Theorem: The product of every pair of even integers is even. Proof: 1. Suppose there are two even integers m an n whose sum is odd 2. m = 2k1, for some integer k₁ 3. n = 2k2, for some integer k2 4. m + n = 2k1, + 2k2 5. m + n = 2(k1, + K2), where k₁ + k2 is an integer 6. m +n is even, which is contradiction Which of the following best describe the contradiction in the above proof by contradiction? Lines 1 and 2 contradict line 1 Line 6 contradicts line 1 Line 6 contains the entire contradiction Line 4 contradicts line 1

Answer:

70 + 67 + 3x + 7 = 180

3x + 144 = 180

3x = 36

x = 12

Let A = [2 4 0 -3 -5 0 3 3 -2] Find an invertible matrix P and a diagonal matrix D such that D = P^-1 AP.In order to find the diagonal matrix D and the invertible matrix P such that D = P^-1 AP, we need to follow the following steps:

STEP 1: The first step is to find the eigenvalues of matrix A. We can find the eigenvalues of the matrix by solving the determinant of the matrix (A - λI) = 0. Here I is the identity matrix of order 3.

[tex](A - λI) = \begin{bmatrix} 2-λ & 4 & 0 \\ -3 & -5-λ & 0 \\ 3 & 3 & -2-λ \end{bmatrix}[/tex]

Let the determinant of the matrix (A - λI) be equal to zero, then:

[tex](2 - λ) [(-5 - λ)(-2 - λ) - 3.3] - 4 [(-3)(-2 - λ) - 3.3] + 0 [-3.3 - 3(-5 - λ)] = 0 (2 - λ)[λ^2 + 7λ + 6] - 4[6 + 3λ] = 0 2λ^3 - 9λ^2 - 4λ + 24 = 0[/tex] The cubic equation above has the roots [tex]λ1 = 4, λ2 = -2 and λ3 = 3[/tex].

STEP 2: The second step is to find the eigenvectors associated with each eigenvalue of matrix A. To find the eigenvector associated with each eigenvalue, we can substitute the eigenvalue into the equation

[tex](A - λI)x = 0 and solve for x. We have:(A - λ1I)x1 = 0 => \begin{bmatrix} 2-4 & 4 & 0 \\ -3 & -5-4 & 0 \\ 3 & 3 & -2-4 \end{bmatrix} x1 = 0 => \begin{bmatrix} -2 & 4 & 0 \\ -3 & -9 & 0 \\ 3 & 3 & -6 \end{bmatrix} x1 = 0 => x1 = \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}[/tex]

Let x1 be the eigenvector associated with the eigenvalue λ1 = 4.

STEP 3: The third step is to form the diagonal matrix D. To form the diagonal matrix D, we place the eigenvalues λ1, λ2 and λ3 along the main diagonal of the matrix and fill in the other entries with zeroes. [tex]D = \begin{bmatrix} 4 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 3 \end{bmatrix}[/tex]

STEP 4: The fourth and final step is to compute [tex]P^-1 AP = D[/tex].

We can compute [tex]P^-1[/tex] using the formula

[tex]P^-1 = adj(P)/det(P)[/tex] , where adj(P) is the adjugate of matrix P and det(P) is the determinant of matrix P.

[tex]adj(P) = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 1 & 2 \\ -2 & 0 & 2 \end{bmatrix} and det(P) = 4[/tex]

Simplifying, we get:

[tex]P^-1 AP = D = \begin{bmatrix} 4 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 3 \end{bmatrix}[/tex]

The invertible matrix P and diagonal matrix D such that [tex]D = P^-1[/tex]AP is given by:

P = [tex]\begin{bmatrix} 2 & -2 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix} and D = \begin{bmatrix} 4 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 3 \end{bmatrix}.[/tex]

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The last digit in the product of the given expression is 3.

Here, we have,

To find the last digit in the product of the given expression, we can observe a pattern in the last digit of powers of 3:

3¹ = 3 (last digit is 3)

3² = 9 (last digit is 9)

3³ = 27 (last digit is 7)

3⁴ = 81 (last digit is 1)

3⁵ = 243 (last digit is 3)

3⁶ = 729 (last digit is 9)

From the pattern, we can see that the last digit of the powers of 3 repeats every 4 powers.

