a particle's motion is described by parametric equations x(t) and y(t) for t ≥ 0 such that
dx/dt = -1/t dan dy/dt = 2/t^2.
if at t = 2 the particle is at (1,-2), which of the following represents the equation of the tangent line to the path of the particle at that time?
a. y = -x - 1
b. y = x - 3
c. y = -x + 1
d. y = x - 1

Triangle KVM is similar to triangle BVG because angle M = angle G = 90° and angle V is common to both triangles.

Two triangles are similar if the angles are the same size or the corresponding sides are in the same ratio.

For two triangles to be similar, the corresponding angles must be congruent i.e equal.. Also the ratio of the corresponding sides of similar triangles are equal.

angle M and G are both 90° , this means they are equal.

angle KVM = BVG

therefore angle K = angle B

Since all the corresponding angles are equal, we can say triangle KVM is similar to triangle BVG

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The Fourier series of the function f(x) on the interval −8 < x < 8 is given by the following expression: f(x) = A0 + Σ(Akcos(kπx/8) + Bksin(kπx/8)). The series consists of a constant term A0 and an infinite sum of cosine and sine terms, where k represents the harmonic frequencies.

To find the Fourier series of f(x), we need to decompose the function into a sum of harmonically related sinusoidal functions. The interval given is divided into two parts: −8 < x < 0 and 0 ≤ x < 8. In the first interval, −8 < x < 0, f(x) is a constant function with a value of 1. The constant term A0 in the Fourier series represents the average value of the function and is given by A0 = 1/2.

In the second interval, 0 ≤ x < 8, f(x) is a linear function with a slope of 1. This part of the function can be expressed as f(x) = x. The coefficients Ak and Bk in the Fourier series represent the amplitudes of the cosine and sine terms, respectively. Ak is given by 1/(kπ), and Bk is given by (2/π)*sin(kπ/2).

By combining the constant term A0 with the cosine and sine terms, we obtain the Fourier series representation of f(x) on the interval −8 < x < 8: f(x) = A0 + Σ(Akcos(kπx/8) + Bksin(kπx/8)). This series represents the function f(x) as an infinite sum of harmonics, which can be used to approximate the original function over the given interval.

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Answer:

4x^3 and -12x^3

Step-by-step explanation:

4x^3 and -12x^3 because they both have x^3 in the expressions which means you can add or take them away from each other to simplify it

The number of boys with heights between 122 and 162 cm  is  499.

We first find  the z-scores for these heights using the formula:

z = (x - μ) / σ

where x =  height,

μ = mean height,

σ =  standard deviation.

case where x = 122 cm:

z = (122 - 148) / 12 = -2.1667

case where  x = 162 cm:

z = (162 - 148) / 12 = 1.1667

We then make use of a standard normal distribution table and determine  area under the curve between these z-scores:

Area under  z = -2.1667 and z = 1.1667 is 0.8315.

Hence, the number of boys with heights between 122 cm and 162 cm is:

600 * 0.8315 = 498.9 or 499 boys.

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#complete question

The heights of 600 boys are found to approximately follow such a distribution, with a mean height of 148 cm and a standard deviation of 12 cm. Find the number of boys with heights between: 122 cm and 162 cm

Answer:

A ≈ 32.03 m²

Step-by-step explanation:

since the 3 sides are congruent then the triangle is equilateral.

the area (A) of an equilateral triangle is calculated as

A = [tex]\frac{s^2\sqrt{3} }{4}[/tex] ( s is the side length )

  = [tex]\frac{8.6^2\sqrt{3} }{4}[/tex]

  = [tex]\frac{73.96\sqrt{3} }{4}[/tex] ( divide numerator/denominator by 4 )

  = 18.49 × [tex]\sqrt{3}[/tex]

  ≈ 32.03 m² ( to 2 decimal places )

Answer:

-2 slope

Step-by-step explanation:

when 2 lines are parallel they have the same slope. line a will have a slope of -2.

the correct option is:y = x² + 2.

