14

                                             15         21

Your friend will take 21 minutes to complete the 15 laps.

Given is that you complete 5 laps in 7 minutes.

Since you and your friend skate at the same rate, your friend will take the same time to complete 15 laps as you will take. Now -

To complete 5 laps it takes 7 minutes.

To complete 1 lap it will take (7/5) minutes.

So, to complete 15 laps it will take (7/5) x 15 or 21 minutes

Therefore, your friend will take 21 minutes to complete the 15 laps.

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Answer:

$160.16

Step-by-step explanation:

Well first you need to find 12% of 143.

100%=143

10%=14.3

2%=2.86

12%=17.16

Now we add 17.16 to 143 to get 160.16. This means he pays $160.16.

Answer:

7/25

Step-by-step explanation:

A child of a parent with a mutation in collagen and displaying OI will have 50% probability of also having OI.

Osteogenesis imperfecta (OI) is an inherited or genetic bone disorder which is present at birth. It is also known as brittle bone disease. A child born with OI may have soft bones which fracture easily, bones which are not formed normally, and other problems. Signs and symptoms may range from mild to severe.

Life expectancy for OI babies vary greatly depending on OI type. Babies with Type II generally die soon after birth. Children with Type III may live longer, but mostly only until around age 10.

About 90% of OI cases are caused by autosomal dominant mutations in the type I collagen genes. Mutations in one or the other of these genes cause the body to make either abnormally formed collagen or too little collagen. Mutations in these genes cause OI Types I through IV.

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The probability that the artist chosen will be a painter or a photographer is 3/4

Probability is the function that measures the chances that an outcome of a random event will be as expected. In this way you can measure how likely it is that an event will happen.

To solve this exercise the formula and the procedure that we have to use for probability is:

probability= (achieving success / possible outcomes)

Achieving success = 7 painters + 5 photographer = 12

Possible outcomes = 7 painters + 5 photographer + 4 sculptors = 16

Applying the formula to calculate the probability we get:

probability= (achieving success / possible outcomes)

probability= 12/16

Simplifying:

probability= 3/4

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The correct identification of the dependent variable and the independent variable for the experiment is valid experiment.

Independent variable: The variable that is being changed or manipulated in an experiment is called the independent variable. The experimenter modifies the independent variable to observe its effect on the dependent variable.

Dependent variable: The dependent variable is the variable that is being measured and observed in an experiment. It is dependent on the independent variable because it changes as a result of the manipulation of the independent variable.

Therefore, To carry out a valid experiment, it is crucial to accurately identify the dependent variable and independent variable.

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Step-by-step explanation:

5 sandwiches plus 7 pies =12

12×7 days (week) =84 were sold

26

Step-by-step explanation:

39/2÷3/4

39/2×4/3

26

Answer:

Okay she purchased 19.5 pints of ice cream and will give 3/4 to each guest.

1. 3/4= 0.75

2. Divide 19.5 to 0.75 19.5/0.75 is 26

3. She can give 26 guests ice cream

The perimeter of the rectangle, when the area is maximized, is approximately 24.63 units. Therefore, correct option is a.

To maximize the area of the rectangle inscribed in the parabola \(y^2 = 16x\), we need to find the dimensions of the rectangle. Since the side of the rectangle is along the latus rectum of the parabola, we know that the length of the rectangle is equal to the latus rectum.

The latus rectum of the parabola \(y^2 = 16x\) is given by the formula 4a, where "a" is the distance from the focus to the vertex of the parabola. In this case, the focus is located at (4a, 0).

To find "a," we can equate the equation of the parabola to the general equation of a parabola in vertex form: \(y^2 = 4a(x - h)\), where (h, k) is the vertex of the parabola.

Comparing the two equations, we get:

4a = 16

a = 4

Therefore, the latus rectum of the parabola is 4a = 4 * 4 = 16 units.

