The solution to a radical expression numbers in pictures ​

The line integral of F(8y, 8x) over the circle 32 + y2 = 81 oriented

clockwise is 9.

To compute the line integral of the vector field F(8y, 8x) over the circle

32 + y2 = 81 oriented clockwise, we can use the line integral formula:

∫CF · ds = ∫ab F(r(t)) · r'(t) dt

where C is the curve we are integrating over, F is the vector field, and r(t)

is a parametrization of the curve.

We can parametrize the circle by setting x = 4cos(t) and y = 9sin(t), with t

ranging from 0 to 2π. Then, the curve C becomes:

r(t) = (4cos(t), 9sin(t))

and the unit tangent vector is given by:

T(t) = r'(t)/|r'(t)| = (-4sin(t)/3, 3cos(t)/2)

Note that we divide by the magnitude of r'(t) to get a unit tangent vector.

Then, we can compute the line integral as:

[tex]\int CF ds = \int 0^2 \pi F(r(t)) r'(t) dt[/tex]

= ∫[tex]0^2[/tex]π (8y, 8x) · (-4sin(t)/3, 3cos(t)/2) dt

= ∫[tex]0^2[/tex]π (-32sin(t)cos(t)/3 + 36cos(t)sin(t)/2) dt

= (-16/3)∫[tex]0^2[/tex]π sin(2t) dt + (18)∫[tex]0^2[/tex]π cos(t)sin(t) dt

= (-16/3)[cos(2t)][tex]0^2[/tex]π + (18)[(-1/2)cos2(t)][tex]0^2[/tex]π

= (-16/3)(1-1) + (18)(0+1/2)

= 9

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Answer:

43rd term

Step-by-step explanation:

Given arithmetic sequence:

We can use the formula for the nth term of an arithmetic sequence to find which term the number 203 corresponds to in the given arithmetic sequence.

The formula for the nth term of an arithmetic sequence is:

[tex]\boxed{a_n = a_1 + (n-1)d}[/tex]

where:

For the given sequence, we know that the first term is a₁ = -7.

The common difference is d = 5, since each term is 5 more than the previous term.

Substitute these values into the formula to create an equation for the nth term:

[tex]\begin{aligned} \implies a_n &= -7 + (n-1)5\\&=-7+5n-5\\&=5n-12 \end{aligned}[/tex]

To find the position of number 203 in the sequence, substitute aₙ = 203 into the equation for the nth term and solve for n:

[tex]\begin{aligned} a_n &= 203\\ \implies5n-12&=203\\5n-12+12&=203+12\\5n&=215\\\dfrac{5n}{5}&=\dfrac{215}{5}\\n&=43 \end{aligned}[/tex]

Therefore, the number 203 corresponds to the 43rd term in the given arithmetic sequence.

Therefore, the solutions of the equation 15x⁴ + x³ - 52x² + 20x + 16 = 0 Reducible to Quadratic Equations are:

x₁ = (±2/√5

x₂ = (±2/√3)

x₃ = 0

x₄ = (±1/20)

To solve 15x⁴ + x³ - 52x² + 20x + 16 = 0 by Equations Reducible to Quadratic Equations, we can use the substitution method. Let's first substitute x² = y and rewrite the equation as:

15y² + y - 52y + 20√y + 16 = 0

Now, let's group the terms:

(15y² - 52y + 16) + (y + 20√y) = 0

Let's solve the first quadratic equation:

15y² - 52y + 16 = 0

We can factor this quadratic equation as:

(3y - 4)(5y - 4) = 0

So, the solutions are:

y₁ = 4/5 and y₂ = 4/3

Now, let's solve the second quadratic equation:

y + 20√y = 0

We can factor out y:

y(1 + 20√y) = 0

So, the solutions are:

y₃ = 0 and y₄ = (-1/20)² = 1/400

Now, let's substitute back y = x²:

x₁ = √(y₁) = ±√(4/5) = ±(2/√5)

x₂ = √(y₂) = ±√(4/3) = ±(2/√3)

x₃ = √(y₃) = 0

x₄ = √(y₄) = ±(1/20)

Therefore, the solutions of the equation 15x⁴ + x³ - 52x² + 20x + 16 = 0 are:

x₁ = (±2/√5)

x₂ = (±2/√3)

x₃ = 0

x₄ = (±1/20)

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There are 182 buttons that have to be sold to make a profit of $500. And, the equation that represents the given situation is [tex]\$500 = \$2b + \$0.75b[/tex] or [tex]\$500 - \$0.75b = \$2b[/tex] . Therefore, options B and D are true.