Given:  fy= x²+2

To compute: y We know that,

fy = x²+2

By putting the value of fy we get;

y = f(x) = x² + 2

We need to substitute x in the equation to get y.

Therefore, y = x² + 2.

Hence, the correct option is:  y = x² + 2.

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The derivative of a function in calculus is a measure of how quickly the function alters in relation to its independent variable. It calculates the function's slope or rate of change at every given point.

The limit of the difference quotient as the interval approaches 0 is known as the derivative of a function f(x), denoted as f'(x) or dy/dx:

Using the notion of a derivative, we can show that f'(x) = -sin(x) for the function f(x) = cos(x):

lim(h0) = f'(x) [f(x + h) − f(x)] / h

First, let's calculate f(x + h) and f(x):

cos(x + h) = f(x + h).

x = cos(f(x))

We now change these values in the derivative definition to read:

lim(h0) = f'(x) [cos(h + x) - cos(x)] / h

The trigonometric formula cos(a + b) = cos(a)cos(b) - sin(a)sin(b) is then used:

lim(h0) = f'(x) [sin(x)sin(h) − cos(x)cos(h)] / h

Making the numerator simpler:

lim(h0) = f'(x) Sin(x)sin(h) = [cos(x)(cos(h) - 1)] / h

Using the formula cos(0) = 1, say:

lim(h0) = f'(x) Sin(x)sin(h) = [cos(x)(cos(h) - 1)] / h

Next, we divide the numerator's two terms by h:

lim(h0) = f'(x) Sin(x)sin(h) = [cos(x)(cos(h) - 1) / h - h]

As h gets closer to 0, we now take the bounds of each term:

lim(h)[cos(h) - 1][h 0] By applying L'Hôpital's rule and the limit definition of cos(h), / h = 0

According to the limit definition of sin(h), lim(h0) sin(h) / h = 1.

Replacing these restrictions in the derivative expression:

cos(x)(0) = f'(x) - sin(x)(1)

F'(x) = sin(x).

By applying the notion of a derivative, we have demonstrated that if f(x) = cos(x), then f'(x) = -sin(x).

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hello

the answer to the question is B)

explanation:

a point shown on the g(x) graph is (3,3)

if x = 3 and y = 3, therefore:

─ answer A) is incorrect

─ answer B) is the answer since:

(1/3)(x²) = (1/3)(9) = 3

─ answer C) is incorrect since:

((1/3)(x))² = ((1/3)(9))² = 9

─ answer D) is incorrect since:

3x² = 3 × 3² = 27

The Volume of sphere is 3,589.543 ft³.

We have,

Diameter of sphere = 19 ft

Radius of sphere= 19/2

So, the formula for Volume of sphere

= 4/3 πr³

= 4/3 x 3.14 x 19/2 x 19/2 x 19/2

= 86,149.04 / 24

= 3,589.543 ft³

Thus, the Volume of sphere is 3,589.543 ft³.

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The point using Mean Value Theorem for function f(x) = x⁻¹ and interval [1, 7] is,

c = √7.

Mean Value Theorem states that if f(x) is continuous on [a, b] and is differentiable on (a, b) so there is at least one point a < c < b such that

f'(c) = (f(b) - f(a))/(b - a)

Given the function is,

f(x) = x⁻¹ and the interval is = [1, 7]

f(1) = 1⁻¹ = 1 and f(7) = 7⁻¹ = 1/7

Differentiating the function with respect to 'x' we get,

f'(x) = -1 x⁻¹⁻¹ = - x⁻²

Clearly the function f(x) is continuous and differentiable on [1, 7] and (1, 7) respectively since it is polynomial and exists for all points of [1, 7].

So by Mean Value Theorem there exist 1 < c < 7 such that

f'(c) = (f(7) - f(1))/(7 - 1)

- c⁻² = (1/7 - 1)/6 = (-6/7)/6 = - 1/7

- 1/c² = - 1/7

c² = 7

c = ± √7

Since 1 < c < 7 so, c = √7.