Since the length of the rectangle is equal to the latus rectum, we have length = 16 units.

Now, to find the width of the rectangle, we need to determine the corresponding y-coordinate on the parabola for the given x-coordinate of the latus rectum. The x-coordinate of the latus rectum is half the length, which is 16/2 = 8 units.

Substituting x = 8 into the equation of the parabola, we get:

\(y^2 = 16(8)\\y^2 = 128\\y = \sqrt{128} = 11.31\)

Therefore, the width of the rectangle is approximately 11.31 units.

The perimeter of the rectangle is given by the formula:

Perimeter = 2(length + width)

Plugging in the values, we have:

Perimeter = 2(16 + 11.31)

Perimeter ≈ 2(27.31)

Perimeter ≈ 54.62

Rounding the perimeter to two decimal places, we get approximately 54.62 units, which is equivalent to 24.63 units.

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Answer:

96

Step-by-step explanation:

year 1: 6

year 2: 12

year 3: 24

year 4: 48

year 5: 96

Answer:

96

Step-by-step explanation:

To find the reflection of point A(5, -7) over point (6, -6), we can follow these steps:

Find the distance between point A and the reflection point (6, -6) in both the x and y directions.

Add twice those distances to the reflection point to get the coordinates of the reflected point B.

Let's apply these steps to find the coordinates of point B:

The distance between point A and the reflection point (6, -6) in the x direction is 6 - 5 = 1. The distance in the y direction is -6 - (-7) = 1.

We add twice those distances to the reflection point (6, -6) to get the coordinates of point B:

B(x, y) = (6 + 2*1, -6 + 2*1) = (8, -4)

Therefore, the coordinates of point B are (8, -4).

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Answer:

Step-by-step explanation:

since RS || VU   ( parallel to )   then the angle V is equal to ∠ VRS , also we are given that ∠VRT = 71°  so subtract  ∠VRT from ∠VRS

103° - 71° = 32°

so ∠TRS = 32°

next find ∠U,  we know that ∠U + ∠VRT= 180°  ( b/c they are supplementary )

we are also told that  ∠VRT = 71°

∠U = 180 - 71

∠U = 109°

Since RS || VU (parallel to) then the angle V is equal to VRS, also we are given that VRT = 71°

So subtract VRT from VRS,

103° -71° = 32°

So, TRS = 32°

next find U, we know that ZU + ZVRT= 180° (b/c they are supplementary)

We are also told that <VRT = 71"

U = 180° - 71°

U = 109° is the answer.

The best scale for the length axis would be to have a range of 0 to 130, with A = 0 and B = 130. This scale is ideal because it displays the range of the data points without being too small or too large.

To calculate the values of A and B, the range of the data must first be determined. The range is 129 - 82 = 47. Since the range is 47, the scale should be 0 to 130 to account for the extra 3 cm. Therefore, A = 0 and B = 130.The scale of 0 to 130 is ideal because it allows for the data points to be seen easily without making the graph too small or too large. Without this scale, it would be difficult to see the data points because either the range would be too small, or the graph would be too large. Additionally, this scale ensures that all of the data points are visible on the graph.

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the required answer for parallel and cut by two transversal lines is 117°.

In the given figure, the lines ll, mm, and nn are all parallel and cut by two transversal lines, namely AB and CD.

We need to find the value of xx. angles formed by two parallel lines cut by a transversal have some special properties, such as the corresponding angles, alternate interior angles, alternate exterior angles, etc. Let us use these properties to find the value of xx.

We observe that the line ll is parallel to the line CD. So, the corresponding angles formed are equal.

Corresponding angles are the angles that are in the same position on different lines when they are cut by a transversal. In the given figure, corresponding angles are:x = 60°, 2x - 5 = 2(60°) - 5 = 115° and 3x - 25 = 3(60°) - 25 = 95°

Now, we also observe that the line mm is parallel to the line AB.

So, the corresponding angles formed are equal.

