Two siblings Edwin and Rhea are both going skiing but choose different payment plans. Edwin's plan charges $45 for rentals and $5.25 per lift up the mountain. Rhea's plan was a bundle where her entire day cost $108. Set up an inequality that models the number of trips, n, up the mountain for which Edwin will pay more than Rhea. Solve the inequality. Also, what is the greatest amount of trips that Edwin can take up the mountain and still pay less than Rhea? Explain how you arrived at your answer.

The system of equations is found and the number of large boxes is 6 and the number of small boxes is 2.

What is meant by the system of equations?

A group of equations comprising one or more variables is known as a system of equations. The variable mappings that satisfy each component equation, or the points where all of these equations intersect, are the solutions of systems of equations.

Given,

Number of nails in small box = 200

Number of nails in large box = 400

Let,

number of small boxes = x

number of large boxes = y

It is said that the contractor bought 3 times as many large boxes as small boxes.

That is,

y = 3x

also,

200x + 400y = 2800

200x + 400*3x = 2800

We can solve this question using the above system of equations.

Substituting for x in the second equation.

200x  + 1200x = 2800

x = 2

y = 3*2 = 6

Graphically solving, the point of intersection of the lines y = 3x and

200x + 400y = 2800 is the point which gives the value of x and y.

It is shown below.

Therefore the system of equations is found and the number of large boxes is 6 and the number of small boxes is 2.

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Answer:

4.56, 4,65, 5,46, 5.64, 6.45 and 6.54

Step-by-step explanation:

start with the whole number and find the smallest one!

It would be 4 then find the number with the smallest amount

Keep doing this to all of the numbers

Answer:

D) (5x - 3) (25x^2+ 15x + 9)

Step-by-step explanation:

\(125 {x}^{3} - 27 \\ \\ = ( {5x)}^{3} - {(3)}^{3} \\ \\ = (5x - 3) \{(5x) ^{2} + 5x \times 3 + ( {3})^{2} \}\\ \\ = (5x - 3) \{25 {x}^{2} + 15x + 9\}\)

The correct response is c. 8√2 . The length of the diagonal if the side of the square is 8√2.

A polygon's opposed vertices (or corners) are connected by a line segment known as a diagonal. Or to put it another way, a diagonal is a line segment joining two polygonal vertices that are not neighbouring. With the exception of the figure's edges, it connects a polygon's vertices. For instance, a diagonal line or movement runs slopingly from one corner of a square to the other corner. a type of straight line called a diagonal. Straight up, down, or across are not possible on a diagonal line. It is a line that joins the corners of a form at two points. Instead of running straight up or across, a diagonal is formed by a straight line that is positioned at an angle.

Right isosceles triangles with both legs 8 feet long would make up each half.

The hypotenuse that both of those triangles would share is the diagonal.

We must determine the hypotenuse's length, d, in feet.

According to the Pythagorean theorem, the lengths of the legs' squares sum up to the length of the hypotenuse's square, so

\(8^{2} +8^{2} =8\sqrt{2}\)

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A gardener wants to divide a square piece of lawn in half diagonally. What is the length of the diagonal if the side of the square is 8ft? Leave your answer in simplest radical form.

a. 16√8

b. 2√8

c. 8√2

d. 4

The net income for Griffin's Goat Farm, Inc., was \(\$184,340\).

The operating cash flow (OCF) for Kerber's Tennis Shop, we can use the following formula:

\(OCF = EBIT + Depreciation - Taxes + \triangle Working Capital\)

where EBIT is earnings before interest and taxes.

Calculate EBIT by subtracting interest expense from operating income:

EBIT = Operating Income - Interest Expense

We do not have the information for operating income, but we can use the fact that interest expense was. \(\$165,000\) to calculate EBIT.

EBIT = Interest Expense / (1 - Tax Rate)

= \(\$165,000 / (1 - 0)\)

= \(\$165,000\)

Calculate the change in working capital (Δ Working Capital). We are told that the company reduced its net working capital investment by \(\$69,000\), which means that working capital increased by  \(\$69,000\).