So, if we calculate 3²⁰²¹, we can determine the last digit in the product.

3²⁰²¹ can be written as

(3⁴)⁵⁰⁵ × 3

= 1⁵⁰⁵ × 3

= 3.

Therefore, the last digit in the product of the given expression is 3.

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The solution to the given initial value problem dy/dt = -4y + 2e^3t, y(0) = 5 is y = e^(6t) + 4e^(4t).

To solve the given initial value problem (IVP) dy/dt = -4y + 2e^3t, y(0) = 5, we can use the method of integrating factors.

Write the differential equation in the form dy/dt + P(t)y = Q(t).
  In this case, P(t) = -4 and Q(t) = 2e^3t.

Determine the integrating factor (IF), denoted by μ(t).
  The integrating factor is given by μ(t) = e^(∫P(t)dt).
  Integrating P(t) = -4 with respect to t, we get ∫P(t)dt = -4t.
  Therefore, the integrating factor μ(t) = e^(-4t).

Multiply the given differential equation by the integrating factor μ(t).
  We have e^(-4t) * dy/dt + e^(-4t) * (-4y) = e^(-4t) * 2e^3t.

Simplify the equation and integrate both sides.
  The left-hand side simplifies to d/dt (e^(-4t) * y) = 2e^(-t + 3t).
  Integrating both sides, we get e^(-4t) * y = ∫2e^(-t + 3t)dt.
  Simplifying the right-hand side, we have e^(-4t) * y = 2∫e^(2t)dt.
  Integrating ∫e^(2t)dt, we get e^(-4t) * y = 2 * (1/2) * e^(2t) + C, where C is the constant of integration.

Solve for y by isolating it on one side of the equation.
  e^(-4t) * y = e^(2t) + C.
  Multiplying both sides by e^(4t), we have y = e^(6t) + Ce^(4t).

Apply the initial condition y(0) = 5 to find the value of the constant C.
  Substituting t = 0 and y = 5 into the equation, we get 5 = e^0 + Ce^0.
  Simplifying, we have 5 = 1 + C.
  Therefore, C = 5 - 1 = 4.

Substitute the value of C back into the equation for y.
  So, y = e^(6t) + 4e^(4t).

Therefore, the solution to the given initial value problem is y = e^(6t) + 4e^(4t).

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Answer:

29,400,000 = 2.94 × 10⁷

Starting at the far right (29400000.), move the decimal point 7 places to the left.

The probability that an item produced by this company is defective is 0.03 or 3%.

To find the probability that an item produced by this company is defective, we can use conditional probability. Let's break down the problem step by step:

Let's assume that the company has only one machine that produces 30% of the products.

Probability of selecting a product from this machine: P(Machine) = 0.3

Probability of a product being defective given it was produced by this machine: P(Defective | Machine) = 0.10

Now, we need to find the probability that any randomly selected item from the company is defective. We can use the law of total probability to calculate it.

Probability of selecting a defective item: P(Defective) = P(Machine) * P(Defective | Machine)

Substituting the values, we get:

P(Defective) = 0.3 * 0.10 = 0.03

Therefore, the probability that an item produced by this company is defective is 0.03 or 3%.

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Step-by-step explanation:

if you're calculating the area of that shape?

first, you calculate the area of triangle

Area of triangle =1/2(8-(-4))(9-5)=1/2(12)(4)=6×4=24

Area of rectangle =(8-(-4))(5-(-5))=(12)(10)=120

the total area will be 120+24=144

The series [tex]$\displaystyle \sum _{ n=17}^{\infty }\dfrac{ 8n\ln( n)}{ n+1}$[/tex] cannot be determined by the limit comparison test and requires another test for convergence.

The limit comparison test is inconclusive in this case. The limit comparison test is typically used to determine the convergence or divergence of a series by comparing it to a known series. However, in this case, it is not possible to find a known series that can be used for comparison. The series [tex]$\displaystyle \sum _{ n=17}^{\infty }\dfrac{ 8n\ln( n)}{ n+1}$[/tex] does not have a clear pattern or a simple known series to compare it with. Therefore, the limit comparison test cannot provide a definitive conclusion.

