Solve the two simultaneous equations. You must show all your working. 3t+2p=15. 5
5t+4p=28. 5

So for the population which is normally distributed using central limit theorem we get,

(A) For sample size n = 100, Pr(Ӯ <101) = 0.937.

(B) For random sample size n = 64, Pr(101< Ӯ <103) = 0.110.

(C) For sample size of n = 165, Pr(Ӯ >98) = 0.893

Given that the population mean (µY) = 100 and the population standard deviation σ2Y = 43.

For the sample size of n = 100.

Now the sample mean (Ӯ) = µY = 100 also.

The sample standard deviation (σ) = square root of (σ2Y/n) = square root of (43/100) = square root of (0.43) = 0.66 (Rounding off  to two decimal places).

Now, (101 - Ӯ)/σ = (101 - 100)/0.66 = 1.53

From the normal distribution table we can find P(Z < 1.53) = 0.937 (approximately).

Hence, Pr(Ӯ < 101) = 0.937 (approximately).

For random sample size n = 64.

sample standard deviation = (σ) = square root of (σ2Y/n) = square root of (43/64) = 0.82

Now,

(101 - 100)/0.82 = 1.22

(103 - 100)/0.82 = 3.66

From standard normal table we can get,

P(Z < 3.66) = 0.999

P(Z < 1.22) = 0.889

So, P(1.22 < z < 3.66) = 0.999 - 0.889 = 0.110

Hence, Pr(101< Ӯ <103) = 0.110.

Again for sample size of n = 165.

Sample standard deviation = square root of (σ2Y/n) = square root of (43/165) = 0.51

Now,

(98 - 100)/0.51 = - 1.24

From the standard normal distribution table we get, P(Z < - 1.24) = 0.107.

Now, Pr(Ӯ >98) = 1 - Pr(Ӯ < 98) = 1 - 0.107 = 0.893.

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The expression which is equivalent to a given expression 24 + 18 is given by option a. 6 ( 4 + 3 ).

The expression is equal to,

24 + 18

verification of equivalent expression is as follow,

6 ( 4 + 3 )

Using the distributive law multiplication over addition is ,

A (B + C ) = AB + AC

Apply it on 6 ( 4 + 3 ) we have

= 6 × 4 + 6 × 3

= 24 + 18

It is correct option and equivalent to 24 + 18.

6 ( 4 + 4 )

Using the distributive law multiplication over addition is ,

A (B + C ) = AB + AC

Apply it on 6 ( 4 + 4 ) we have

= 6 × 4 + 6 × 4

= 24 + 24

It is not correct option and not equivalent to 24 + 18.

2 ( 22 + 9 )

Using the distributive law multiplication over addition is ,

A (B + C ) = AB + AC

Apply it on 2 ( 22 + 9 ) we have

= 2 × 22 + 2 × 9

= 44 + 18

It is not correct option and not equivalent to 24 + 18.

6 ( 4 + 12 )

Using the distributive law multiplication over addition is ,

A (B + C ) = AB + AC

Apply it on 6 ( 4 + 12 ) we have

= 6 × 4 + 6 × 12

= 24 + 72

It is not correct option and not equivalent to 24 + 18.

Therefore, the equivalent expression of 24 + 18 is equal to option a. 6 ( 4 + 3 ).

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This statement is false. Total variability is the calculated by dividing the within-group variance by the between-group variance.

The total variability, also known as the total sum of squares (TSS), is the total variation in the response variable of a dataset. The total of the squared variances of each data point from the overall mean is used to calculate it.

The within-group variance, also known as the residual sum of squares (RSS), measures the variability of the data within each group or category. It is calculated as the sum of the squared deviations of each data point from its group mean.

The between-group variance, also known as the explained sum of squares (ESS), measures the variability of the data between the groups or categories. It is calculated as the sum of the squared deviations of each group mean from the overall mean.

To calculate the total variability or TSS, we add the between-group variance or ESS to the within-group variance or RSS. The formula for TSS is:

TSS = ESS + RSS

Therefore, the statement "Total variability is calculated by dividing the within-group variance by the between-group variance" is not accurate.