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To estimate the area under the graph of f(x) = 20x from x = 0 to x = 4, we can use the concept of numerical integration, specifically the trapezoidal rule.

The trapezoidal rule approximates the area under a curve by dividing the interval into small trapezoids and summing up their areas.

Here's how we can estimate the area using the trapezoidal rule:

Since we don't know the specific value of n, let's assume we divide the interval into 4 subintervals, resulting in Δx = (4 - 0) / 4 = 1.

Now, let's calculate the estimated area using the trapezoidal rule:

Area ≈ [(f(0) + f(1)) * Δx / 2] + [(f(1) + f(2)) * Δx / 2] + [(f(2) + f(3)) * Δx / 2] + [(f(3) + f(4)) * Δx / 2]

Substituting the values of f(x) = 20x:

Area ≈ [(20(0) + 20(1)) * 1 / 2] + [(20(1) + 20(2)) * 1 / 2] + [(20(2) + 20(3)) * 1 / 2] + [(20(3) + 20(4)) * 1 / 2]

= [(0 + 20) * 1 / 2] + [(20 + 40) * 1 / 2] + [(40 + 60) * 1 / 2] + [(60 + 80) * 1 / 2]

= [10] + [30] + [50] + [70]

= 160

Therefore, the estimated area under the graph of f(x) = 20x from x = 0 to x = 4 is approximately 160 square units.

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The value of the given expression 6(1 + 7j) equal to 6 + 42j.

If we have given a vector v of initial point A and terminal point B

v = ai + bj

then the components form will be

AB = xi + yj

Here, xi and yj are the components of the vector.

We can calculate the expression 6(1 + 7j) as;

6 + 42j

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Answer:

Step-by-step explanation:

In testing the hypotheses H0: p = 0.5 vs Ha: p > 0.5, the test statistic is found to be 1.83. We need to determine the correct p-value. From the options provided, the correct p-value is d) 0.0336

The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed test statistic, assuming that the null hypothesis is true. Since this is a right-tailed test (Ha: p > 0.5), we are interested in the probability of observing a test statistic larger than 1.83. Looking at the given options, the correct p-value would be the smallest value that corresponds to a probability larger than 1.83. From the options provided, the correct p-value is d) 0.0336, as it represents a probability smaller than 1.83. Therefore, 0.0336 is the correct p-value for this hypothesis test.

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Let's suppose that the given polynomial equation is P(z), and it is a real and monic polynomial of degree n. We are supposed to find the real, monic polynomial of the lowest possible degree that has zeros 2-2i, -3i and 2i, using z as the variable.

Given zeros are as follows:

2 - 2i-3i2iTherefore, the complex conjugates of the first and third zeros will also be roots of the given polynomial, so we also have:2 + 2iand-2ias roots of the given polynomial.

The polynomial that has roots 2 - 2i, 2 + 2i, 2i, and -3i is: (z - (2 - 2i))(z - (2 + 2i))(z - 2i)(z + 3i)

Expanding it we get;= (z - (2 - 2i))(z - (2 + 2i))(z - 2i)(z + 3i)= (z - 2 + 2i)(z - 2 - 2i)(z - 2i)(z + 3i)

Now let us multiply and simplify the above expression to get the polynomial in a monic form by expanding the product of first two terms as follows:

=(z - 2)² - (2i)² (z - 2i)(z + 3i)=(z - 2)² - 4(z - 2i)(z + 3i)

By expanding and simplifying the above expression we get;= z4 - 2z³ - 7z² + 12z + 40The required real, monic polynomial of the lowest possible degree is z⁴ - 2z³ - 7z² + 12z + 40.

Therefore, the answer is  z⁴ - 2z³ - 7z² + 12z + 40.

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There can be 6,760,000 different license plates manufactured for this state.

To calculate the number of different license plates that can be manufactured for this state, we need to consider the number of options for each character position.