To determine the convergence or divergence of the series [tex]$\displaystyle \sum _{ n=17}^{\infty }\dfrac{ 8n\ln( n)}{ n+1}$[/tex], one must employ another convergence test. There are several convergence tests available, such as the integral test, ratio test, or root test, which can be applied to this series to determine its convergence or divergence. It is necessary to explore alternative methods to establish the convergence or divergence of this series since the limit comparison test does not yield a conclusive result.

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The table of data below shows the sales ($), cost of goods sold ($), and accounts receivable for the years 2017, 2018, 2019, 2020, and 2021. To compute trend percents for the above accounts, using 2017 as the base year.

For each of the three accounts, state the situation as revealed by the trend percents appears to be favorable or unfavorable. Here are the calculations:

Trend Percent for Net Sales: Numerator: / Denominator: / = Trend percent2021: (507222/126300) x 100 = 401%2020: (333699/126300) x 100 = 264%2019: (260702/126300) x 100 = 206%2018: (175557/126300) x 100 = 139%2017: (126300/126300) x 100 = 100%Is the trend percent for Net Sales favorable or unfavorable?

The trend percent for Net Sales is favorable since it is increasing over time. Trend Percent for Cost of Goods Sold: Numerator: / Denominator: / = Trend percent2021: (261133/64413) x 100 = 405%2020: (171736/64413) x 100 = 267%2019: (136208/64413) x 100 = 211%2018: (91284/64413) x 100 = 142%2017: (64413/64413) x 100 = 100% Is the trend percent for Cost of Goods Sold favorable or unfavorable?

The trend percent for Cost of Goods Sold is unfavorable since it is increasing over time.

Trend Percent for Accounts Receivable: Numerator: / Denominator: / = Trend percent2021: (24702/8664) x 100 = 285%2020: (19555/8664) x 100 = 225%2019: (17910/8664) x 100 = 207%2018: (10253/8664) x 100 = 118%2017: (8664/8664) x 100 = 100%

Is the trend percent for Accounts Receivable favorable or unfavorable? The trend percent for Accounts Receivable is unfavorable since it is increasing over time.

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The general solution of y" + y = cotx is cos⁡x+c_2sin⁡x-(ln|cos⁡x|+C)sin⁡x.

a) The general solution of y″+y=cot⁡x

We can find the general solution of y″+y=cot⁡x by finding the complementary solution of y″+y  and then apply the method of variation of parameters.

So, the complementary solution of y″+y=0 is given by

c = c_1cos⁡x+c_2sin⁡xwhere c1 and c2 are constants of integration.

Then the particular solution of y″+y=cot⁡x is given by

y_p = -(ln|cos⁡x|+C)sin⁡x

where C is the constant of integration.

The general solution of y″+y=cot⁡x is

y = y_c + y_p

= c_1

cos⁡x+c_2sin⁡x-(ln|cos⁡x|+C)sin⁡x

The above solution is in the form of implicit solution.

We cannot find the constants of integration until initial or boundary conditions are given.

b) Solve the given equation using the method of undetermined coefficients.

Here, the homogeneous equation is given byy″+4y=0and the characteristic equation is

r^2+4=0

r^2=-4r

=±2i

So, the complementary solution of y″+4y=0 is

y_c=c_1cos⁡(2t)+c_2sin⁡(2t)where c1 and c2 are constants of integration.

Now, we find the particular solution of y″+4y = 4cos⁡2tusing the method of undetermined coefficients.

Let's assume that the particular solution of

y″+4y = 4cos⁡2t is

y_p=Acos⁡(2t)+Bsin⁡(2t)

where A and B are constants.

Now,y_p'=−2Asin⁡(2t)+2Bcos⁡(2t)y_p''

=−4Acos⁡(2t)−4Bsin⁡(2t)

Therefore,y_p''+4y_p

=−4Acos⁡(2t)−4Bsin⁡(2t)+4Acos⁡(2t)+4Bsin⁡(2t)

=4(cos⁡2tA+sin⁡2tB)=4cos⁡2t

Let's compare the coefficients.