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As x approaches infinity, the limit of function [(2x-1)(3-x)]/[(x-1)(x+3)] is equal to -2.

What is function?

In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the codomain) with the property that each input is related to exactly one output.

To find the limit of [(2x-1)(3-x)]/[(x-1)(x+3)] as x approaches infinity, we need to consider the highest power of x in the numerator and the denominator.

In the numerator, the highest power of x is [tex]x^2[/tex], which comes from the product of (2x)(3). In the denominator, the highest power of x is also [tex]x^2[/tex], which comes from the product of (x)(x).

Thus, we can use the rule that when the highest powers of x in the numerator and denominator are equal, the limit is the ratio of the coefficients of these highest powers. Therefore:

lim [(2x-1)(3-x)]/[(x-1)(x+3)]

= lim [([tex]-2x^2[/tex] + 7x - 3)/([tex]x^2[/tex] + 2x - 3)]

= lim [-2 + (7/x) - (3/[tex]x^2[/tex])] / [1 + (2/x) - (3/[tex]x^2[/tex])]

= -2/1

= -2

Therefore, as x approaches infinity, the limit of [(2x-1)(3-x)]/[(x-1)(x+3)] is equal to -2.

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The correct answer is either (a) if the p-value is less than 0.05, or (b) if the p-value is greater than 0.05.

What is the equivalent expression?

Equivalent expressions are expressions that perform the same function despite their appearance. If two algebraic expressions are equivalent, they have the same value when we use the same variable value.

Without knowing the specific p-value for the relationship between sales rep age and customer satisfaction rating, it is not possible to determine the correct answer to this question.

In general, a p-value less than 0.05 indicates that there is a statistically significant relationship between the predictor variable and the outcome variable.

However, it is important to interpret the p-value in the context of the specific analysis and research question.

If the p-value for the relationship between sales rep age and customer satisfaction rating is greater than 0.05, it would suggest that the relationship is not statistically significant and may be due to chance.

However, if the p-value is less than 0.05, it would suggest that there is a statistically significant relationship between sales rep age and customer satisfaction rating, and the relationship is not likely due to chance.

Therefore, the correct answer is either (a) if the p-value is less than 0.05, or (b) if the p-value is greater than 0.05.

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The correct answer is “This is a right-tailed test because a direction is specified. The population parameter is greater than the specified value.”

What is hypothesis testing?

In hypothesis testing, we test the null hypothesis against the alternative hypothesis. The null hypothesis (H0) is the default assumption, which states that there is no significant difference between the sample data and the population parameter. The alternative hypothesis (H1) contradicts the null hypothesis and suggests that there is a significant difference between the sample data and the population parameter.

In this scenario, the null hypothesis would be that the average number of keywords per web page is less than or equal to 10, and the alternative hypothesis would be that the average number of keywords per web page is greater than 10.

Since the alternative hypothesis specifies a direction, it is a one-tailed or one-directional test. Moreover, as the alternative hypothesis states that the average number of keywords per web page is greater than 10, the critical region is in the right tail of the distribution. Therefore, this is a right-tailed test.

So, the correct answer is: This is a right-tailed test because a direction is specified. The population parameter is greater than the specified value.

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Answer: We can use the formula for sample size calculation for estimating a population proportion:

n = (z^2 * p * (1 - p)) / E^2

where:

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

p = the estimated proportion from the population (0.13 in this case)

E = the desired margin of error (0.05 in this case)

Substituting the given values, we get:

n = (z^2 * p * (1 - p)) / E^2

n = (2.326^2 * 0.13 * (1 - 0.13)) / 0.05^2

n ≈ 319.8

We need a sample size of at least 320 to be 98% confident that the estimated proportion of women-owned businesses providing profit-sharing and/or stock options is within 5 percentage points of the true population proportion.

(b) 734.0 ft-lbs of work is done.

To find the work done by the force, we need to use the formula:

Work = force x distance x cos(θ)

where:

force = 56 pounds (the given constant force)

distance = 16 feet (the distance the door is being pulled)

θ = 35 degrees (the angle between the force and the displacement of the door)

We need to convert the angle to radians to use it in the formula:

theta = 35 degrees x (pi/180) = 0.6109 radians

Now we can substitute the values into the formula:

Work = 56 pounds x 16 feet x cos(0.6109 radians)

Work = 734.0 ft-lbs (rounded to one decimal place)

Therefore, the answer is (b) 734.0 ft-lbs.