Using the idea behind the equation that says,

A set of numerical variables and functions coupled via the use of operations like addition, subtraction, multiplication, and division make form an equation.

Given that,

For homecoming, the Booster Club at Martin Middle School is selling pride buttons.

And, The buttons cost $0.75 to make and will be sold for $2 each.

Let us assume that,

Number of buttons = b

Since we have to calculate the number of buttons to make a profit of $500.

Hence the equation can be written as,

[tex]\$500 = \$2b + \$0.75b[/tex]

Simplify the equation for b,

[tex]\$500 = \$2.75b[/tex]

Divide both sides by 2.75,

[tex]b = \dfrac{500}{2.75}[/tex]

[tex]b = 182[/tex]

Therefore, the correct option is B and D.

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Answer:

8x

Step-by-step explanation:

f(x) + g(x) = (3x-2) + (5x+2)

f(x) + g(x) = 3x + 5x - 2 + 2

f(x) + g(x) = 8x

A. Decimal squares and C. Base ten pieces would be the foremost reasonable manipulative materials for instructing decimal documentation to the hundredths put.

Decimal squares can offer assistance to understudies to visualize the relationship between tenths, hundredths, and thousandths, as each square can be separated into 10 little squares. Understudies can utilize squares to construct and compare decimal numbers to the hundredths put.

Base ten squares can too be utilized to speak to decimals to the hundredths put, with one level speaking to one entirety, one bar speaking to one-tenth, and one unit speaking to one-hundredth. Understudies can construct and compare decimal numbers utilizing the pieces, as well as utilize them to show operations with decimals.

The other manipulative materials recorded (Design pieces, Tangrams, Color Tiles, and Geoboards) are not particularly planned to speak to decimals and would likely not be as viable in instructing decimal documentation to the hundredths put. 

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Answer:

60

Step-by-step explanation:

a) The Formula for A(t) is A(t) = 5000 [tex]e^{0.075t[/tex]

b) The differential equation is satisfied by A(t) is

dA/ dt = 375  [tex]e^{0.075t[/tex]

c) Amount after 2 year is $5, 809.

d) t= 4.48 years

We have,

R= 7.5%

P= $5000

a) The Formula for A(t) is

A(t) = P[tex]e^{rt[/tex]

Where P is Principal , t is time.

So, A(t) = 5000 [tex]e^{0.075t[/tex]

b) The differential equation is satisfied by A(t) is

dA/ dt = 375  [tex]e^{0.075t[/tex]

c) Amount after 2 year

A(2) = 5000 (2.71828[tex])^{0.15[/tex]

A(2) = 5000 x 1.1618

A(2)= $5, 809.

d) 7000 = 5000   [tex]e^{0.075t[/tex]

 [tex]e^{0.075t[/tex]= 1.4

Taking log on both side

0.075t log e= log 1.4

0.075t=   0.14612803567/0.4342944819

0.075t= 0.3364

t= 4.48 years

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Answer:

$0.78

Step-by-step explanation:

You divide the $7.80 by 10 to get the cost of one tulip.

The answer of the given question based on the commutative property is ,  choice B: c + (a + b).

The commutative property is a fundamental property of some mathematical operations, which states that the order of the operands (inputs) can be changed without affecting the result. In other words, the commutative property means that the operation is independent of the order in which the operands are presented.

The commutative property of addition states that the order of the addends can be changed without changing the sum. In other words, a + b = b + a.

Using this property, we can rearrange the terms in the equation (a + b) + c = A to get:

c + (a + b) = A

This is the same as choice B: c + (a + b). Therefore, the choice that illustrates the commutative property for (a + b) + c is B.