We have cos⁡2t coefficient equal to 4 and sin⁡2t coefficient equal to 0.

So, A=2 and B=0.

Substituting A=2 and B=0, the particular solution isy_p=2cos⁡(2t)

Therefore, the general solution of y″+4y=4cos⁡2t is given by

y=y_c+y_p

=c_1cos⁡(2t)+c_2sin⁡(2t)+2cos⁡(2t)

Simplifying this, we have

y= (c1+2)cos⁡(2t)+c2sin⁡(2t)

Therefore, the solution to the given differential equation with the initial conditions

y(0)=0 and

y′(0)=1 is

y = 2cos⁡(2t)−\dfrac{1}{2}sin⁡(2t)

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The graph of the inequality y > x² - 9 is given by the image presented at the end of the answer.

The inequality for this problem is given as follows:

y > x² - 9.

For the curve y = x² - 9, we have that:

Due to the > sign, the values greater than the inequality, that is, above the inequality, are shaded.

As the inequality does not have an equal sign, the parabola is dashed.

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The simplified expression of the division (4x⁵y³x * 3x³y²) / (6x⁴y¹⁰) is  

2x² / y⁵

To simplify the expression (4x⁵y³x * 3x³y²) / (6x⁴y¹⁰), we can combine the terms and simplify the coefficients and variables separately.

First, let's simplify the coefficients: 4 * 3 / 6 = 12 / 6 = 2.

Now, let's simplify the variables. For the variable x, we subtract the exponents when dividing: 5 + 1 - 4 = 2. For the variable y, we subtract the exponents: 3 + 2 - 10 = -5.

Therefore, the simplified expression is:

2x² * y⁻⁵

However, we can simplify the expression further by simplifying the negative exponent of y. Recall that y⁻⁵ is equivalent to 1/y⁵, indicating that y is in the denominator. So, we can rewrite the expression as:

2x² / y⁵

Hence, the simplified expression is 2x² / y⁵

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The simple interest made for 200 days is approximately 4.44%.

Given that the principal (P) is Php 25,000 and the accumulated amount (A) is Php 26,111.11, we need to find the rate (R) for 200 days of time (T).

Rearranging the formula, we have: Rate = (Simple Interest * 100) / (Principal * Time).

Substituting the given values, we have: Rate = ((26,111.11 - 25,000) * 100) / (25,000 * 200).

Simplifying the equation, we have: Rate = (1,111.11 * 100) / (25,000 * 200) = 4.44444%.

Converting the rate to a percentage, we have: Rate ≈ 4.44%.

Therefore, the simple interest made for 200 days is approximately 4.44%.

None of the options provided in the answer choices match the calculated simple interest, so there doesn't seem to be a suitable option available.

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The altitudes of a triangle are concurrent. This is known as the concurrency of altitudes in a triangle.

In Euclidean geometry, the altitudes of a triangle are lines drawn from each vertex of the triangle perpendicular to the opposite side. The main property of altitudes is that they are concurrent, meaning they intersect at a single point called the orthocenter.

To prove this, we can use various geometric methods such as triangle similarity, the properties of right angles, and the concept of perpendicularity. By considering each pair of altitudes, we can demonstrate that they intersect at a common point. This point, the orthocenter, is the unique intersection of the altitudes.

The concurrency of the altitudes is a fundamental property of triangles and has many implications in triangle geometry, such as the existence of orthocenters and the relationships between the sides and angles of a triangle.

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The roots of the equation are;

a. (n +2)(n -8)

b. (x-5)(x-3)

From the information given, we have the expressions as;

f(x) = n² - 6n - 16

Using the factorization method, we have to find the pair factors of the product of the constant and x square, we have;

a. n² -8n + 2n - 16

Group in pairs, we have;

n(n -8) + 2(n -8)

Then, we get;

(n +2)(n -8)

b. y = x² - 8x + 15

Using the factorization method, we have;

x² - 5x - 3x + 15

group in pairs, we have;

x(x -5) - 3(x - 5)

(x-5)(x-3)

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The exact value of tan(θ/2) given sinθ = -24/25 and 180° < θ < 270° is ±(4/3). This is obtained by applying the half-angle identity for tangent and finding the value of cosθ using the given value of sinθ.

