If f(6)=7, find f⁻¹(7)
Assume that the function f is a one-to-one function.

1/3 is the rate of change (slope) given the set of data points below

To find the rate of change (slope) given the set of data points, we can use the formula for slope:

slope = (change in y) / (change in x)

Using the given points:

Point 1: (10, 19)

Point 2: (4, 17)

Point 3: (-2, 15)

Point 4: (-8, 13)

We can calculate the slope between each pair of points:

Slope between Point 1 and Point 2:

slope = (17 - 19) / (4 - 10) = -2 / -6 = 1/3

Slope between Point 2 and Point 3:

slope = (15 - 17) / (-2 - 4) = -2 / -6 = 1/3

Slope between Point 3 and Point 4:

slope = (13 - 15) / (-8 - (-2)) = -2 / -6 = 1/3

The rate of change (slope) between these points is consistent and equal to 1/3.

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The inequality 6x - 13 < 6 (x - 2) is always true. 0 < 1 is always true so, the inequality is always true for any value of x.

When we simplify the inequality, we have 6x - 13 < 6x - 12.

6x - 13 < 6x - 12

or, 6x - 6x < 13 - 12

or, 0 < 1

Notice that the x-terms cancel out, resulting in 0 < 1.

In the comparison 0 < 1, the left side 0 is indeed less than the right side 1, making the inequality true. This holds true for all values of x.

Since the inequality is true regardless of the value of x, we can conclude that the original inequality 6x - 13 < 6 (x - 2) is always true.

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There is no such relation between ∠9 and ∠10, except they are adjacent pair of interior and exterior angles, the sum of the measure of ∠9 and ∠10 is 180°.

We know that,

⇒Transversal lines cut two parallel or non-parallel lines and made pair of angles such as interior angles, exterior angles, and alternate angles.

Here in the picture with some examples, we can understand this.

∠1 and ∠14 - This pair of angles lie on opposite sides of the transversal line "op". Those are known as alternate exterior angles.

∠1 and ∠3 - This pair of angles lie on opposite sides of the transversal line "as". Those are known as  nonadjacent interior angles

∠10 and ∠11 - This pair of angles lie on the same side of the transversal "tu". Those are known as consecutive interior angles.

∠5 and ∠7 - This pair of angles lie on the same side of the transversal "as", and the same side of lines "op", and "qr". Those are known as corresponding angles.

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The tenth term of the sequence 4, 5, 6, 7, 8, ... is 13.

Using this formula, we can find the tenth term as follows:

a10 = a1 + (10-1)

= 4 + 9

= 13

The given sequence is an arithmetic sequence where each term is obtained by adding 1 to the previous term. The first term, a1, is 4. To find the nth term, we use the explicit formula an = a1 + (n-1), which represents the pattern of the sequence.

By substituting the value of n as 10, we can calculate the tenth term, which is 13. This means that the tenth term in the sequence is obtained by adding 9 to the first term, 4.

Therefore, the tenth term of the sequence is 13.

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There are 30° between every two consecutive numbers on an analog clock.

Given that on an analog clock, the minute hand moved 128° in an hour we need to find the numbers it has pass next,

On an analog clock, the minute hand completes a full revolution of 360° in 60 minutes.

This means that each minute corresponds to a movement of 360°/60 = 6°.

The minute hand has moved 128° from the hour, we can determine the number of minutes it has traveled by dividing 128° by 6°:

128° / 6° = 21.33 minutes

Since the minute hand has moved beyond 21 minutes, it will pass the next number on the clock.

If we consider the current number to be n, then it will pass the number (n+1) next.

To find the number of degrees between every two consecutive numbers on an analog clock, we divide the total degrees in a circle (360°) by the number of divisions on the clock face (12 numbers):

360° / 12 = 30°

Therefore, there are 30° between every two consecutive numbers on an analog clock.

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The value of x is approximately 9.8.

In this question, we are given an equation, and we need to find the value of x in the equation.

The equation that is given to us is as follows: 19 cos 20° + x = 29

To find the value of x, we need to isolate it on one side of the equation.

First, we will subtract 19 cos 20° from both sides of the equation.

This gives us: x = 29 - 19 cos 20°Now we can use a calculator to evaluate the right-hand side of the equation.

Rounding to the nearest tenth, we get: x ≈ 9.8

Therefore, the value of x is approximately 9.8.