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Vertex: (1/5, -5), Axis of Symmetry: x = 1/5, Transformations: vertical stretch by 10, horizontal shift 1/5 right, and vertical shift 7 down.

Let's analyze the given function, Y = |10x - 2| - 7, and compare it to the parent function, Y = |x|.

Vertex: The vertex of the given function can be found by determining the x-coordinate that will make the expression inside the absolute value equal to zero. In this case, 10x - 2 = 0. Solving for x, we get x = 1/5. The corresponding y-coordinate is found by substituting x back into the function: Y = |10(1/5) - 2| - 7, which simplifies to Y = |-2| - 7 = 2 - 7 = -5. Thus, the vertex is at the point (1/5, -5).

Axis of Symmetry: Since the given function is an absolute value function, the axis of symmetry is vertical and goes through the x-coordinate of the vertex. In this case, the axis of symmetry is x = 1/5.

Transformation: Comparing the given function to the parent function, there are a few transformations applied:
1. Vertical stretch by a factor of 10 (due to the "10x" term)
2. Horizontal shift of 1/5 units to the right (due to the "-2" term inside the absolute value)
3. Vertical shift of 7 units down (due to the "-7" term outside the absolute value)

So, to summarize:
- Vertex: (1/5, -5)
- Axis of Symmetry: x = 1/5
- Transformations: Vertical stretch by 10, horizontal shift 1/5 right, and vertical shift 7 down.

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Complete Question

Compare the function with the parent function. Without graphing, what are the vertex, axis of symmetry, and transformation of the given function? Y=|10x-2|-7

The president recommends legislation to Congress shows how branches of government interact with each other. The correct answer is C.

The interaction between the executive and legislative branches of government is essential in the process of making laws in the United States. The president has the power to recommend legislation to Congress, but Congress ultimately decides whether or not to pass the bill into law.

This interaction is an example of the system of checks and balances in the US government, which ensures that no one branch becomes too powerful.

Option A is an example of the judicial branch acting alone, as the Supreme Court has the power to interpret the law and make decisions on cases. Option B is an example of the legislative branch acting alone, as the House of Representatives has the power to draft and pass bills into law.

Option D is an example of the president acting alone in foreign affairs, as the president has the power to negotiate and sign treaties with foreign countries. While these actions may have an impact on the other branches, they do not represent an example of direct interaction between branches.

The correct answer is C.

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For a normal random variable, the probability of an observation being less than the median is 0.5 or 50%.

This is because the median is the middle value in a set of data, and for a normal distribution, the probability of being below or above the median is equal. Therefore, half of the observations will be below the median and half will be above.

For a normal random variable, the probability of an observation being less than the median is 0.5 or 50%. This is because, in a normal distribution, the median is the value that divides the distribution into two equal halves, with 50% of the observations falling below it and 50% above it.

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The surface area of the cone with radius 4 inches and height 6.5 inches is equal to 146.07 square inches.

Radius of the cone = 4 inches

height of the cone = 6.5 inches

Let us consider 'r' be the radius of the cone and 'h' be the height of the cone.

Formula to calculate surface area of the cone

= πr ( r  + √ h² + r² )

Substitute the value of radius and height of the cone we have,

⇒ Surface area of the cone = π × 4 ( 4 + √ ( 6.5 )² + ( 4 )² )

⇒ Surface area of the cone =4π ( 4 + √58.25 )

⇒ Surface area of the cone = 4 × 3.14 ( 4 + 7.63 )

⇒ Surface area of the cone =  12.56 × 11.63

⇒ Surface area of the cone = 146.0728 square inches

⇒ Surface area of the cone = 146.07 in²

Therefore, the surface area of the cone is equal to 146.07 square inches.

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The solution to the equation is t = log(146)/8, and the approximate value of t is 0.270.