Therefore:

\(\triangle Working Capital = \$69,000\)

Now, we can calculate OCF:

OCF = EBIT + Depreciation - Taxes + Δ Working Capital

=\(\$165,000 + 0 - 0 +\ $69,000\)

=\(\$234,000\)

Therefore, the firm’s 2018 operating cash flow (OCF) was \(\$234,000\) .

To calculate the net income for Griffin's Goat Farm, we can use the following formula:

Net Income = EBIT - Interest Expense - Taxes

where EBIT is earnings before interest and taxes. We can calculate EBIT as:

EBIT = Sales - Costs - Depreciation

= \(\$686,000 - \$332,000 - \$65,000\)

=\(\$289,000\)

Next, we need to calculate taxes. We are given a tax rate of 24%, so:

Taxes = Tax Rate x (EBIT - Depreciation)

= \(0.24 \times (\$289,000 - \$65,000)= \$56,160\)

Now, we can calculate net income:

Net Income = EBIT - Interest Expense - Taxes

= \(\$289,000 - \$48,500 - \$56,160= \$184,340\)

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cot(θ) = -3/4: The cotangent of angle θ, when the terminal side passes through the point (-3, -4), is -3/4. .

Given that the terminal side of an angle θ passes through the point (-3, -4), we can determine the value of cot(θ), which is the ratio of the adjacent side to the opposite side in a right triangle. To find cot(θ), we need to identify the adjacent and opposite sides of the triangle formed by the point (-3, -4) on the terminal side of angle θ.

The adjacent side is represented by the x-coordinate of the point, which is -3. The opposite side is represented by the y-coordinate, which is -4. Using the definition of cotangent, cot(θ) = adjacent/opposite, we substitute the values:

cot(θ) = -3/-4

Simplifying the fraction gives us:

cot(θ) = 3/4 . Therefore, the exact value of cot(θ) when the terminal side of angle θ passes through the point (-3, -4) is 3/4.

In geometric terms, cotangent is a trigonometric function that represents the ratio of the adjacent side to the opposite side of a right triangle. By identifying the appropriate sides using the given point, we can evaluate the cotangent of the angle accurately.

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• Domain: All real numbers

• Range: (-∞, ∞)

• To find the local minimum/maximum and intervals of increase/decrease, we need to take the first and second derivatives of the function:

First derivative: -4x^3 + 6x + 2

Setting this equal to zero and solving for x, we get critical points at x ≈ -1.23, x ≈ 0.65, and x ≈ 1.23.

Second derivative: -12x^2 + 6

• At x ≈ -1.23, the second derivative is negative, so we have a local maximum.

• At x ≈ 0.65, the second derivative is positive, so we have a local minimum.

• At x ≈ 1.23, the second derivative is negative, so we have a local maximum.

• As x —> -∞, the function approaches ∞.

• As x —> ∞, the function approaches ∞.

• To find the x-intercepts, we set the function equal to zero and solve for x:

-x^4+3x^2+2x+2 = 0

This is a quartic equation that can be solved using various methods, such as factoring or using the quadratic formula. One solution is x ≈ -1.38. The other three solutions are complex numbers.

• To find the y-intercept, we set x = 0:

-y^4 + 2 = 0

Solving for y, we get y ≈ ±1.19. So the y-intercepts are (0, 1.19) and (0, -1.19).

Answer:

Domain- (−∞,∞),{x|x∈R}

Range-  (−∞,17+6√3(over)4], {y∣y≤17+6√3(over)4}

Local Minimum/Maximum- (−1,2) is a local maxima

(1+√3(over)2 ,17+6√3(over)4) is a local maxima

(1−√3(over)2 ,17−6√3(over)4)is a local minima

Intervals of Increase/Decrease- Increasing on: (−∞,−1)(1−√3(over)2,1+√3(over)2)

Decreasing on: (−1,1−√3(over)2),(1+√3(over)2,∞)

Step-by-step explanation:

i hope this helps !

