Let X be a random variable with the following probability distribution:x −2 3 5f(x) 0.3 0.2 0.5(a) Find the standard deviation of X. (b) Find the expected value of X^2. Round off your answer to four decimal places.

Step-by-step explanation:

if x=4. then it be |4-2|-1= 1

x=5 5-2-1 so 2

and so on

The numerical values of the coefficients A and B are:

A = -1/6
B = 1/6

To perform partial fraction decomposition, we need to break down the fraction into simpler terms.

The form of the partial fraction decomposition of the given function is:

5/(5x - 72) = A/(5x - 66) + B/(5x - 72)

Here, A and B are the coefficients we need to find. We can find them by cross-multiplying and equating the numerators of both sides of the equation:

5 = A(5x - 72) + B(5x - 66)

Now, we can substitute some values of x to get two equations in terms of A and B:

For x = 14:

5 = A(5(14) - 72) + B(5(14) - 66)

Simplifying and solving for A and B, we get:

A = 1/6
B = -1/6

For x = 12:

5 = A(5(12) - 72) + B(5(12) - 66)

Simplifying and solving for A and B again, we get:

A = -1/6
B = 1/6



Generally, for a function f(x) with a rational expression in the integral, we can use partial fraction decomposition to rewrite the expression as a sum of simpler fractions. This makes it easier to find the integral.

The coefficients A and B are constants in the simpler fractions, and their values can be determined by solving a system of linear equations.

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Your answer: b. Reject the null hypothesis.

Explanation:

In a repeated-measures study using a sample of n = 8 participants, the researcher obtained a t-value of 2.98. For a two-tailed test with an alpha of .05.

To determine the correct decision for a two-tailed test with an alpha level of 0.05 based on a t-value of 2.98 for a repeated-measures study with a sample size of n = 8, we need to compare the t-value to the critical t-value for a two-tailed test at alpha level of 0.05 with 7 degrees of freedom, which is n - 1.

Using a t-table or a t-distribution calculator with 7 degrees of freedom and an alpha level of 0.05, we can find the critical t-value for this sample size is approximately 2.365(rounded to three decimal places).

Since the obtained t-value (2.98) is greater than the critical t-value (2.365), we would reject the null hypothesis.

Therefore, the correct decision for a two-tailed test with an alpha of 0.05 based on the given t-value of 2.98 is:

b. Reject the null hypothesis.

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The simplified expression is just the set s. First, we can use the associative property of intersection to rearrange the parentheses: (r∩s) ∩(s∩(r∩s)) = (r∩s) ∩((r∩s)∩s)

We can use the commutative property of intersection to switch the order of r and s in the first set:

(r∩s) ∩((r∩s)∩s) = (s∩r) ∩((r∩s)∩s)

Now, we can use the distributive property of intersection over intersection to expand (r∩s)∩s:

(s∩r) ∩((r∩s)∩s) = (s∩r) ∩r∩s

Finally, we can use the associative and commutative properties of intersection to rearrange the sets again and simplify:

(s∩r) ∩r∩s = s∩r∩r∩s = s∩(r∩r)∩s = s∩s = s

Therefore, the simplified expression is just the set s.

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The value of the constant k such that the function f(x) = k √x [0, 1] is a probability density function is k = 3/2.

To find the value of the constant k such that the function f(x) = k √x [0, 1] is a probability density function, we need to ensure that the integral of f(x) over the interval [0,1] equals 1.

So, we need to find k such that ∫0^1 k √x dx = 1.

Integrating, we get:
∫0^1 k √x dx = k(2/3)x^(3/2)|0^1 = k(2/3)

Setting this equal to 1, we have:
k(2/3) = 1

Solving for k, we get:
k = 3/2√1

Therefore, the value of the constant k such that the function f(x) = k √x [0, 1] is a probability density function is k = 3/2.

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Answer:

78.274 (Steps shown below)

Step-by-step explanation:

Rectangle Area

6 x 12 =50

Circle (put the 2 half circles together)

Circle area = π * r² = π * 9 [inch²] ≈ 28.274 [in²]

π ≈ 3.14159265 ≈ 3.14

d = r * 2 = 3 [inch] * 2 = 6 [inch]

Area of Circle= 28.274

The standard deviation of the sample proportion will be 0.14. So, the correct option is option D) 0.14.