To solve the equation 0.5 *[tex]10^{8t}[/tex] = 73 for t, we can first simplify the left side of the equation by dividing both sides by 0.5 * 10^8:

[tex]10^{8t}[/tex] = 146

Next, we can take the logarithm of both sides of the equation using base 10:

[tex]log(10^{8t}) = log(146)[/tex]

Using the property of logarithms that says [tex]log(a^{b} ) = b*log(a)[/tex], we can simplify the left side of the equation:

8t * log(10) = log(146)

Since log(10) = 1, we can further simplify the equation:

8t = log(146)

Finally, we can solve for t by dividing both sides by 8:

t = log(146)/8

t = 2.164/8

The approximate the value of t as:

t ≈ 0.27054410

Therefore, the solution to the equation is t = log(146)/8, and the approximate value of t is 0.270.

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The GCF of the number is 12 and the LCM of the number is 24.

Let's start by finding the greatest common factor (GCF) of two whole numbers less than or equal to 100. The GCF is the largest number that divides both of the given numbers without leaving any remainder. We can start by listing all the factors of each number and finding the largest one they have in common.

Let's say we have the numbers 60 and 72. We can find their factors as follows:

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

From this list, we can see that the largest factor that 60 and 72 have in common is 12. Therefore, the GCF of 60 and 72 is 12.

Let's say we have the numbers 6 and 8. We can list their multiples as follows:

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96, 102

Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104

From this list, we can see that the smallest multiple that both 6 and 8 share is 24. Therefore, the LCM of 6 and 8 is 24.

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Complete Question:

Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12.

The frequency table for the set of data is given below:

A frequency table is a crafted arrangement and visual representation of data in a tabular shape, to show the number of times each distinct value or classification within a dataset appears.

It contains all possible values or classes of a variable in one category, along with the respective number of episodes or occurrences of each one in a second section.

Here is an example of how you could sort this data.

Interval:     Frequency:

0 - 1                  7

2 - 3                  7

4 - 5                  1

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The matrix exponential M(t) = exp(t*A) are: M11(t) = exp(1*t)*(-1/2)*(-1/2) + exp(2*t)*(1/2)*(1/2) = (1/4)*exp(t) + (1/2)*exp(2*t)
M12(t) = exp(1*t)*(-1/2)*(1) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) - exp(t))
M21(t) = exp(1*t)*(1)*(1/2) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) + 1)
M22(t) = exp(1*t)*(1)*(1) + exp(2*t)*(1)*(1) = exp(t) + exp(2*t)

To find the matrix exponential M(t) = exp(t*A), we first need to find the eigenvectors of A corresponding to the eigenvalues X1 = 1 and X2 = 2.

For X1 = 1, we solve the equation (A - I)*v = 0, where I is the identity matrix:

(A - I)*v = (1 1; 2 2 - 1)*v = 0
RREF([A - I, zeros(2,1)])
ans =
   0    0   -1
   0    0    0

So we have the equation -v2 = 0, which means v2 can be any non-zero value. Let's choose v2 = 1, then v1 = -1/2. So the eigenvector corresponding to X1 is v1 = (-1/2; 1).

For X2 = 2, we solve the equation (A - 2*I)*v = 0:

(A - 2*I)*v = (-1 1; 2 -2)*v = 0
RREF([A - 2*I, zeros(2,1)])
ans =
   0    0   -1
   0    0    0

So we have the equation -v2 = 0, which means v2 can be any non-zero value. Let's choose v2 = 1, then v1 = 1/2. So the eigenvector corresponding to X2 is v2 = (1/2; 1).

Now we can construct the matrix exponential M(t) = exp(t*A) using the formula:

M(t) = c1*exp(X1*t)*v1*v1' + c2*exp(X2*t)*v2*v2'

where c1 and c2 are constants determined by the initial conditions. Since we don't have any initial conditions given, we can choose c1 = 1 and c2 = 0 for simplicity.