Answer: The answer that you’re looking for is D. \(p(x)=\frac{1}{3}x-\frac{5}{3}\). Brainliest?

Step-by-step explanation:

The expected number of dollars you will win or lose playing this game once is -$0.50. To calculate the expected value, we multiply each outcome by its probability and sum them up.

In the first round, the probability of rolling a 1 is 1/6, and the corresponding outcome is $1.

Expected value for the first round = (1/6) * $1 = $1/6.

In the second round, the probability of rolling a 2 is 1/6, and the corresponding outcome is $2.

Expected value for the second round = (1/6) * $2 = $2/6 = $1/3.

In the third round, the probability of rolling a 3 is 1/6, and the corresponding outcome is $3.

Expected value for the third round = (1/6) * $3 = $3/6 = $1/2.

If you lose all three rounds, the outcome is -$3 with a probability of (5/6) * (5/6) * (5/6) = 125/216.

The expected value for losing all three rounds = (125/216) * (-$3) = -$125/72.

Now, we sum up the expected values for each round and the outcome of losing all three rounds:

Expected value = ($1/6) + ($1/3) + ($1/2) + (-$125/72)

             = $12/72 + $24/72 + $36/72 - $125/72

             = ($12 + $24 + $36 - $125)/72

             = -$53/72

             ≈ -$0.50.

Therefore, the expected number of dollars you will win or lose playing this game once is -$0.50.

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Based on the given distribution of grades, the model grade of the statistics class is B.

This is because the highest percentage of grades (35%) is in the B range, followed by 30% in the C range, 25% in the A range, 9% in the D range, and only 1% in the F range.


To determine the model grade, we need to identify the grade range that has the highest percentage of students. In this case, the B range has the highest percentage with 35%. This means that the majority of students in the class received a B grade. The C range comes in second with 30%, followed by the A range with 25%. The D range has a smaller percentage with 9%, while the F range has the smallest percentage with only 1%. Therefore, the model grade of the statistics class is B.


In conclusion, the model grade of the statistics class with the given distribution of grades is B, indicating that the majority of students received a B grade. This information can be useful for analyzing the performance of the class and making decisions about teaching strategies and assessment methods.

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Answer:

$24

Step-by-step explanation:

Let the full price of $30 be 100%.

If the discount allowed is 20%, then the percentage of the original price that a customer will pay is:

\(P=100\%-20\%\\P=80\%\)

80% of a marked price of $30 is:

\(80\%*\$30\\\$24\)

A customer will pay $24.

Answer: The side length is 3 yards.

Step-by-step explanation:

To find the area of a square, you have to square one of it's side lengths since they all share the same length.  Since we know the area, and we need to find the side length so we just need to find the square root of the area.

\(\sqrt{9}\) = 3    

The side length is 3 yards.

The length of each side of a square is, 3 yards.

A rectangle is a two dimension figure with 4 sides, 4 corners and 4 right angles. The opposite sides of the rectangle are equal and parallel to each other.

We have to given that;

⇒ An area of square = 9 yards²

Now, We know that;

⇒ Area of square = Side²

Let the side of square = a

So, We get;

⇒ a² = 9

⇒ a = √ 9

⇒ a = 3 yards

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The age of Barbra is 22 years.

The algebraic method for solving simultaneous linear equations is the substitution method. A quantity of one variable through one expression is replaced in the second equation, as the name implies.

Now, according to the question.

Let 'a' be the age of Alice.

Let 'b' be the age of Barbra

Let 'c' be the age of Carol.

The sum of age of all three sisters is 68 years.

Thus,  a + b + c = 68

Now, as Alice is three years younger than Barbra.

a = b - 3

and, Barbra is 5 years younger than Carol.

b = c - 5

c = b + 5

By substitution of values of a and c in total age.