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Answer:

$30

Step-by-step explanation:

17+13

Answer:

[tex]\text{g}^{-1}(x)=\boxed{\dfrac{x-13}{2}}[/tex]

[tex]\left(\text{g}^{-1} \circ \text{g}\right)(-4)=\boxed{-4}[/tex]

[tex]h^{-1}(9)=\boxed{-3}[/tex]

Step-by-step explanation:

To find the inverse of function g(x) = 2x + 13, begin by replacing g(x) with y:

[tex]y=2x+13[/tex]

Swap x and y:

[tex]x=2y+13[/tex]

Rearrange to isolate y:

[tex]\begin{aligned}x&=2y+13\\\\x-13&=2y+13-13\\\\x-13&=2y\\\\2y&=x-13\\\\\dfrac{2y}{2}&=\dfrac{x-13}{2}\\\\y&=\dfrac{x-13}{2}\end{aligned}[/tex]

Replace y with g⁻¹(x):

[tex]\boxed{\text{g}^{-1}(x)=\dfrac{x-13}{2}}[/tex]

[tex]\hrulefill[/tex]

As g and g⁻¹ are true inverse functions of each other, the composite function (g⁻¹ o g)(x) will always yield x. Therefore, (g⁻¹ o g)(-4) = -4.

To prove this algebraically, calculate the original function g at the input value x = -4, and then evaluate the inverse function of g at the result.

[tex]\begin{aligned}\left(\text{g}^{-1} \circ \text{g}\right)(-4)&=\text{g}^{-1}\left[\text{g}(-4)\right]\\\\&=\text{g}^{-1}\left[2(-4)+13\right]\\\\&=\text{g}^{-1}\left[5\right]\\\\&=\dfrac{(5)-13}{2}\\\\&=\dfrac{-8}{2}\\\\&=-4\end{aligned}[/tex]

Hence proving that (g⁻¹ o g)(-4) = -4.

[tex]\hrulefill[/tex]

The inverse of a one-to-one function is obtained by reflecting the original function across the line y = x, which swaps the input and output values of the function. Therefore, (x, y) → (y, x).

Given the one-to-one function h is defined as:

[tex]h=\left\{(-3,9), (1,0), (3,-7), (5,2), (9,6)\right\}[/tex]

Then, the inverse of h is defined as:

[tex]h^{-1}=\left\{(9,-3),(0,1),(-7,3),(2,5),(6,9)\right\}[/tex]

Therefore, h⁻¹(9) = -3.

The difference between solving |x+3|>4 and |x+3|<4 lies in the direction of the inequality and the resulting solution set.

In the first inequality, |x+3|>4, we are looking for values of x that make the absolute value of x+3 greater than 4. In the second inequality, |x+3|<4, we are looking for values of x that make the absolute value of x+3 less than 4. When solving |x+3|>4, we consider two cases: one where x+3 is positive and one where x+3 is negative. For x+3 > 0, we solve x+3 > 4, which gives x > 1.

For x+3 < 0, we solve -(x+3) > 4, which gives x < -7. Combining these two cases, the solution to |x+3|>4 is x < -7 or x > 1. On the other hand, when solving |x+3|<4, we consider two cases as well. For x+3 > 0, we solve x+3 < 4, which gives x < 1. For x+3 < 0, we solve -(x+3) < 4, which gives x > -7. Combining these two cases, the solution to |x+3|<4 is -7 < x < 1.

The difference between solving |x+3|>4 and |x+3|<4 lies in the direction of the inequality and the resulting solution set. The former involves values of x that make the absolute value greater than 4, while the latter involves values that make the absolute value less than 4. The solution set for |x+3|>4 is x < -7 or x > 1, while the solution set for |x+3|<4 is -7 < x < 1.

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The expression √-24 can be simplified using the imaginary unit i as 2i√6.

To simplify √-24, we can break it down into two parts: the square root of -1 (which is represented by the imaginary unit i) and the square root of 24.

The square root of -1 is i, and the square root of 24 can be simplified as 2√6.

Combining these two parts, we get 2i√6 as the simplified form of √-24.

The imaginary unit i is defined as the square root of -1. It is used to represent imaginary numbers, which are numbers that involve the square root of a negative number. In this case, the expression involves the square root of -24, which is a negative number. By using the imaginary unit i, we can simplify √-24 as 2i√6, where 2 is the coefficient and √6 is the remaining radical term.

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The average flow time of a caller at this help desk is 5 minutes.