For the two letters, there are 26 options for each letter (A-Z), so the total number of combinations is 26 × 26 = 676.

For the four numbers, there are 10 options for each number (0-9), so the total number of combinations is 10 × 10 × 10 × 10 = 10,000.

To find the total number of different license plates, we multiply the number of combinations for the letters by the number of combinations for the numbers:

676 × 10,000 = 6,760,000.

Therefore, there can be 6,760,000 different license plates manufactured for this state.

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The order of the product ab in the abelian group is lcm(m, n).

In an abelian group, the order of the product of two elements can be determined using the concept of the least common multiple (LCM) of their individual orders.

Let a and b be elements of an abelian group, where the order of a is m and the order of b is n. The order of an element in a group is defined as the smallest positive integer k such that the element raised to the power of k yields the identity element.

In this case, the order of the product ab can be determined by considering the LCM of m and n, denoted as lcm(m, n). The LCM is the smallest positive integer that is divisible by both m and n.

Therefore, the order of the product ab in the abelian group is lcm(m, n).

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Their yearly contributions should be $54,193.29. To pay for this, they will contribute 12 equal yearly payments to an account bearing interest at the APR of 6.3%

To pay for this, they will contribute 12 equal yearly payments to an account bearing interest rate  at the APR of 6.3%, compounded annually. Five years after their last contribution, they will begin the first of five, yearly, withdrawals of $34,700 to pay the university's bills. We have to determine the size of their yearly contribution. We can use the formula for the future value of an annuity to solve this problem.  

Formula used:FV = P × ((1 + i)n - 1) / iWhere, FV is the future value,P is the payment amount per period,  i is the interest rate per period, andn is the number of periods.  As given, Interest rate (i) = 6.3%, compounded annually.N = 12 years and 5 yearsWe have to find the value of P, which is the payment amount per period. From the formula of the future value of an annuity, we can write the formula as:    

FV = P × ((1 + i)n - 1) / i  where, FV is the future value of the annuity. We need to calculate FV at the end of 12 years, which will be the present value of their yearly contributions to the university fund. Then, we will use this present value to calculate the payment amount per year. We have n = 12, i = 0.063, and P = Not known FV = P × ((1 + i)n - 1) / i   = P × ((1 + 0.063)12 - 1) / 0.063   = P × 9.5425

Therefore, P = FV / 9.5425 We know that the value of their yearly withdrawals will be $34,700, starting from the end of the 17th year. Therefore, we need to calculate the present value of these withdrawals, which will be the future value of their yearly contributions over the next 17 years. We have n = 17, i = 0.063, and P = $pmt (calculated above) FV = P × ((1 + i)n - 1) / i   = P × ((1 + 0.063)17 - 1) / 0.063   = P × 14.8921  

The present value of the withdrawals = $34,700 × 14.8921 = $516,781.07 This present value should be equal to the future value of their contributions. So, we can equate the two present values and solve for P. Present value of their contributions = FV of the withdrawals = $516,781.07 P = FV / 9.5425 = $516,781.07 / 9.5425 = $54,193.29 Therefore, their yearly contributions should be $54,193.29.

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7.54 and 0.46 are the solutions to the given quadratic equations

Given the quadratic equation below:

x^2-18x+8=0

We need to determine the solutions to the given quadratic expression. Using the general formula below:

x = -b±√b²-4ac/2a

From the equation

a = 1

b = -18

c = 8

Substitute

x = 18±√18²-4(1)(8)/2(1)
x= 18±√324-32/2

x =18± 17.08/2

x = 35.08/2 and 0.92/2

x = 17.54 and 0.46

Hence the two solutions to the given quadratic equation are 17.54 and 0.46

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The two tow trucks are exerting forces of 2,850 N and 2,655 N on a stuck truck via 12 m long towing straps attached to its hitch. The angle between the two trucks is 42. We have to determine the magnitude of the resultant force.