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It is true that when conducting a t-test for two independent samples with a two-tailed example and α = 0.05, you divide the α by 2 (0.05/2 = 0.025) to find the critical t-scores that form the boundaries of the critical region. This is because you are considering both tails of the distribution.

True. When conducting a t test for two independent samples with a two-tailed example, we need to find the critical t-scores that form the boundaries of the critical region for α = 0.05. To do this, we divide the α value of 0.05 by 2 to get 0.0250, and then use this value to find the t-scores using a t-distribution table or calculator. This is necessary because we are looking at the possibility of a significant difference in either direction, hence the two-tailed example.

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The probability of selecting apple, a peach and a pear in that order is [tex]4.85%[/tex]%.

In this basket, we have 4 apples, 4 peaches, and 4 pears and will select three pieces of fruit without replacement.

The probability of selecting an apple first is 4/12.

The probability of selecting a peach second is 4/11.

The probability of selecting a pear third is 4/10.

P(Apple, Peach, & Pear) = (4/12) * (4/11) * (4/10)

P(Apple, Peach, & Pear) = 64/1320

P(Apple, Peach, & Pear) = 8/165

P(Apple, Peach, & Pear) = 0.04848484848

P(Apple, Peach, & Pear) = 4.85%

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By squaring the deviations from the mean, we make them positive numbers, and the sum of the squared deviations will also be a positive number.

In mathematics and statistics, deviation refers to the difference between a value and a reference value or an expected value. More specifically, deviation is a measure of how far a set of numbers is spread out from their average value or the central point.

By squaring the deviations from the mean, we make them positive numbers, and the sum of the squared deviations will also be a positive number. This is because squaring any number makes it positive, regardless of whether the original number was positive or negative.

Additionally, squaring the deviations before summing them up allows us to give more weight to larger deviations from the mean. This is because squaring a larger deviation will result in a much larger value than squaring a smaller deviation, which helps to highlight the effect of outliers or extreme values in the data set.

The sum of the squared deviations is used in many statistical calculations, including calculating the variance and standard deviation of a data set, which are measures of the spread of the data.

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Answer:

57,310

Step-by-step explanation:

So you can either round up to 57,310 or round down to 57,300 and 57,309 is one away from 57,310 and 9 away from 57,300 so it’s closer to 57,310 therefore 57,310 is the answer

To round 57,309 to the nearest ten, we look at the ones place digit and round up if the digit is 5 or greater. In this case, since the ones place digit is 9, we round up the tens digit to 1.

To round 57,309 to the nearest ten, we look at the digit in the ones place, which is 9. Since 9 is greater than or equal to 5, we round up the tens digit to the next number. Therefore, 57,309 rounded to the nearest ten is 57,310.

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It is not required to calculate the standard error of a sample mean or the standard error of an estimate in general.

False. The standard error is a measure of the variability of the sample mean and is calculated using the sample standard deviation and sample size, without necessarily requiring the calculation of the pooled variance.

The pooled variance, on the other hand, is a statistic used in hypothesis testing when comparing means from two independent samples, assuming that the two populations have equal variances. It is calculated by pooling the variances of the two samples, weighted by their degrees of freedom, and is used to calculate the standard error of the difference between the means.

While the pooled variance can be used to calculate the standard error of the difference between two means, it is not required to calculate the standard error of a sample mean or the standard error of an estimate in general.

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Answer:

Step-by-step explanation:

Complementary Angles add up to 90°

90°-46°=44°

∠m = 44°

1. The dimensions of the printed area are approximately:

4.22 in (smaller value)

6.89 in (larger value).

2. An approximation for t, we can use it to find the point on the graph of f(x) that is closest to (x, y):

t = approximate solution from Newton's method

x-coordinate: t

y-coordinate: [tex]t^2 (21).[/tex]

Let the width of the printed area be x and the length of the printed area be y.

Then the total area of the page, including margins, is:

A = (x + 2)(y + 2)

The area of the printed portion is:

xy = 29

We want to minimize the total area, subject to the constraint that the printed area has an area of 29 square inches.