To calculate the average flow time of a caller at the help desk, we need to sum up the individual call durations and divide it by the total number of callers. In this case, there are four callers with call durations of 2, 5, 3, and 10 minutes.

Adding up the call durations: 2 + 5 + 3 + 10 = 20

To find the average, we divide the sum by the total number of callers, which is 4 in this case.

Average flow time = 20 minutes ÷ 4 callers = 5 minutes

The average flow time of a caller at this help desk is 5 minutes. This represents the average time a caller spends on their call during the given one-hour period from 5 a.m. to 6 a.m.

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P(B∣A) is approximately 0.310, rounded to three decimal places.

To find the probability P(B∣A), we can use Bayes' theorem:

P(B∣A)= P(A) / P(A∣B)⋅P(B)

Given information:

P(B)=0.2

P(A∣B)=0.9

P(B')=0.8 (probability of not B)

P(A∣B′)=0.5 (probability of A given not B)

First, we need to calculate

P(A), the probability of event A. We can use the law of total probability to express P(A) in terms of the probabilities related to B and not B:

P(A)=P(A∣B)⋅P(B)+P(A∣B′)⋅P(B′)

Substituting the given values:

P(A)=0.9⋅0.2+0.5⋅0.8=0.18+0.4=0.58

Now, we can substitute the known values into Bayes' theorem:

P(B∣A)= P(A)/ P(A∣B)⋅P(B)

= 0. 9.0.2 / 0.58

Calculating this expression:

P(B∣A)≈ 0.5 80.18 ≈0.310

Therefore, P(B∣A) is approximately 0.310, rounded to three decimal places.

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The expression 10 / ³√5x² can be simplified by rationalizing the denominator to obtain 10√(5x²) / 5x.

To simplify the expression 10 / ³√5x², we need to rationalize the denominator.

The cube root (∛) of 5x² can be rewritten as (5x²)^(1/3).

To rationalize the denominator, we multiply both the numerator and denominator by the cube root of the denominator raised to the power that will cancel out the radical. In this case, we multiply by (∛(5x²))^2 or (∛(5x²))^2 / (∛(5x²))^2.

Applying the multiplication, we get (10 / ³√5x²) * (∛(5x²))^2 / (∛(5x²))^2 = 10 * (∛(5x²))^2 / (5x²).

Simplifying further, we have 10 * √(5x²) / (5x²) = 10√(5x²) / 5x.

Therefore, the simplified expression, after rationalizing the denominator, is 10√(5x²) / 5x.

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The dimensions of the rectangle are width = 7/2 meters and length = 4 meters.

Let's assume that the width of the rectangle is x meters. According to the given information, the length of the rectangle is 3 meters less than double the width, which can be expressed as 2x - 3.

The area of a rectangle is given by the formula: Area = Length × Width. In this case, the area is given as 14 m². Therefore, we can write the equation:

(x)(2x - 3) = 14

Expanding the equation:

2x² - 3x = 14

Rearranging the equation to standard form:

2x² - 3x - 14 = 0

To solve this quadratic equation, we can factor it or use the quadratic formula. In this case, let's use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

In our equation, a = 2, b = -3, and c = -14. Plugging in these values into the quadratic formula:

x = (-(-3) ± √((-3)² - 4(2)(-14))) / (2(2))

x = (3 ± √(9 + 112)) / 4

x = (3 ± √121) / 4

x = (3 ± 11) / 4

Simplifying further:

x = (3 + 11) / 4 or x = (3 - 11) / 4

x = 14 / 4 or x = -8 / 4

x = 7/2 or x = -2

Since the width cannot be negative, we discard the negative solution. Therefore, the width of the rectangle is 7/2 meters.

Now, we can substitute the value of x into the expression for the length:

Length = 2x - 3

Length = 2(7/2) - 3

Length = 7 - 3

Length = 4 meters

Thus, the dimensions of the rectangle are width = 7/2 meters and length = 4 meters.

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The real solutions of the polynomial equation x⁴ = 16 are x = ±2.To find the real or imaginary solutions of the polynomial equation x⁴ = 16, we can start by rewriting it as x⁴ - 16 = 0.

We can then factor the equation as a difference of squares: (x²)² - 4² = 0. Now, we have a quadratic equation in the form a² - b² = 0, which can be factored using the difference of squares formula: (x² - 4)(x² + 4) = 0. From this equation, we get two possible cases: Case 1: x² - 4 = 0. Solving for x, we have: x² = 4; x = ±2. Case 2: x² + 4 = 0.  