To find the exact value of tan(θ/2) given sinθ = -24/25 and 180° < θ < 270°, we can use the half-angle identity for tangent. The half-angle identity for tangent is: tan(θ/2) = ±√((1 - cosθ) / (1 + cosθ))

First, we need to find the value of cosθ using the given value of sinθ. Since sinθ = -24/25, we can use the Pythagorean identity for sine and cosine: sin^2θ + cos^2θ = 1. Substituting sinθ = -24/25, we have: (-24/25)^2 + cos^2θ = 1

Simplifying the equation, we get:

576/625 + cos^2θ = 1

cos^2θ = 1 - 576/625

cos^2θ = 49/625

cosθ = ±√(49/625) = ±7/25. Since 180° < θ < 270°, we know that cosθ is negative. Therefore, cosθ = -7/25.

Now, substituting the value of cosθ into the half-angle identity for tangent, we get:

tan(θ/2) = ±√((1 - cosθ) / (1 + cosθ))

tan(θ/2) = ±√((1 - (-7/25)) / (1 + (-7/25)))

tan(θ/2) = ±(4/3). Therefore, the exact value of tan(θ/2) given sinθ = -24/25 and 180° < θ < 270° is ±(4/3).

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The solution to the initial value problem is y(x) = e^x. The maximum domain of the solution is (-∞, ∞).

To solve the initial value problem, we start by rearranging the given differential equation: ye^(-xy') = -x. Next, we differentiate both sides of the equation with respect to x using the chain rule. The derivative of ye^(-xy') with respect to x is y'e^(-xy') - xye^(-xy')y''.

Plugging these values back into the original equation, we get y'e^(-xy') - xye^(-xy')y'' = -x. Simplifying further, we divide through by e^(-xy') to obtain y' - xy'' = -xe^(xy').

We now have a linear homogeneous second-order differential equation. To solve it, we assume a power series solution of the form y = ∑(n=0 to ∞) a_nx^n. Substituting this series into the equation and equating the coefficients of like powers of x, we find that the coefficients satisfy the recurrence relation a_n = (n+1)a_(n+2).

Since the equation is homogeneous, it implies that the coefficient a_0 must be nonzero for nontrivial solutions. By solving the recurrence relation, we find that all coefficients a_n are proportional to a_0.

Therefore, the general solution to the differential equation is y(x) = a_0e^x. To determine the value of a_0, we substitute the initial condition y(0) = 1 into the general solution, giving a_0e^0 = 1. Thus, a_0 = 1.

Hence, the solution to the initial value problem is y(x) = e^x, and its maximum domain is (-∞, ∞).

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The given equation is 1 / (b+1) + 1 / (b-1) = 2 / (b² - 1) and it has no solutions.

To solve this equation, we'll start by finding a common denominator for the fractions on the left-hand side. The common denominator for (b+1) and (b-1) is (b+1)(b-1), which is also equal to b² - 1 (using the difference of squares identity).

Multiplying the entire equation by (b+1)(b-1) yields (b-1) + (b+1) = 2.

Simplifying the equation further, we combine like terms: 2b = 2.

Dividing both sides by 2, we get b = 1.

To check if this solution is valid, we substitute b = 1 back into the original equation:

1 / (1+1) + 1 / (1-1) = 2 / (1² - 1)

1 / 2 + 1 / 0 = 2 / 0

Here, we encounter a problem because division by zero is undefined. Hence, b = 1 is not a valid solution for this equation.

Therefore, there are no solutions to the given equation.

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Answer:

The correct answer is:

B. The equation that represents this situation is 3x = 21. The second number is 7.

Since the product of two numbers is 21 and the first number is given as -3, we can represent this situation using the equation 3x = 21. Solving for x, we find that x = 7. Therefore, the second number is 7.

Step-by-step explanation:

To accurately construct triangle ABC using the given information, follow these steps:

Draw a line segment AB of length 7 cm.

Place the compass at point A and draw an arc with a radius of 4 cm, intersecting the line segment AB. Label this intersection point as C.

Without changing the compass width, place the compass at point C and draw another arc intersecting the previous arc. Label this intersection point as D.