Using the constraint, we can solve for y in terms of x:

y = 29/x

Substituting this into the equation for A, we get:

A = (x + 2)(29/x + 2)

Expanding this expression, we get:

A = 29 + 2x + 58/x + 4

A = 2x + 58/x + 33.

To find the minimum value of A, we take the derivative with respect to x and set it equal to zero:

[tex]dA/dx = 2 - 58/x^2 = 0[/tex]

Solving for x, we get:

[tex]x = \sqrt{(58)}[/tex]

Substituting this back into the equation for y, we get:

[tex]y = 29/\sqrt{(58) }[/tex]

Therefore, the dimensions of the printed area are approximately:

4.22 in (smaller value)

6.89 in (larger value)

To find the point on the graph of the function [tex]f(x) = x^2 (21)[/tex]that is closest to the point (x, y), we can use the distance formula:

[tex]d = \sqrt{((x - t)^2 + (y - f(t))^2) }[/tex]

where t is the value of x that corresponds to the closest point on the graph, and[tex]f(t) = t^2 (21)[/tex]

We want to minimize d, so we take the derivative of d with respect to t and set it equal to zero:

[tex]dd/dt = (x - t) - 2t(21)(y - f(t)) = 0[/tex]

Expanding f(t), we get:

f(t) = 21t^2

Substituting this into the equation for dd/dt, we get:

[tex]dd/dt = (x - t) - 42ty + 42t^3 = 0[/tex]

Solving for t is difficult, but we can use an iterative numerical method, such as Newton's method, to approximate the solution.

We can start with an initial guess, [tex]t_0[/tex] , and use the iteration:

[tex]t_{n+1} = t_n - dd/dt(t_n) / d^2d/dt^2(t_n)[/tex]

where [tex]dd/dt(t_n)[/tex]  is the value of dd/dt at [tex]t_n[/tex], and [tex]d^2d/dt^2(t_n)[/tex] is the second derivative of d with respect to t evaluated at[tex]t_n.[/tex]

We can continue this iteration until the value of [tex]t_n[/tex] stops changing or until we reach a desired level of accuracy.

Once we have an approximation for t, we can use it to find the point on the graph of f(x) that is closest to (x, y):

t = approximate solution from Newton's method.

x-coordinate: t

y-coordinate: [tex]t^2 (21).[/tex]

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The total distance traveled by the airplane is 15.3 feet.

The Pythagorean theorem asserts that the square of the length of the hypotenuse in a right triangle equals the sum of the squares of the lengths of the two legs in a right triangle. This theorem is the basis for the distance formula. In the distance formula, the hypotenuse is the distance between the two points, and the two legs are the differences between the x- and y-coordinates of the two points. Any two locations in a two-dimensional coordinate system can have their distance between them calculated using the formula.

Let us suppose the starting point, that is, the point for jack as (0, 0).

Now, according to the given placements the position of the other students are:

Destiny: (0,2)

Barbara: (-9,2)

Now, using the distance formula we have:

The distance between Jack and Destiny is:

√[(0-0)² + (2-0)²] = √(4) = 2

The distance between Destiny and Barbara is:

√[(-9-0)² + (2-2)²] = √(81) = 9

The distance between Barbara and Jack is:

√[(0-(-9))² + (0-2)²] = √(85)

So the total distance traveled by the paper airplane is:

2 + 9 + √(85) ≈ 15.3 feet

Hence, the total distance traveled by the airplane is 15.3 feet.

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According to the histogram, a total of five participants had a heart rate between 120 and 130 bpm.

Review the histogram: Look at the histogram and locate the section that represents heart rates between 120 and 130 bpm.

Count the bars: Count the number of bars within that section.

Interpret the bars: Each bar represents one participant, so the total number of bars counted in the previous step represents the number of participants with heart rates between 120 and 130 bpm.

Identify the answer: The total number of bars counted is the answer to the question, which is five.

Therefore, according to the histogram, five participants had a heart rate between 120 and 130 bpm.

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Answer:

5+(-9)=14

Step-by-step explanation:

Answer:

14 and 23

Step-by-step explanation:

To find two arithmetic means between 5 and 23, we need to first find the common difference between consecutive terms.