This equation has no real solutions because the square of a real number is always positive. Therefore, the real solutions of the polynomial equation x⁴ = 16 are x = ±2.

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The mean acrylamide level in nanograms/gram for the given data is 320.

To calculate the mean, we sum up all the data values and divide the sum by the number of values:

(346 + 296 + 334 + 276 + 248) / 5 = 1500 / 5 = 300

Therefore, the mean acrylamide level is 300 nanograms/gram.

To calculate the **deviation from the mean** for each data value, we subtract the mean from each value:

346 - 300 = 46

296 - 300 = -4

334 - 300 = 34

276 - 300 = -24

248 - 300 = -52

The deviations from the mean for each data value are as follows:

46, -4, 34, -24, -52

The positive deviations indicate values higher than the mean, while the negative deviations indicate values lower than the mean.

Understanding the deviations from the mean allows us to assess the variability of the acrylamide levels within the dataset. Positive deviations suggest higher acrylamide levels compared to the mean, while negative deviations indicate lower levels. Analyzing the spread of these deviations can provide insights into the consistency or variability of acrylamide content among different brands of biscuits.

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(a) The student repays $6200 ($5000 principal + $1200 interest) over 4 years. (b) Varies; paying interest is generally ethical, compensating the lender. Loan amount and duration impact interest and commitment.

(a) To calculate the amount repaid, we need to add the principal amount ($5000) to the interest accrued over the 4 years. The simple interest is calculated as I = P * R * T, where P is the principal amount, R is the interest rate (6% or 0.06), and T is the time period (4 years).

Interest accrued = 5000 * 0.06 * 4 = $1200

Amount repaid = Principal amount + Interest accrued = $5000 + $1200 = $6200

The student repays $6200 in total.

(b) Personal experiences and financial arrangements may vary for individuals. Regarding the ethical aspect, it is generally considered fair and ethical to pay interest when borrowing money, as the lender is sacrificing the opportunity cost of utilizing that money elsewhere. The amount and duration of the loan matter as they determine the interest accrued and the level of financial commitment.

Disclosure: As an AI, I don't have personal experiences or engage in financial transactions with family and friends.

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The expression tan(θ/4) can be written as (1 - ±√[(1 + cos θ) / 2]) / ±√[(1 - cos θ) / 2], where the ± symbol represents both possibilities of the sign depending on the quadrant of θ

To express the expression tan(θ/4) using half-angle identities, we can use the following steps:

1. Start with the half-angle identity for tangent:

  tan(θ/2) = (1 - cos θ) / sin θ

2. Substitute θ/2 with θ/4 in the above identity:

  tan(θ/4) = (1 - cos(θ/2)) / sin(θ/2)

3. Now, we need to express cos(θ/2) and sin(θ/2) in terms of trigonometric functions of θ. We can use the half-angle identities for cosine and sine:

  cos(θ/2) = ±√[(1 + cos θ) / 2]

  sin(θ/2) = ±√[(1 - cos θ) / 2]

  Note: The sign depends on the quadrant of θ. Since we are not given the specific quadrant of θ, we use the ± symbol to represent both possibilities.

4. Substitute the half-angle identities for cosine and sine into the expression:

  tan(θ/4) = (1 - ±√[(1 + cos θ) / 2]) / ±√[(1 - cos θ) / 2]

  Note: The ± symbol appears in both the numerator and the denominator to represent both possibilities of the sign.

Therefore, using the half-angle identities, the expression tan(θ/4) can be written as (1 - ±√[(1 + cos θ) / 2]) / ±√[(1 - cos θ) / 2], where the ± symbol represents both possibilities of the sign depending on the quadrant of θ.

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Each employee earns $33.50. Since there are 4 employees and they earn a total of $134, dividing the total by the number of employees gives us $33.50 per employee.

To determine how much each employee earns in a store where there are 4 employees in total, and they all earn the same amount, we can divide the total amount earned by the number of employees.

Given that the total amount earned by the 4 employees is $134, we need to divide this amount equally among them to find the individual earnings.

Let's calculate the amount earned by each employee:

Amount earned by each employee = Total amount earned / Number of employees

Amount earned by each employee = $134 / 4

Dividing $134 by 4, we find that each employee earns $33.50.