The formula to find the magnitude of the resultant force is given below:[tex]F = √(F₁² + F₂² + 2F₁F₂cosθ) where, F₁ = 2,850 NF₂ = 2,655 Nθ = 42 degrees F = √(2,850² + 2,655² + 2(2,850)(2,655)cos(42))F = 4,325 N (rounded off to th[/tex]e nearest whole number) Hence, the magnitude of the resultant force is 4,325 N.

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A perceptron is a simple linear classifier used in machine learning to make predictions based on the given inputs.

In this case, the perceptron has three inputs: the always 1 feature (bias term), and two binary features x1 and x2. The output y is also binary, either 0 or 1. The perceptron takes the input features and calculates a weighted sum of these values. If the sum is above a certain threshold, the perceptron outputs a 1, otherwise, it outputs a 0. The weights for the input features, as well as the threshold, are determined through a training process. The always 1 feature acts as a bias term that allows the decision boundary to be shifted away from the origin.

To summarize, the given perceptron has three inputs (always 1 feature and two binary features x1, x2) and a binary output y. It calculates a weighted sum of the input features and compares it to a threshold to determine the output. This model can be used to classify data into two classes based on the input features x1 and x2.

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Based on the linear best-fit model, when Mitchell is 62 inches tall, So, approximately he will weigh 312 pounds.

To estimate Mitchell's weight when he is 62 inches tall using a linear best-fit model, we need to determine the equation of the line that best represents the relationship between height and weight based on the given data.

We can use the least squares method to find the equation of the line. By fitting a line to the data points, we can determine the slope (m) and y-intercept (b) of the line.

Using statistical software or calculations, the equation of the best-fit line for the given data is estimated to be:

Weight = 4.96 * Height + 4.48

To find Mitchell's estimated weight when he is 62 inches tall, we substitute 62 for Height in the equation:

Weight = 4.96 * 62 + 4.48

Weight = 307.52 + 4.48

Weight = 312 pounds

Therefore, based on the linear best-fit model, Mitchell is estimated to weigh approximately 312 pounds when he is 62 inches tall.

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Answer:

Step-by-step explanation:

Which ones have four sides?

A

B

D

The simplified form of the given expression is (3x - 4x^2 - 15x - 20)/(7x^2 - 33x - 70).

o simplify the expression (-4)/(x-5) + 3x/(7 (x+2)), we need to find a common denominator and combine the fractions.

   The first term (-4)/(x-5) has the denominator (x-5), while the second term 3x/(7 (x+2)) has the denominator 7(x+2). To find a common denominator, we multiply the first term by 7(x+2) and the second term by (x-5).

   After multiplying, we get (-4)(7(x+2))/(7(x+2)(x-5)) + (3x)(x-5)/(7(x+2)(x-5)).

   Simplifying the numerator, we have -28x - 56 + 3x^2 - 15x.

   Combining like terms, the numerator becomes -4x^2 - 43x - 56.

   The denominator remains as 7(x+2)(x-5).

   The final simplified expression is (-4x^2 - 43x - 56)/(7(x+2)(x-5)).

Now, let's simplify the second expression: 1/(x^2+7x) + 2/(49-x^2).

   The denominators are x^2+7x and 49-x^2. To find a common denominator, we multiply the first term by (49-x^2) and the second term by (x^2+7x).

   After multiplying, we get (49-x^2)/(x^2+7x)(49-x^2) + (2)(x^2+7x)/(x^2+7x)(49-x^2).

   Simplifying the numerator, we have (49-x^2) + 2x^2 + 14x.

   Combining like terms, the numerator becomes 51 + x^2 + 14x.

   The denominator remains as (x^2+7x)(49-x^2).

   The final simplified expression is (51 + x^2 + 14x)/[(x^2+7x)(49-x^2)].

Therefore, the simplified form of the given expression is (3x - 4x^2 - 15x - 20)/(7x^2 - 33x - 70) + (51 + x^2 + 14x)/[(x^2+7x)(49-x^2)].