Connect points A and D to form the line segment AD.

Using a protractor, measure and draw an angle of 80° at point A, with AD as one of the rays. Label the intersection point of the angle and the line segment AD as B.

Draw the line segments BC and AC to complete the triangle ABC.

To measure the size of angle ACB to the nearest degree, use a protractor and align the baseline of the protractor with the line segment BC. Read the degree measure where the other ray of angle ACB intersects the protractor.

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The mean value of the radius (r) at which you would find the electron, given the H atom wave function is 100(r), is 0.

The wave function of an electron in the hydrogen atom, denoted by Ψ, describes the probability distribution of finding the electron at different positions around the nucleus. In this case, the given wave function is 100(r), where r represents the radius.

To calculate the mean value of the radius, we need to evaluate the integral of r multiplied by the absolute square of the wave function, integrated over all possible values of r. However, the wave function 100(r) does not provide a valid description of the hydrogen atom's electron distribution. The wave function should be normalized, meaning that the integral of the absolute square of the wave function over all space should equal 1. In this case, the given wave function lacks normalization.

Since the wave function is not properly normalized, we cannot accurately calculate the mean value of the radius. Without normalization, the probability distribution described by the wave function does not provide meaningful information about the electron's position.

In summary, based on the given wave function, the mean value of the radius cannot be determined without proper normalization of the wave function. A properly normalized wave function is necessary to obtain accurate information about the electron's position.

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By using the given information and properties of lines, we can prove that AQ + BR + CS = 30P.

In order to prove the equation AQ + BR + CS = 30P, we need to utilize the given information that OA + OB + OC = 0 and PQ + PR + PS = 0.

Let's consider the points A, B, C, P, Q, R, and S that lie on the same line. The equation OA + OB + OC = 0 implies that the sum of the distances from point O to points A, B, and C is zero. Similarly, the equation PQ + PR + PS = 0 indicates that the sum of the distances from point P to points Q, R, and S is zero.

Now, let's examine the expression AQ + BR + CS. We can rewrite AQ as (OA - OQ), BR as (OB - OR), and CS as (OC - OS). By substituting these values, we get (OA - OQ) + (OB - OR) + (OC - OS).

Considering the equations OA + OB + OC = 0 and PQ + PR + PS = 0, we can rearrange the terms and rewrite them as OA = -(OB + OC) and PQ = -(PR + PS). Substituting these values into the expression, we have (-(OB + OC) - OQ) + (OB - OR) + (OC - OS).

Simplifying further, we get -OB - OC - OQ + OB - OR + OC - OS. By rearranging the terms, we have -OQ - OR - OS.

Since PQ + PR + PS = 0, we can rewrite it as -OQ - OR - OS = 0. Therefore, AQ + BR + CS = 30P is proven.

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The probability that a worker selected at random makes between $350 and $450 is given as follows:

68%.

The Empirical Rule states that, for a normally distributed random variable, the symmetric distribution of scores is presented as follows:

350 and 450 are within one standard deviation of the mean of $400, hence the probability is given as follows:

68%.

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The probability that a worker selected at random makes between $350 and $450 is approximately 0.6827.

To calculate this probability, we need to use the concept of the standard normal distribution. Firstly, we convert the given values into z-scores, which measure the number of standard deviations an individual value is from the mean.

To find the z-score for $350, we subtract the mean ($400) from $350 and divide the result by the standard deviation ($50). The z-score is -1.

Next, we find the z-score for $450. By following the same process, we obtain a z-score of +1.

We then use a z-table or a calculator to find the area under the standard normal curve between these two z-scores. The area between -1 and +1 is approximately 0.6827, which represents the probability that a worker selected at random makes between $350 and $450.

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Using Fermat's Little Theorem  83^98 mod 13 is 2.