Therefore, each employee in the store earns $33.50.

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Bailey's season batting average is approximately 0.317, while Janson's season batting average is approximately 0.314.

To determine each player's batting average for the entire season, we can use the formula for batting average, which is calculated by dividing the number of hits by the number of at-bats. Let's calculate the batting averages for Bailey and Janson based on the data provided in the table.

For Bailey:

In the first half of the season, Bailey had 17 hits and 56 at-bats, resulting in a batting average of 17/56 ≈ 0.304.

In the second half of the season, Bailey had 85 hits and 265 at-bats, giving a batting average of 85/265 ≈ 0.321.

To find Bailey's season batting average, we can add up the total hits and total at-bats from both halves of the season:

Total hits = 17 + 85 = 102

Total at-bats = 56 + 265 = 321

Bailey's season batting average = Total hits / Total at-bats

= 102 / 321 ≈ 0.317.

For Janson:

In the first half of the season, Janson had 107 hits and 345 at-bats, resulting in a batting average of 107/345 ≈ 0.310.

In the second half of the season, Janson had 50 hits and 155 at-bats, giving a batting average of 50/155 ≈ 0.323.

To find Janson's season batting average, we can add up the total hits and total at-bats from both halves of the season:

Total hits = 107 + 50 = 157

Total at-bats = 345 + 155 = 500

Janson's season batting average = Total hits / Total at-bats

= 157 / 500 ≈ 0.314.

In summary, Bailey's season batting average is approximately 0.317, while Janson's season batting average is approximately 0.314.

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Questions:The table shows the number of hits and at bats for two baseball players during the first and second halves of the season.

First half of season| Second half of season

Player & Hits & At bats & Batting average| Hits & At bats & Batting average

Bailey &17 & 56 & 0.304| 85& 265 &0.321

Janson & 107 & 345 & 0.310|50& 155 &0.323

Click to download the data in your preferred format. Crunch It! CSV Excel JMP Mac Text Minitab PC Text R SPSS TI Calc Batting average is determined by dividing the number of hits by the number of at-bats. Use the number of hits and at bats from the proceeding table to determine each player's batting average for the entire season. Enter the values in the following table. Give your answers precise to three decimal places. Player| Season batting average

Bailey |  

Janson|  

The values of g(-4), g(-2), and g(3) are -5, undefined, and -11, respectively. The function g is defined differently for x = -2 and other real numbers, resulting in different output values.

Given the function g(x) = {(-4/1)(x-2) - (1/1)(x+2), if x ≠ -2; undefined, if x = -2}, we can evaluate g(-4), g(-2), and g(3) as follows:

1. g(-4):

Since -4 ≠ -2, we use the first part of the definition of g(x). Plugging in x = -4, we have:

g(-4) = (-4/1)(-4-2) - (1/1)(-4+2)

      = (-4/1)(-6) - (1/1)(-2)

      = 24 + 2

      = 26

Therefore, g(-4) = 26.

2. g(-2):

Since x = -2 matches the condition in the second part of the definition of g(x), g(-2) is undefined.

3. g(3):

Since 3 ≠ -2, we use the first part of the definition of g(x). Plugging in x = 3, we have:

g(3) = (-4/1)(3-2) - (1/1)(3+2)

     = (-4/1)(1) - (1/1)(5)

     = -4 - 5

     = -9

Therefore, g(3) = -9.

In summary, g(-4) = 26, g(-2) is undefined, and g(3) = -9. The function g(x) has different output values depending on whether x is equal to -2 or not.

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The simplified expression is:
[tex]6x + 6 - (5x^2 / 18) - 5y[/tex]

To divide the expression 6x + 6y / y - x ÷ 18/5x - 5y, we need to follow the order of operations, which is known as PEMDAS (Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction).
Let's break it down step by step:
Step 1: Simplify the expression inside the parentheses, if any.
The expression does not have any parentheses, so we can move to the next step.
Step 2: Divide the expression from left to right.
We have:
6x + 6y / y - x ÷ 18/5x - 5y
First, let's divide 6y by y:
6x + (6y/y) - x ÷ 18/5x - 5y
= 6x + 6 - x ÷ 18/5x - 5y
Now, let's divide -x by 18/5x:
6x + 6 - (x ÷ 18/5x) - 5y
= 6x + 6 - (x * 5x / 18) - 5y
=[tex]6x + 6 - (5x^2 / 18) - 5y[/tex]
Finally, we have:
[tex]6x + 6 - (5x^2 / 18) - 5y[/tex]
Step 3: Simplify the expression further, if possible.
There are no additional simplifications we can make, so we can move on to the next step.
Step 4: State any restrictions on the variables.
In this expression, there are no restrictions on the variables x and y. They can take any real values.