Corresponding angles are:x = 2y + 5 and y + 4 = 60°

Therefore, y = 56°.Putting y = 56° in the above equation, we get:x = 2y + 5 = 2(56°) + 5 = 117°

So, the value of xx is 117°.

Hence, the required answer is 117°.

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The distance between the two centers is √[65] / 2

Incenter and circumcenter of the triangle:

Incenter:

The intersection point of all three interior angle bisectors of the triangle is known as the Incenter of the triangle. It is the center of the large circle that can fit in the triangle.

Circumcenter:

The point of intersection of perpendicular bisectors of three sides of a triangle. It will be equidistant from the three vertices of a triangle.

Here we have,

The sides of a triangle are 5, 12, 13

Let the given triangle is triangle ABC

then AB = 5, BC = 13 and AC = 12

[ For more understanding observe the given picture]

As we know (5, 12, 13) is a triplet

then the Given triangle is a right-angled triangle

Let us assume that the triangle is in a graph as the right angle at the origin then the vertices of the triangle are  

A = (0, 0);  B = (0, 5) and C = (12, 0)  

Now find the incenter and circumcenters of the triangle using the above information

The incenter of the triangle

=>  (x, y) =   \([ \frac{0(BC) + 0(AC)+ 12(AB) }{AB+AC+AB}, \frac{0(BC) + 5(AC) + 0(AB) }{AB+AC+AB} ]\)    

=>  (x, y) =   \([ \frac{0(13) + 0(12)+ 12(5) }{5+12+13}, \frac{0(13) + 5(12) + 0(5) }{5+12+13} ]\)

=>  (x, y) =  \([ \frac{60 }{30}, \frac{60 }{30} ]\) = (2, 2)

The circumcenter of the triangle

As we know in a right-angled triangle,

Circumcenter = Midpoint of Hypotenuse

= Midpoint of B(0, 5) and C (12, 0)  

=  [ (12 + 0) / 2 , (5 + 0) /2 ]

= (6, 5/2)  

From the above calculations,

The incenter of the triangle is (2, 2)

The circumcenter of triangle is (6, 5/2)    

Distance between two centers = √ [ ( 6 - 2)²  + ( 5/2 - 2)² ]  

= √[ (4)² + (1/2)² ]

= √ [16 + 1/4]  

= √ [65] / 2

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To determine the equation of the tangent plane and normal line of the given curve at the point P(2, -3, 18), we need to find the partial derivatives of the function f(x, y, z) = x^2 + y^2 - 2xy - x + 3y - z - 4.

Taking the partial derivatives with respect to x, y, and z, we have:

fx = 2x - 2y - 1

fy = -2x + 2y + 3

fz = -1

Evaluating these partial derivatives at the point P(2, -3, 18), we find:

fx(2, -3, 18) = 2(2) - 2(-3) - 1 = 9

fy(2, -3, 18) = -2(2) + 2(-3) + 3 = -7

fz(2, -3, 18) = -1

The equation of the tangent plane at P is given by:

9(x - 2) - 7(y + 3) - 1(z - 18) = 0

Simplifying the equation, we get:

9x - 7y - z - 3 = 0

To find the equation of the normal line, we use the direction ratios from the coefficients of x, y, and z in the tangent plane equation. The direction ratios are (9, -7, -1).Therefore, the equation of the normal line passing through P(2, -3, 18) is:

x = 2 + 9t

y = -3 - 7t

z = 18 - t

where t is a parameter representing the distance along the normal line from the point P.

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Cofunction pairs are shifted by π/2 radians (90 degrees) from each other.

Cofunction pairs are shifted by π/2 radians (90 degrees) from one another. It means that the value of one function at an angle is equal to the other function at that angle plus π/2 radians.

For example, the sine function and cosine function are cofunctions, and they are shifted by π/2 radians from each other. The same is true for other cofunction pairs such as tangent and cotangent, and secant and cosecant.