So we have:

M11(t) = exp(1*t)*(-1/2)*(-1/2) + exp(2*t)*(1/2)*(1/2) = (1/4)*exp(t) + (1/2)*exp(2*t)
M12(t) = exp(1*t)*(-1/2)*(1) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) - exp(t))
M21(t) = exp(1*t)*(1)*(1/2) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) + 1)
M22(t) = exp(1*t)*(1)*(1) + exp(2*t)*(1)*(1) = exp(t) + exp(2*t)

So the matrix exponential M(t) is:

M(t) = ( (1/4)*exp(t) + (1/2)*exp(2*t)    (1/2)*(exp(2*t) - exp(t));
       (1/2)*(exp(2*t) + 1)              exp(t) + exp(2*t) )

To find the matrix exponential M(t) = exp(tA) given that the eigenvalues of matrix A are λ1 = 1 and λ2 = 2, we first need to find the eigenvectors corresponding to each eigenvalue, and then form the matrix P of eigenvectors and the diagonal matrix D of eigenvalues. Finally, we can compute M(t) using the formula:

M(t) = P * exp(tD) * P^(-1)

After finding the eigenvectors and forming the matrices P and D, compute exp(tD) by taking the exponentiation of each diagonal element:

exp(tD) = | exp(tλ1)   0      |
              | 0          exp(tλ2) |

Now, compute M(t) by multiplying P, exp(tD), and the inverse of P. The resulting matrix M(t) will have the following components:

M11(t) = exp(1*t)*(-1/2)*(-1/2) + exp(2*t)*(1/2)*(1/2) = (1/4)*exp(t) + (1/2)*exp(2*t)
M12(t) = exp(1*t)*(-1/2)*(1) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) - exp(t))
M21(t) = exp(1*t)*(1)*(1/2) + exp(2*t)*(1/2)*(1) = (1/2)*(exp(2*t) + 1)
M22(t) = exp(1*t)*(1)*(1) + exp(2*t)*(1)*(1) = exp(t) + exp(2*t)

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To organize the results in a two-way table, we can create a table with rows for Spanish class (Yes/No) and columns for French class (Yes/No). The two-way table is shown below.

The intersection of each row and column will show the number of students who have taken both classes, only Spanish, only French, or neither.

Using the given information, we can fill in the table as follows:

        French Class No French Class Total

Spanish            45                           64                    109

No                     0                             82                    82

Total                 45                           146                   245

The marginal frequencies are included in the last row and column of the table. The marginal frequency for the Spanish class is 109 (45 + 64) and for the French class is 45 (45 + 0). The marginal frequency for students who have not taken either class is 82.

This table provides a clear visual representation of the survey results and allows for easy comparison between the number of students who have taken each class or neither. The information in this table could be useful for making decisions about language class offerings or analyzing student language learning trends.

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Answer:

  C.  CD

Step-by-step explanation:

You want the type of savings vehicle that offers the highest interest rate, possibly with a requirement the deposit be for a specific period.

As of today, my local savings institution offers these choices:

The highest interest rate is for a CD, choice C, which requires the money stay deposited for a specific time.

__

Additional comment

This institution also offers an interest rate of 0.10% on checking account deposits, with no monthly fees. Rates vary with the institution and over time. You will likely find different rates and/or charges if you explore the marketplace.

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The value of expression "4x⁵ + e²ˣ + 6" at x=0 is 7 and at x=1 is 10 + e².

In mathematics, an expression is a combination of symbols and numbers that represents a value.

To find the value of the expression "4x⁵ + e²ˣ + 6" at x=0 and x=1, we simply substitute 0 and 1 for "x" and simplify:

When x=0:

We have : 4x⁵ + e²ˣ + 6 ⇒  4(0)⁵ + e⁰ + 6 = 1 + 6 = 7

So, the value of the expression at x=0 is 7.

When x=1:

we have : 4x⁵ + e²ˣ + 6 ⇒ 4(1)⁵ + e² + 6

⇒ 4 + e² + 6,

⇒ 10 + e²,

So, the value of the expression at x=1 is 10 + e².

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The given question is incomplete, the complete question is

Find the value of the expression at "4x⁵ + e²ˣ + 6" at x=0 and x=1.

The probability for the event of your coin is a dime and your friend's coin is a quarter is P = 1/12.

We assume that all the coins have the same probability of being randomly drawn.