(b - 3) + b + (b + 5) = 68

3b - 3 + 5 = 68

On solving the expression.

b = 22

Therefore, the age of Barbra is 22 years.

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Answer:

2 quarts= 4 pints

Step-by-step explanation:

multiply the volume value by 2

Answer:

5/2, 4/5 are terminating decimals

Step-by-step explanation:

5/2 = 2.5  this is a terminating decimal

4/5 = .8 this is a terminating decimal

2/7 = .285714(repeating)  This is a repeating decimal

4/3 = 1.3(repeating)  this is a repeating decimal

Answer:

Hey There!!

4/5 and 5/2 are your answers!

Step-by-step explanation

The question is incomplete. Here is the complete question.

Pete is choosing his clothes for the day. His bottoms will be either cago shorts (C) or jeans (J). His top will be T-shirt (T), sweatshirt (S) or long sleeved shirt (L).

Which is a complete list of the possilbe outfits he can choose from?

CT, CS, CL, JT, JS, JL

CJ, CT, CS, CL

CJT, CJS, CJL

CT, TC, JS, SJ, CL, LC

Answer: CT, CS, CL, JT, JS, JL

Step-by-step explanation: The Fundamental Counting Principle is a method of determining a number of ways independents events can occur, i.e., it states that if there are t ways of doing something and w ways of doing another, then there are t x w ways of doing both things.

For example:

Peter has 2 choices of bottom and 3 choices of top. So, according to the principle, he will have 2 x 3 choices of clothing.

And the choices are:

shorts and t-shirt = CT

shorts and sweatshirt = CS

shorts and long sleeved = CL

jeans and T-short = JT

jeans and sweatshirt = JS

jeans and long sleeved = JL

The order he chooses doesn't matter.

So, the right alternative is CT, CS, CL, JT, JS, JL.

After solving the percentage, the height of flower stem after two months is found to be 10 feet.

Percentage is a number expressed as a fraction of 100. The % sign means to divide the number by 100.

To solve this question, we will do 40% x 25.

= (40/100) x 25

= 10 feet

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In a proportional relationship between two quantities, the constant of proportionality, often denoted by the letter "k," represents the value that relates the two quantities. The equation y = kx is the standard form for expressing a proportional relationship, where "y" and "x" are the variables representing the two quantities.

Here's a breakdown of the components in the equation:

y: Represents the dependent variable, which is the quantity that depends on the other variable. It is usually the output or the variable being measured.

x: Represents the independent variable, which is the quantity that determines or influences the other variable. It is typically the input or the variable being controlled.

k: Represents the constant of proportionality. It indicates the ratio between the values of y and x. For any given value of x, multiplying it by k will give you the corresponding value of y.

The constant of proportionality, k, is specific to the particular proportional relationship being considered. It remains constant as long as the relationship between x and y remains proportional. If the relationship is linear, k also represents the slope of the line.

For example, if we have a proportional relationship between the distance traveled, y, and the time taken, x, with a constant of proportionality, k = 60 (representing 60 miles per hour), the equation would be y = 60x. This equation implies that for each unit increase in x (in hours), y (in miles) will increase by 60 units.

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a) The line is called a level curve because all points on this line have the same function value of ln(3x + y) = 0.

b) Circle is called a level curve because all points on this circle have the same function value of g(x, y).

(a) We are given the function f(x, y) = ln(3x + y).

We need to find the level curve of the function for the given c-value. Here, c = 0.

Now, the level curve of the function for c = 0 can be found by setting f(x, y) = 0.

Therefore,

ln(3x + y) = 0

We know that ln eˣ = x, where e is a constant known as Euler's number.

Therefore, ln(3x + y) = 0 ⇒ e⁰ = 3x + y, which gives 3x + y = 1 as the level curve.

(b) We are given the function g(x, y) = x/x^2 + y^2.

We need to find the level curve of the function for the given c-value. Here, c = − 1/2.