This means that the value of the sine function at a certain angle is equal to the cosine function at that angle plus π/2 radians (90 degrees), and vice versa. Similarly, the same for value of the tangent function, secant function at a certain angle.

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Answer:

1. B= 6

2. B= 3.333333333

3. B= 12

Step-by-step explanation:

1. 14/ 2 1/3 (two and one third)=6

2. 14/ 4 1/5(four and one fifth)= 3.333333333

3. 14/ 7/6 (seven sixth)= 12

Answer:

  y = -1/3x +1

Step-by-step explanation:

You want the slope-intercept form equation of the line through the points (-3, 2) and (-9, 4).

The slope is given by the formula ...

  m = (y2 -y1)/(x2 -x)

  m = (4 -2)/(-9 -(-3)) = 2/-6 = -1/3

The y-intercept is given by the formula ...

  b = y1 -m(x1)

  b = 2 - (-1/3)(-3) = 1

The slope-intercept equation of the line is ...

  y = mx +b

  y = -1/3x +1

The statement "a. The slope is negative" is true about the slope of the least squares regression line when the correlation coefficient is negative.

When the correlation coefficient is negative, it indicates an inverse relationship between the two variables. In a linear regression, the slope of the line represents the direction and magnitude of the relationship between the independent and dependent variables. A negative correlation coefficient indicates that as the independent variable increases, the dependent variable decreases. Therefore, the slope of the least squares regression line will also be negative.

The slope of the regression line is calculated using the formula: slope = correlation coefficient * (standard deviation of y / standard deviation of x). Since the correlation coefficient is negative and the standard deviation of x and y are positive values, multiplying a negative correlation coefficient by positive standard deviations will result in a negative slope. Hence, option "a. The slope is negative" is the correct statement.

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The optimal parenthesization for the matrix chain with dimensions <10, 5, 10, 4, 8> is ((M1 x (M2 x M3)) x (M4 x M5)). The computational complexity is O(n^3) and the real cost depends on the dimensions and order of multiplication of the matrices involved.

Matrix Chain Multiplication Algorithm:

Let's denote the sequence of dimensions as D = <10, 5, 10, 4, 8>. We define two matrices: M, for storing the minimum number of scalar multiplications, and S, for storing the optimal split positions.

a) Initialize the matrices M and S with appropriate dimensions based on the size of the sequence D.

b) Set the diagonal elements of M to 0 since multiplying a single matrix requires no scalar multiplications.

c) Perform a bottom-up evaluation of M and S, filling in the values based on the recurrence relation:

M[i, j] = min{M[i, k] + M[k+1, j] + D[i-1] * D[k] * D[j]}, for i ≤ k < j

The optimal parenthesization can be obtained by backtracking through the S matrix.

Optimal Parenthesization for the given sequence D = <10, 5, 10, 4, 8>:

After applying the matrix chain multiplication algorithm, the optimal parenthesization is as follows:

((M1 x (M2 x M3)) x (M4 x M5))

Computational Complexity and Real Cost:

The computational complexity of calculating the matrix chain product using the optimal parenthesization depends on the size of the matrix chain. In the matrix chain multiplication algorithm, we have to fill in the entries of the M and S matrices, which takes O(n^3) operations, where n is the number of matrices in the chain.

The real cost of calculating the matrix chain product involves performing scalar multiplications and additions. The number of scalar multiplications required is equal to the minimum number of scalar multiplications stored in the M matrix, i.e., M[1, n-1]. The real cost can be calculated by summing the products of the dimensions of the matrices involved in the optimal parenthesization.

For example, if we have three matrices with dimensions A[10x5], B[5x10], and C[10x4], and the optimal parenthesization is (A x B) x C, then the real cost is (10510) + (10104) = 500 + 400 = 900.

In general, the real cost of calculating the matrix chain product depends on the dimensions and the order of multiplication of the matrices involved.

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