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For θ = π/6 radians, cosθ = √3/2 and sinθ = 1/2 is obtained by using trigonometric functions.

The measure θ of an angle in standard position is π/6 radians. To find the exact values of cosθ and sinθ for this angle measure, we can use the unit circle.

Step 1: Draw the unit circle, which is a circle with a radius of 1 centered at the origin (0, 0) on the coordinate plane.

Step 2: Locate the angle θ = π/6 radians on the unit circle. This angle is formed by the positive x-axis and a line segment from the origin to a point on the unit circle.

Step 3: To find the exact value of cosθ, look at the x-coordinate of the point where the angle intersects the unit circle. In this case, the x-coordinate is √3/2. Therefore, cos(π/6) = √3/2.

Step 4: To find the exact value of sinθ, look at the y-coordinate of the point where the angle intersects the unit circle. In this case, the y-coordinate is 1/2. Therefore, sin(π/6) = 1/2.

So, for θ = π/6 radians, cosθ = √3/2 and sinθ = 1/2.

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A person weighing 173 pounds can burn 10.7 calories per minute.

The given equation is ŷ=2.2+0.05x.

To solve this question, we can use the equation given, ŷ=2.2+0.05x. We can insert the known weight value, 173 pounds, to calculate the anticipated calories burned per minute.

y = 2.2 + 0.05x

y = 2.2 + 0.05 × 173

y = 10.7

Therefore, a person weighing 173 pounds can burn 10.7 calories per minute.

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"Your question is incomplete, probably the complete question/missing part is:"

By observing a set of data values, Thomas used a calculator for the weight (in pounds) and predicted the number of calories burned per minute to get an equation for the least-squares line: ŷ=2.2+0.05x

Based on the information gathered by Thomas, select the statement that is true.

a) A person weighing 149 pounds can burn 9.8 calories per minute.

b) A person weighing 134 pounds can burn 8.9 calories per minute.

c) A person weighing 125 pounds can burn 8.3 calories per minute.

d) A person weighing 173 pounds can burn 10.7 calories per minute.

Answer:

A person weighing 134 pounds can burn 8.9 calories a minute.

Step-by-step explanation:

If a triangle is equilateral, then it is equiangular.

The statement "Equilateral triangles are equiangular" is already in if-then form.

In if-then form, the statement can be written as "If a triangle is equilateral" (the "if" part), followed by "then it is equiangular" (the "then" part).

An equilateral triangle is a triangle in which all three sides are equal in length. Equiangular refers to a triangle having all three angles equal. The statement asserts that if a triangle is equilateral (the "if" condition), then it is also equiangular (the "then" consequence). This relationship holds true for equilateral triangles, making the statement accurate.

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For n = 50, the standard deviation of the sampling distribution is σY/√n = 7.21. So, Pr(ˉY<91) = 0.8365, and for n = 166, the standard deviation of the sampling distribution is σY/√n = 2.82. So, Pr(91<ˉY<94) = 0.5987.

The central limit theorem states that the sampling distribution of the sample mean, ˉY, will be normally distributed with mean μY and standard deviation σY/√n, where n is the sample size.

The theorem states that, as the sample size increases, the sampling distribution of the sample mean will approach a normal distribution, regardless of the shape of the population distribution.

This means that we can use the normal distribution to calculate probabilities about the sample mean, even when we don't know the shape of the population distribution.

In this problem, we were able to use the central limit theorem to calculate the probability that the sample mean would be less than 91 and the probability that the sample mean would be between 91 and 94. These probabilities were calculated using the standard normal distribution, which is a table of probabilities for the normal distribution.

In this problem, we are given that μY = 90 and σY2 = 52. We are asked to find Pr(ˉY<91) and Pr(91<ˉY<94) for two different sample sizes, n = 50 and n = 166.

For n = 50, the standard deviation of the sampling distribution is σY/√n = 7.21. So, Pr(ˉY<91) = 0.8365.

For n = 166, the standard deviation of the sampling distribution is σY/√n = 2.82. So, Pr(91<ˉY<94) = 0.5987.