Then the probability of drawing a dime is equal to the quotient between the number of dimes and the total number of coins, here we will get:

p  = 10/16

Now the total number of coins is 15, because you take one, now the probability that your friend takes a quarter is:

q = 2/15

The joint probability (for the two events happening one after the other) is equal to the product of the individual ones, so we will get:

P  = p*q = (10/16)*(2/15) = (5/8)*(2/15) = 1/12

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The correct answer is (D) AB is 4 x 4, BA is 2 x 2.

What is order of a matrix?

The order of a matrix refers to the number of rows and columns in the matrix. If a matrix has m rows and n columns, we say that it is an m x n matrix. The order of the matrix is written as "m x n".

In general, if A is an m x n matrix and B is an n x p matrix, then the product AB is an m x p matrix and the product BA is an n x n matrix.

In this case, A is a 4 x 2 matrix and B is a 2 x 4 matrix. Therefore, the product AB is a 4 x 4 matrix, and the product BA is a 2 x 2 matrix.

So the answer is (D) AB is 4 x 4, BA is 2 x 2.

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An equation that relates the lengths of the sides of the triangle is (2 + y)² + y² = 15².

The dimensions of this triangle are 9.56 cm by 11.56 cm by 15 cm.

In Mathematics and Geometry, Pythagorean's theorem is modeled or represented by the following mathematical equation (formula):

x² + y² = z²

Where:

x, y, and z represents the length of sides or side lengths of any right-angled triangle.

Based on the information provided about the side lengths of this right-angled triangle (one leg is 2 feet longer than the other leg), we have the following equation:

x = 2 + y

By substituting the side lengths and solving the quadratic equation, we have:

x² + y² = z²

(2 + y)² + y² = 15²

4 + 4y + y² + y² = 225

2y² + 4y - 221 = 0

y = 9.56 cm or y = -11.56 cm

x = 2 + y = 2 + 9.56 = 11.56 cm.

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For function represented as "f(x) = 3x(x+5)", the value of f(-2) is -18.

The 'Function" is a rule which assigns unique output value for each input value for given set. A function takes one or more inputs, and produces a "single-output", The "input-values" are called domain, and "output-values" are named as range.

In order to find the value of function at "-2", we substitute x as "-2" in the given function f(x) and then evaluate the expression:

We get,

⇒ f(x) = 3x(x+5),

⇒ f(-2) = 3(-2)(-2+5),

⇒ f(-2) = 3(-2)(3),

⇒ f(-2) = -18,

Therefore, the value of f(-2) is -18.

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The given question is incomplete, the complete question is

For the function f(x) = 3x(x+5), find the value of f(-2).

The value of k for the given relation of the function represented by attached graph is equal to 3.

Two function f(x) and g(x) .

Relation between f(x) and g(x) is equal to,

g ( x ) = f ( x ) + k

From the attached graph of the function f(x) and g(x) we have,

Slope of both the function f(x) and g(x) is equal to

Slope of f(x) = ( 2 -1 )/( 0 - (-3))

                    = 1/3

Slope of g(x) = (5 - 4)/ ( 0 - (-3))

                     = 1/3

Both the lines are parallel to each other with different value of intercept.

y-intercept of f(x) is equal to 2.

y-intercept of g(x) is equal to 5.

As g(x) = f(x)  + k

Substitute the value we have,

⇒ 5 = 2 + k

⇒ k = 3

Therefore, for the given attached graph the value of k is equal to 3.

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The above question is incomplete, the complete question is:

If g ( x ) = f ( x ) + k , what is the value of k using the attached graph of the function?

Maximum = n/n+1 and Minimum = 1/n+1

What is the uniform distribution?

Probability distributions with uniform distributions have outcomes that are all equitably likely. Results are discrete and have the same probability in a discrete uniform distribution. Results are continuous and infinite in a continuous uniform distribution. Data near the mean occur more frequently in a normal distribution.