The common difference (d) between consecutive terms in an arithmetic sequence can be found using the formula:

d = (an - a1) / (n - 1)

where a1 is the first term, an is the last term, and n is the number of terms.

In this case, a1 = 5, an = 23, and n = 3 (since we want to find two means, there will be a total of 4 terms in the sequence). Plugging these values into the formula, we get:

d = (23 - 5) / (3 - 1) = 9

So the common difference between consecutive terms is 9. To find the first mean, we add the common difference to the first term:

First mean = 5 + 9 = 14

To find the second mean, we add the common difference to the first mean:

Second mean = 14 + 9 = 23

Therefore, the two arithmetic means between 5 and 23 are 14 and 23.

The mean number of American adults who disapprove of the overall job that President Bush is doing, based on a sample size of 140, is approximately 93.8.

To calculate the mean, we need to first understand what it represents. The mean is a measure of central tendency, which represents the average value of a set of data. In this case, we want to find the average number of American adults who disapprove of President Bush's overall job.

Since we know that 67% of respondents disapproved of President Bush's overall job, we can assume that this percentage also applies to the entire population of American adults.

To find the mean number of American adults who disapprove, we can use the formula:

mean = total / sample size

In this case, the total number of American adults who disapprove can be calculated as:

total = sample size x percentage who disapprove

total = 140 x 0.67

total = 93.8

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Measures of m∠1 = 67.4°, m∠2 = 104.5°

What are the measures of angle?

When two lines or rays intersect at a single point, an angle is created. The vertex is the term for the shared point. An angle measure in geometry is the length of the angle created by two rays or arms meeting at a common vertex.

Here, we have

The exterior angle 121.8° is the sum of the remote interior angles 17.3° and angle 2. Then ...

angle 2 = 121.8° -17.3° = 104.5° . . . . . . . matches the second choice

Angle 2 is the exterior angle of the top triangle. It, too, is the sum of the remote interior angles:

104.5° = angle 1 + 37.1°

angle 1 = 104.5° -37.1° = 67.4°

Hence, the measures of m∠1 = 67.4°, m∠2 = 104.5°.

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The mean slope of the function on the interval [-6, 10] is:

Mean slope is -62.

To find the mean slope of the function f(x) on the interval [-6, 10], we

need to calculate the total change in the function over that interval and

divide by the length of the interval:

Mean slope = (f(10) - f(-6)) / (10 - (-6))

We can simplify this by first finding the derivative of the function f(x):

[tex]f'(x) = 6x^2 - 18x - 108[/tex]

Then, we can use the mean value theorem to find the two values of c

where the instantaneous slope of the function equals the mean slope:

[tex]c1 = (-6 + \sqrt{(783)} ) / 3\\c2 = (-6 - \sqrt{ (783)} ) / 3[/tex]

Plugging these values into the derivative of f(x) gives the instantaneous

slope at each value of c:

f'(c1) = 165

f'(c2) = -33

Therefore, the mean slope of the function on the interval [-6, 10] is:

Mean slope = (f(10) - f(-6)) / (10 - (-6)) = (57 - 1015) / 16 = -62

And we can conclude that the mean slope of the function f(x) on the

interval [-6, 10] is equal to -62.

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Answer:

4

Step-by-step explanation:

80 divided by 4 is 20 then you take 20 and divide it by 5 to get 4 so in conclusion X = 4

The null hypothesis (H0) is that there is no difference in the average time taken to read a book before and after the program.

A middle school teacher conducts a study to evaluate the impact of a reading rewards program on the time taken by students to read a book. To analyze the data, the teacher will use a two-sample t-test. The population parameters being compared are the average time taken to read a book before and after the program.

The level of significance is a threshold value that determines the probability of making a Type I error (rejecting a true null hypothesis). It is usually denoted by  (alpha) and is typically set at 0.05, which means there is a 5% chance of making a type I error. However, the specific level of significance will depend on the teacher's chosen value.

In this case, the t-test should be a paired t-test because the same group of students is being compared before and after the program, making the two samples dependent on each other.