Fermat's Little Theorem states that if p is a prime number, and a is a positive integer less than p, then a^(p−1) ≡ 1 mod p. Now we can use this theorem to compute 83^98 mod 13.

a = 83 and p = 13

Since 83 is not divisible by 13, we can use Fermat's Little Theorem. Here, we have to find the exponent (p-1), which is 12 because 13-1=12.Therefore, we can use a^(p-1) ≡ 1 mod p to simplify the expression:

83^(12) ≡ 1 mod 13

Now we can use this equivalence to find the remainder when 83^98 is divided by 13.83^(12) = 1 mod 1383^96 = (83^12)^8 = 1^8 = 1 mod 1383^98 = 83^96 * 83^2 = 1 * 83^2 mod 13

Now, we need to calculate the remainder when 83^2 is divided by 13.83^2 = 6889 = 13 * 529 + 2

Hence, 83^98 ≡ 83^2 ≡ 2 mod 13.

Therefore, 83^98 mod 13 is 2.

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The value for s is 7.

Equations are mathematical statements containing two algebraic expressions on both sides of an 'equal to (=)' sign. It shows the relationship of equality between the expression written on the left side with the expression written on the right side.

Given:

Rearrange unknown terms to the left side of the equation:

[tex]\sf 5s-3s=9+5[/tex]

Combine like terms:

[tex]\sf 2s=9+5[/tex]

Calculate the sum or difference:

[tex]\sf 2s=14[/tex]

Divide both sides of the equation by the coefficient of variable:

[tex]\sf s=\dfrac{14}{2}[/tex]

[tex]\rightarrow \bold{s=7}[/tex]

Hence, the value for s is 7.

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The line y = k, where k is a constant, does not have an inverse.

For a function to have an inverse, it must pass the horizontal line test, which means that every horizontal line intersects the graph of the function at most once. However, for the line y = k, every point on the line has the same y-coordinate, which means that multiple x-values will map to the same y-value.

Since there are multiple x-values that correspond to the same y-value, the line y = k fails the horizontal line test, and therefore, it does not have an inverse.

In other words, if we were to attempt to solve for x as a function of y, we would have multiple possible x-values for a given y-value on the line. This violates the one-to-one correspondence required for an inverse function.

Hence, the line y = k, where k is a constant, does not have an inverse.

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The unit vector normal to the surface is (√3/3, √3/3, √3/3)

a unit vector normal to the surface defined by the parametric equations x = 7cos(θ)sin(4), y = 5sin(θ)sin(4), and z = cos(4), we need to calculate the gradient vector of the surface and then normalize it to obtain a unit vector.

The gradient vector of a surface is given by (∂f/∂x, ∂f/∂y, ∂f/∂z), where f(x, y, z) is an implicit equation of the surface. In this case, we can consider the equation f(x, y, z) = x - 7cos(θ)sin(4) + y - 5sin(θ)sin(4) + z - cos(4) = 0, as it represents the equation of the surface.

Taking the partial derivatives, we have:

∂f/∂x = 1

∂f/∂y = 1

∂f/∂z = 1

Therefore, the gradient vector is (1, 1, 1).

To obtain a unit vector, we need to normalize the gradient vector. The magnitude of the gradient vector is given by:

|∇f| = √(1^2 + 1^2 + 1^2) = √3.

Dividing the gradient vector by its magnitude, we have:

n = (1/√3, 1/√3, 1/√3).

Simplifying the expression, we get:

n = (√3/3, √3/3, √3/3).

Therefore, the unit vector normal to the surface is (√3/3, √3/3, √3/3).

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The distance between two different nodes of a complete bipartite graph Km,n is (e) between 1 and n+m-1, depending on the nodes.

In a complete bipartite graph Km,n, the nodes are divided into two distinct sets, one with m nodes and the other with n nodes. Each node from the first set is connected to every node in the second set, resulting in a total of m*n edges in the graph.

To find the distance between two different nodes in Km,n, we need to consider the shortest path between them. Since every node in one set is connected to every node in the other set, there are multiple paths that can be taken.

The shortest path between two nodes can be achieved by traversing directly from one node to the other, which requires a single edge. Therefore, the minimum distance between any two different nodes in Km,n is 1.

However, if we consider the maximum distance between two different nodes, it would involve traversing through all the nodes in one set and then all the nodes in the other set, resulting in a path with n+m-1 edges. Therefore, the maximum distance between any two different nodes in Km,n is n+m-1.

In conclusion, the distance between two different nodes in a complete bipartite graph Km,n is between 1 and n+m-1, depending on the specific nodes being considered.

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