Now, the level curve of the function for c = − 1/2 can be found by setting g(x, y) = − 1/2.

Therefore, x/x² + y² = − 1/2

Multiplying both sides by x² + y², we get

x = − x²/2 − y²/2

Rearranging the terms, we get

x² + 2x/2 + y²/2 = 0

Adding 1/2 on both sides, we get

x² + 2x/2 + y²/2 + 1/2 = 1/2

Simplifying, we get

(x + 1)² + y² = 1

Now, the given equation represents a circle with a center at (-1, 0) and a radius 1. Therefore, the name of the curve is a circle.

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By using the property of division, it can be calculated that M = -21 divided by 0 is not possible

What is division?

Division is the process by which value of single unit can be calculated from the value of multiple unit.

The number to be divided is known as dividend, the number by which the dividend is divided is the divisor, the result obtained is the quotient and the remaining part is the remainder.

There is a well known formula for division

Divisor x Quotient + Remainder = Dividend.

Anything divided by 0 is not possible.

M = -21 divided by 0 is not possible

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The first partial derivative of f(x, y) with respect to x is \(y + 2xy^2\), and the first partial derivative of f(x, y) with respect to y is \(x + 2x^2y\).

To find the first partial derivatives of the function\(f(x, y) = xy + x^2y^2\) with respect to x and y, we differentiate the function with respect to each variable while treating the other variable as a constant.

Partial derivative with respect to x (df/dx):

To find df/dx, we differentiate each term of the function with respect to x:

\(d(xy)/dx = y\\d(x^2y^2)/dx = 2xy^2\)

Therefore, the first partial derivative of f(x, y) with respect to x is:

\(df/dx = y + 2xy^2\)

Partial derivative with respect to y (df/dy):

To find df/dy, we differentiate each term of the function with respect to y:

d(xy)/dy = x

\(d(x^2y^2)/dy = 2x^2y\)

Therefore, the first partial derivative of f(x, y) with respect to y is:

\(df/dy = x + 2x^2y\)

So, the first partial derivative of f(x, y) with respect to x is \(y + 2xy^2\), and the first partial derivative of f(x, y) with respect to y is \(x + 2x^2y\).

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Answer:

3e - \(\frac{3}{2}\) f - \(\frac{3}{8}\)

Step-by-step explanation:

\(\frac{1}{2}\) (6e - 3f - \(\frac{3}{4}\) ) ← multiply each term in the parenthesis by \(\frac{1}{2}\)

= ( \(\frac{1}{2}\) × 6e) - (\(\frac{1}{2}\) × 3f) - (\(\frac{1}{2}\) × \(\frac{3}{4}\) )

= 3e - \(\frac{3}{2}\) f - \(\frac{3}{8}\)

13) The scientific notation is        1.4354125 * 10⁸. 14) The scientific notation is        1.4354125 * 10⁻¹. 15) The scientific notation is 22366.0992. 16) The scientific notation is 1.55 * 10¹⁰.

Let's solve the given expressions and write the answers in scientific notation:

13) \((6.125*10^3) * (2.345*10^4)\)

To multiply the numbers in scientific notation, we multiply the coefficients and add the exponents:

\((6.125 * 2.345) * (10^3 * 10^4) = 14.354125 * 10^(3 + 4) = 14.354125 * 10^7 = 1.4354125 * 10^8\)

\(= 1.4354125 * 10^8\)

14) \((6.1258*10^3) * (2.345*10^{-4})\)

To multiply the numbers in scientific notation, we multiply the coefficients and add the exponents:

\((6.125 * 2.345) * (10^3 * 10^{-4}) = 14.354125 * 10^{3 - 4} = 14.354125 * 10^{-1} = 1.43541258* 10^{-1}\)

\(= 1.4354125 * 10^{-1}\)

15) \(3700.13 * (6.04*10^4)\)

To multiply a decimal number and a number in scientific notation, we simply multiply the decimal by the coefficient of the scientific notation:

3700.13 × 6.04 = 22366.0992

Since the answer does not need to be expressed in scientific notation, the result is 22366.0992.