The null hypothesis (H0) is that there is no difference in the average time taken to read a book before and after the program. In other words, the average difference between the pre- and post-program reading times is equal to zero.

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Logarithmic differentiation to evaluate f′(x)f(x)=(3x)ln3xf′(x) will give f′(x) = 3ln3x + 3/x.

To use logarithmic differentiation to evaluate f′(x) for the given function f(x)=(3x)ln3x, we first take the natural logarithm of both sides:
ln(f(x)) = ln[(3x)ln3x]
Using the properties of logarithmic, we can simplify this expression:
ln(f(x)) = ln(3x) + ln(ln3x)
ln(f(x)) = ln(3) + ln(x) + ln(ln3) + ln(x)
Now we differentiate both sides of this equation with respect to x, using the chain rule on the right-hand side:
(1/f(x))f′(x) = 1/x + 1/x ln3 + 1/ln3 + 1/x
Simplifying this expression, we get:
f′(x) = f(x) [(1/x) + (1/x ln3) + (1/ln3) + (1/x)]
Substituting in the original expression for f(x), we get:
f′(x) = (3x)ln3x [(1/x) + (1/x ln3) + (1/ln3) + (1/x)]
Simplifying this expression, we get:
f′(x) = 3ln3x + 3/x
Therefore, the derivative of f(x) is f′(x) = 3ln3x + 3/x.

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The economic impact of fishing for nearly all great lakes states varies, but according to a report by the U.S. Fish and Wildlife Service, it falls within the range of $1 to $8 billion (in millions of dollars).

This impact includes the economic contributions of recreational fishing, commercial fishing, and related industries such as tourism and boat manufacturing. The exact amount varies from state to state and from year to year depending on factors such as weather, fish populations, and fishing regulations.

The Great Lakes region of the United States is home to some of the largest freshwater bodies in the world and boasts a rich variety of fish species. Fishing is an important economic activity in the region, contributing billions of dollars to the local and national economy. The economic impact of fishing in the Great Lakes region includes not only the direct revenue generated by commercial and recreational fishing, but also the indirect and induced effects of fishing-related industries such as tourism and boat manufacturing.

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a) The probability of having at least one major fault in the next 2 km stretch on the highway is approximately 0.959.

b) The distribution of X, the distance between two successive major faults on the highway, is X ~ Exponential (mean=1/1.6). Therefore, the correct option is B.

a) To find the probability of having at least one major fault in the next 2 km stretch on the highway, we first need to find the probability of having zero major faults in that stretch. The number of major faults follows a Poisson distribution with mean λ = 1.6 for every 1 km.

Since we are looking at a 2 km stretch, the new mean is λ' = 2 * 1.6 = 3.2.

Using the Poisson probability mass function (PMF) formula:

P(X = k) = (e^(-λ) * λ^k) / k!

For zero major faults in the 2 km stretch (k = 0):

P(X = 0) = (e^(-3.2) * 3.2^0) / 0! = e^(-3.2)

Now, we want the probability of having at least one major fault, which is the complement of having zero faults:

P(X >= 1) = 1 - P(X = 0) = 1 - e^(-3.2)

Calculating this value:

P(X >= 1) ≈ 1 - 0.040762 = 0.959

So, the probability is approximately 0.959.

b) The distribution of X, the distance between two successive major faults on the highway, is described by an exponential distribution. In this case, the mean distance between major faults is the inverse of the rate (mean number of faults per km).

The rate is λ = 1.6 faults per km, so the mean distance between faults is 1/1.6 km.

Therefore, the correct answer is: B. X ~ Exponential(mean=1/1.6)

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Answer: C) option

csc theta is hypotenuse/perpendicular

that is 15/12

Solution for the value cosec θ is,

cosec θ = r / 12

We have,

A right triangle is shown in image,

From the figure,

Apply Pythagoras theorem,

r² = 12² + 9²

r² = 144 + 81

r² = 225

r = 15

Hence, We get;

cosec θ = r / 12

cosec θ = 15 / 12

Therefore, Solution for the value cosec θ is,

cosec θ = r / 12

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