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The standard deviation of the duration of this task is 4 days

To determine the standard deviation of the duration of the task, we can use the information about the confidence interval and the properties of the normal distribution.

In a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. Since the confidence interval provided (81 to 85 days) represents the range within one standard deviation from the mean, we can find the standard deviation by calculating the range between the upper and lower limits of the confidence interval.

The range of the confidence interval is given by:

Range = Upper Limit - Lower Limit

= 85 - 81

= 4

Since this range corresponds to one standard deviation, the standard deviation of the duration of the task is also 4 days.

Therefore, the standard deviation of the duration of this task is 4 days.

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The hypothesis test for the claim that the majority of adults are in favor of raising the voting age to 21 is a right-tailed test. So, correct option is C.

In this scenario, the claim is that the majority of adults (more than 50%) are in favor of raising the voting age. This implies a specific directionality in the hypothesis being tested.

A left-tailed test would be appropriate if the claim was that the proportion of adults in favor is less than 50%. The alternative hypothesis would state that the proportion is less than 50%, and the critical region would be on the left side of the distribution.

A right-tailed test would be appropriate if the claim was that the proportion of adults in favor is greater than 50%. The alternative hypothesis would state that the proportion is greater than 50%, and the critical region would be on the right side of the distribution.

Since the claim is that the majority (more than 50%) of adults are in favor, it is a right-tailed test. The alternative hypothesis would be that the proportion is greater than 50%, and the critical region would be on the right side of the distribution.

So, correct option is C.

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Complete question is:

Suppose we want to test the claim that the majority of adults are in favor of raising the voting age to 21. Is the hypothesis test left-tailed, right-tailed, or two-tailed?

A. Left-tailed

B. Two-tailed

C. Right-Tailed

Flowcharts are used to summarize large amounts of statistical data and indicate trends over time. The correct options for the flow charts are to summarize large amounts of statistical data and indicate trends over time

Flowcharts are a powerful tool for both technical and non-technical people, providing a visual representation of complex information. They are used to map out workflows, decision-making processes, and other systems.

By breaking down a process into simple steps and depicting them visually, flowcharts allow users to understand the flow of information and actions, making it easier to identify bottlenecks, inefficiencies, and opportunities for improvement. Flowcharts can also help to standardize processes, ensuring that all stakeholders are aligned on the correct procedures.

Overall, flowcharts are a valuable tool for any organization seeking to streamline processes, improve efficiency, and communicate complex ideas in a simple and easy-to-understand format.

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Answer:

u = -1/5

Step-by-step explanation:

name me brainliest please.

The probability that the sample proportion will be between 0.9115 and 0.946 is 0.1496.

To calculate the probability that the sample proportion will be between 0.9115 and 0.946, we can use the sampling distribution of the sample proportion, assuming that the sample is taken from an infinite population.

The standard deviation of the sample proportion is given by:

σ_p = sqrt((p * (1 - p)) / n)

where p is the population proportion and n is the sample size.

In this case, p = 0.85 and n = 51. Plugging these values into the formula, we get:

σ_p = sqrt((0.85 * (1 - 0.85)) / 51)

= sqrt(0.127275 / 51)

≈ 0.092

Now, we can standardize the interval (0.9115, 0.946) using the sample proportion distribution:

z1 = (0.9115 - p) / σ_p

= (0.9115 - 0.85) / 0.092

≈ 0.667

z2 = (0.946 - p) / σ_p

= (0.946 - 0.85) / 0.092

≈ 1.043

Next, we can calculate the probability using the standard normal distribution:

P(0.9115 < p < 0.946) = P(z1 < Z < z2)

Looking up the values in the standard normal distribution table, we find:

P(0.9115 < p < 0.946) ≈ P(0.667 < Z < 1.043)

≈ 0.1496

Therefore, the probability that the sample proportion will be between 0.9115 and 0.946 is approximately 0.1496.

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