= 22366.0992

16) \((6.2*10^4) / (0.4*10^{-5})\)

To divide numbers in scientific notation, we divide the coefficients and subtract the exponents:

\((6.2 / 0.4) * (10^4 / 10^{-5}) \\= 15.5 * 10^{4 - (-5}) \\= 15.5 * 10^9 \\= 1.55 * 10^{10}\\= 1.55 * 10^{10}\)

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Answer:

C.

yes; y = 1.8x

Step-by-step explanation:

Answer for the question :

Yes, Y varies directly with X, and the equation is :

Y = 1.8X

NOTE : For every value of X, when you multiply them with 1.8, you will get the corresponding Y.  

Y = 1.8 x X

5.4 = 1.8 x 3

12.6 = 1.8 x 7

21.6 = 1.8 x 12   (Q.E.D.)

That’s the answer, good luck in solving the question!

:-)

ideal all you ought to do is to divide the y via potential of x and u get the equation working occasion #a million y=32 / x=4 then u have y/x=32/4 ----> y=8x and do the comparable indoors something so #2 y=-20 & x=-20 so y/x=-20/-20 y=x etc

The linear equation is given as y = 1.8x, Then the correct option is C.

A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.

Let the equation of the line pass through (x₁, y₁) and (x₂, y₂). Then the equation of the line is given as,

\(\rm (y - y_2) = \left (\dfrac{y_2 - y_1}{x_2 - x_1} \right ) (x - x_2)\)

The table is given below.

x         y

3      5.4

7      12.6

12     21.6

Then the linear equation is given as,

\(\rm (y - 5.4) = \left [\dfrac{5.4-12.6}{7-3} \right] (x-3)\)

Simplify the equation, then we have

y - 5.4 = 1.8(x - 3)

y - 5.4 = 1.8x - 5.4

y = 1.8x

The linear equation is given as y = 1.8x, Then the correct option is C.

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The ESE (estimated standard error) in the bottom of the independent-measure t statistic measures the variability in the sample data used to calculate the t statistic and provides information on the precision of the t statistic.

The t statistic is used to compare the difference between two means from independent groups to determine if the difference is significant or not.

The ESE reflects the degree of uncertainty in the difference between the two sample means and is calculated as the standard deviation of the sampling distribution of the difference between the two sample means. The ESE is an estimate of the standard deviation of the population difference and is used to calculate the t statistic.

The ESE is an important measure because it provides information on the precision of the t statistic, which is used to make inferences about the population difference. If the ESE is large, it means that the sample difference is less precise and the t statistic is less significant. If the ESE is small, it means that the sample difference is more precise and the t statistic is more significant.

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Given that the correlation between two variables is r=0.23. We need to find out the new correlation that would exist if the following three changes are made to the existing variables: All values of the x-variable are added by 0.14. All values of the y-variable are doubled Interchanging the two variables.  the correct option is B. 0.37.

The effect of changing the variables on the correlation coefficient between the two variables can be determined using the following formula: `r' = (r * s_x * s_y) / s_u where r' is the new correlation coefficient, r is the original correlation coefficient, s_x and s_y are the standard deviations of the two variables, and s_u is the standard deviation of the composite variable obtained by adding the two variables after weighting them by their respective standard deviations.

If we assume that the x-variable is the original variable, then the new values of x and y variables would be as follows:x' = x + 0.14 (since all values of the x-variable are added by 0.14)y' = 2y (since every value of the y-variable is doubled)Now, the two variables are interchanged. So, the new values of x and y variables would be as follows:x" = y'y" = using these values, we can find the new correlation coefficient, r'`r' = (r * s_x * s_y) / s_u.

To find the new value of the standard deviation of the composite variable, s_u, we first need to find the values of s_x and s_y for the original and transformed variables respectively. The standard deviation is given by the formula `s = sqrt(sum((x_i - mu)^2) / (n - 1))where x_i is the ith value of the variable, mu is the mean value of the variable, and n is the total number of values in the variable.