Here, we have

Given: X1, X2,..., Xn is independent and identically distributed random variables having uniform distributions over (0,1).

a) Z = max{X₁, X₂....Xₙ}

Since Z is maximum so it is greater than X1, X2...Xn so cdf of Z will be

F(Z) = P(Z≤z) = P(X₁, X₂....Xₙ≤z) = P(X₁≤z, X₂≤z.....Xₙ≤z)

= P(X₁≤z)P(X₂≤z)....P(Xₙ≤z) = Fₓ(z)Fₓ(z).....Fₓ(z)

F(Z) = zⁿ

So pdf of Z is

F(z) = F'(z) = nzⁿ⁻¹

Expectation of Z is

F(Z) = [tex]\int\limits^0_1 {zf_Z(z)} \, dz[/tex] = [tex]n\int\limits^0_1 {} \,[/tex]zⁿdz

= n[zⁿ⁺¹/n+1]₀¹ = n/n+1

b) Let Y = min{X₁, X₂....Xₙ}

Since Y is the minimum so it is less than X1, X2...Xn so the cdf of Y will be

F(Y) = P(Y≤y) = 1 - P(Y>y) = 1- P(X₁, X₂....Xₙ≤z) = 1- P(X₁≤z, X₂≤z.....Xₙ>y)

= 1- P(X₁>y)P(X₂>y)....P(Xₙ>y) = 1- Fₓ(y)Fₓ(y).....Fₓ(y)

= 1 - [1-y]ⁿ

So pdf of Y is

F(y) = F'(y) = n[1-y]ⁿ⁻¹

The expectation of Y is

E(Y) = [tex]\int\limits^0_1 {yf_Y(y)} \, dy[/tex] = ∫₀¹ yn(1-y)ⁿ⁻¹dy = 1/n+1

Hence, Maximum = n/n+1 and Minimum = 1/n+1

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We have the proof statements as;

<A ≅ <C is given as the base angles

D is the midpoint of line AC; definition of midpoint

AC ⊥ BD; line of symmetry

ΔABC is an isosceles triangle; two equal sides and angles

It is important to note that properties of an isosceles triangle is given as;

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The variance of the total weight of flour sacks purchased is 36.

Since the moment generating function of a random variable uniquely determines its moments, we can use the moment generating function MW(t) to find the mean and variance of the weight of one sack of flour.

The first derivative of the moment generating function is:

MW'(t) = 6t/(1 - 3t)³

Setting t = 0, we get the mean:

MW'(0) = 6(0)/(1 - 3(0))³ = 0

So the mean weight of one sack of flour is zero.

The second derivative of the moment generating function is:

MW''(t) = 54t²/(1 - 3t)⁴ + 18/(1 - 3t)³

Setting t = 0, we get the second moment:

MW''(0) = 54(0)²/(1 - 3(0))⁴ + 18/(1 - 3(0))³ = 18

So the variance of the weight of one sack of flour is 18.

Now, let X and Y denote the weights of the first and second sacks of flour, respectively. Since the weights are independently distributed, the variance of the total weight of flour sacks purchased is the sum of the variances:

Var(X + Y) = Var(X) + Var(Y) = 18 + 18 = 36

Therefore, the variance of the total weight of flour sacks purchased is 36.

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The  dental hygienist compute "critical value" during the process of hypothesis testing, the dental hygienist computes a value based on the significance level and test type.

During the hypothesis testing process, the dental hygienist would first choose a significance level (such as 0.05) and a test type (such as a one-tailed test in this case). Based on the significance level and degrees of freedom (which depend on the sample size and assumed population standard deviation), the hygienist would then look up the critical value from a t-distribution table.

The critical value represents the cutoff point beyond which the null hypothesis (in this case, that the average number of cavities is not more than 3) would be rejected. If the test statistic (calculated from the sample data) falls within the rejection region (determined by the critical value), the hygienist would reject the null hypothesis and conclude that there is evidence to support the claim that the average number of cavities is more than 3.

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We can expect the mean length of ears in the next generation to be 31.6 cm.

It is given that the narrow sense heritability (h2) is 0.70, which means that 70% of the total variation in maize ear length is due to genetic factors.

Let the mean length of ears in the original population be µ and the mean length of ears in the selected plant be x. Then, we can use the formula for response to selection to find the expected mean length of ears in the next generation:

x' = µ + h2 * (x - µ)

Substituting the given values, we get:

x' = 28 + 0.70 * (34 - 28) = 31.6 cm

Therefore, we can expect the mean length of ears in the next generation to be 31.6 cm.

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