In conclusion, the central limit theorem allows us to use the normal distribution to approximate the sampling distribution of the sample mean, even when the population distribution is not normally distributed.

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The rational equation and radical equation in one variable are x + 1 = 2x/(x - 1) and √x + 3 = 5.

1. x + 1 = 2x/(x - 1)

To solve this equation, we can first subtract x from both sides, and then multiply both sides by (x - 1). This gives us:

1 = x/(x - 1)

We can then cross-multiply to get:

1 * (x - 1) = x

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This simplifies to x - 1 = x, which means that 0 = 1. This is a contradiction, so there are no solutions to this equation.

2. Radical equations

√x + 3 = 5

To solve this equation, we can first subtract 3 from both sides, and then square both sides. This gives us:

√x = 2

Squaring both sides again gives us:

x = 4

However, this solution is extraneous because the original equation only applies when x is greater than or equal to 3. Since 4 is not greater than or equal to 3, this solution is not valid.

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The data can be tallied into a frequency distribution using a class interval of 100 and a starting point of 0.

To create a frequency distribution, we group the data into intervals or classes and count the number of data points falling within each interval. The class interval represents the range of values covered by each class, and the starting point determines the first interval.

Here is an example of how the data can be tallied into a frequency distribution using a class interval of 100 and a starting point of 0:

```

Class Interval    Frequency

0 - 99            12

100 - 199         18

200 - 299         24

300 - 399         15

400 - 499         10

500 - 599         8

600 - 699         5

700 - 799         3

800 - 899         2

900 - 999         1

```

In this frequency distribution, the data is divided into classes based on the class interval of 100. The first class, from 0 to 99, has a frequency of 12, indicating that there are 12 data points falling within that range. The process is repeated for each subsequent class interval, resulting in a frequency distribution table.

By organizing the data into a frequency distribution, we gain insights into the distribution and patterns within the dataset. It provides a summarized view of the data, allowing us to identify the most common or frequent values and analyze the overall distribution.

In summary, the data has been tallied into a frequency distribution using a class interval of 100 and a starting point of 0. The frequency distribution table presents the number of data points falling within each class interval, enabling a better understanding of the distribution of the data.

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If Muenster, inc., has an equity multiplier of 1.35, total asset turnover of 1.87, and a profit margin of 6.1 percent, its ROE, using the Du Pont Identity, is 15.4%.

Under the DuPont equation or model, the ROE (Return on Equity) equals the profit margin multiplied by asset turnover multiplied by financial leverage.

The ROE can also be represented as equal to PM (operating efficiency measured by profit margin) x TAT (asset use efficiency measured by total asset turnover) x EM (financial leverage measured by equity multiplier)

Equity multiplier = 1.35

Total asset turnover = 1.87

Profit margin = 6.1%

ROE (Return on Equity) using the Du Pont Identity = 6.1 x 1.35 x 1.87

= 15.4%

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The provided information appears to be a citation for a research paper titled "Annotation Artifacts in Natural Language Inference Data" by Suchin Gururangan, Swabha Swayamdipta, Omer Levy, Roy Schwartz, Samuel R. Bowman, and Noah A. Smith. The paper was published as an arXiv preprint with the identifier arXiv:1803.02324.

The given citation refers to a research paper titled "Annotation Artifacts in Natural Language Inference Data" published in 2018.

The research paper, authored by Suchin Gururangan, Swabha Swayamdipta, Omer Levy, Roy Schwartz, Samuel R. Bowman, and Noah A. Smith, addresses the issue of annotation artifacts in natural language inference data. Natural language inference (NLI) is a task in natural language processing that involves determining the logical relationship between pairs of sentences, such as whether one sentence contradicts or entails the other.

Annotation artifacts refer to systematic biases or inconsistencies in the process of annotating NLI data, which can impact the performance of NLI models. The authors of the paper investigate and analyze these artifacts, aiming to understand their prevalence and impact on NLI performance.

In the paper, the authors propose a new methodology called "stress test evaluation" to identify and quantify annotation artifacts. They conduct experiments on large-scale NLI datasets and observe that the presence of these artifacts significantly affects the behavior of state-of-the-art NLI models.

The research paper contributes to the field of natural language processing by shedding light on the challenges posed by annotation artifacts in NLI data and providing insights for improving NLI models and datasets. The citation provided allows others to access and refer to this research paper for further study and exploration of the topic.

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