For the original variables, we have:r = 0.23s_x = standard deviation of x variable = s_y = standard deviation of y variable = We do not have any information about the values of x and y variables, so we cannot calculate their standard deviations. For the transformed variables, we have:x' = x + 0.14y' = 2ys_x' = sqrt(sum((x_i' - mu_x')^2) / (n - 1)) = s_x = standard deviation of transformed x variable` = sqrt(sum(((x_i + 0.14) - mu_x')^2) / (n - 1)) = s_x'y' = 2ys_y' = sqrt(sum((y_i' - mu_y')^2) / (n - 1)) = 2s_y = standard deviation of transformed y variable` = sqrt(sum((2y_i - mu_y')^2) / (n - 1)) = 2s_yNow, we can substitute all the values in the formula for the new correlation coefficient and simplify:

r' = (r * s_x * s_y) / s_ur' = (0.23 * s_x' * s_y') / sqrt(s_x'^2 + s_y'^2)r' = (0.23 * s_x * 2s_y) / sqrt((s_x^2 + 2 * 0.14 * s_x + 0.14^2) + (4 * s_y^2))r' = (0.46 * s_x * s_y) / sqrt(s_x^2 + 0.0396 + 4 * s_y^2)Now, we can substitute the value of s_x = s_y = in the above formula:r' = (0.46 * * ) / sqrt( + 0.0396 + 4 * )r' = (0.46 * ) / sqrt( + 0.1584 + )r' = (0.46 * ) / sqrt(r' = (0.46 * ) / sqrt(r' = (0.46 * ) / sqrt(r' = r' = Therefore, the new correlation coefficient, r', would be approximately equal to.

Hence, the correct option is B. 0.37.

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Answer:

12+6K

Step-by-step explanation:

3(4+2k)

removing bracket

3×2+2×2k

12+6k

Answer:

Plot the points on the graphing calculator. Then generate a linear regression model.

f(x) = .6714x + 4.0536

Substituting 12 for x:

f(12) = about 12.1 million

∃x(Tx ∧ ¬Hx) is the predicate logic representation of the English sentence "Some tall people are not happy" and ¬∃x(Hx) is the predicate logic representation of the English sentence "No one is happy."

To translate the following English sentence "No one is happy" into the language of predicate logic (RL) we can write it as, ¬∃x(Hx).In this expression, ∃x is the existential quantifier. It indicates the existence of an object in the given domain that satisfies a specific condition.For the given statement, we know that no one is happy. So, Hx should be false for all individuals in the domain. Therefore, the negation sign ¬ should be used to specify that the condition of being happy is false for every human being. Hence, the final predicate logic representation of the given sentence will be ¬∃x(Hx).To translate the English sentence "Some tall people are not happy" into the language of predicate logic (RL), we can write it as ∃x(Tx ∧ ¬Hx).In this expression, ∃x is the existential quantifier that represents that there is at least one object in the domain that satisfies the given condition, i.e., there exists a person who is both tall and not happy. Therefore, the final predicate logic representation of the given sentence will be ∃x(Tx ∧ ¬Hx).

∃x(Tx ∧ ¬Hx) is the predicate logic representation of the English sentence "Some tall people are not happy" and ¬∃x(Hx) is the predicate logic representation of the English sentence "No one is happy."

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From the calculations, Gerald will now have to pay the sum of $145.50 towards his monthly health insurance.

Note that the first sum being contributed by Gerald is (100%- 65%) of $345. This is 35% of $345 = $120.75

If his employer takes up 50% of $38, this leaves his contribution for dental as 50% of $38 = $19

Also if his employer takes up 75% of $23, his contribtuion for vision premium is 25% of 23 = $5.75

Hence his total contribution is now:
120.75 + 19.00 + 5.75 = $145.50

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