The formula for the standard deviation of a sample proportion is given by:
standard deviation = √[p(1-p)/n]
where p is the proportion of successes in the population (i.e. the proportion of students who cleared the exam), and n is the sample size.

In this case, p = 225/300 = 0.75, since in the school of 300 students 225 students cleared the exam. The sample size is n = 10, as sample of 10 of these students is taken.

Plugging these values into the formula, we get:

standard deviation = √[0.75(1-0.75)/10] = √[0.01875] = 0.1366

Rounding to two decimal places, the answer 0.14.

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The intersection of two sets can be represented using a truth table that shows the input values and the resulting output value that represents the intersection of the two sets.

The purpose of truth tables is to help analyze logical statements and determine their truth values based on the different combinations of inputs or variables. Truth tables display all possible outcomes of a logical operation and allow for a clear visualization of the relationship between inputs and outputs.

Number systems, particularly those derived from binary, are closely related to truth tables because they involve the use of binary digits or bits (0 and 1) to represent numbers and perform logical operations. Truth tables can be used to determine the output of binary logical operations, such as AND, OR, and NOT, based on the input values.

Set theory, on the other hand, is related to truth tables in the sense that it deals with the study of sets, which can be represented using truth tables. Truth tables can be used to determine the membership of elements in a set and to evaluate logical statements involving sets. For example, the intersection of two sets can be represented using a truth table that shows the input values and the resulting output value that represents the intersection of the two sets.

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To determine whether the set of solutions y1, y2, ..., yk is linearly dependent or not, we need to calculate the Wronskian W(y1, y2, ..., yk) and check whether it is zero for some x in (-[infinity], [infinity]). If it is never zero, then the set of solutions is linearly independent. If it is zero for some x, then the set of solutions is linearly dependent.

To determine whether the set of solutions y1, y2, ..., yk is linearly dependent or not, we can use the Wronskian determinant. The Wronskian of a set of k functions is defined as:

W(y1, y2, ..., yk) = det [y1, y2, ..., yk; y1', y2', ..., yk'; ..., ..., ..., ...; y1^(k-1), y2^(k-1), ..., yk^(k-1)]

where y1', y2', ..., yk' are the first derivatives of y1, y2, ..., yk, respectively, and y1^(k-1), y2^(k-1), ..., yk^(k-1) are their (k-1)th derivatives.

If the Wronskian is nonzero for all x in (-[infinity], [infinity]), then the set of solutions is linearly independent. If the Wronskian is zero for some x in (-[infinity], [infinity]), then the set of solutions is linearly dependent.

Therefore, to determine whether the set of solutions y1, y2, ..., yk is linearly dependent or not, we need to calculate the Wronskian W(y1, y2, ..., yk) and check whether it is zero for some x in (-[infinity], [infinity]). If it is never zero, then the set of solutions is linearly independent. If it is zero for some x, then the set of solutions is linearly dependent.

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To calculate the time it would take to reach 50,000 pairs, we can use the average doubling time provided in the table. The shape of the curve is a J-shaped exponential curve.

which means that the number of pairs increases at an accelerating rate over time. Since the average doubling time is 11.05 years, we can determine the number of times we need to double the initial number of breeding pairs (791) to reach 50,000 pairs. By successively doubling the number of pairs and keeping track of the years passed, we reach 50,000 pairs in 235 years (as given in the table).

However, there are unrealistic assumptions made in predicting the time to reach 50,000 pairs. The main assumption is that the doubling time remains constant, while in reality, factors such as environmental limitations, availability of resources, and human intervention could cause the rate of growth to change over time.

Additionally, the model assumes that there are no significant negative events (such as disease outbreaks or natural disasters) that could negatively impact the bald eagle population during this period.

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To show that (βa)^(-1) = 1/β A^(-1), we can use the definition of the inverse of a matrix.

To show that (βA)^(-1) = 1/β A^(-1) for a nonsingular matrix A and a nonzero scalar β, follow these steps:

1. Let's consider a nonsingular matrix A and a nonzero scalar β.
2. Multiply both sides of the equation by (βA).

On the left side, we have:
(βA)(βA)^(-1)

On the right side, we have:
(βA)(1/β A^(-1))

3. Apply the property of inverse matrices:

(βA)(βA)^(-1) = I, where I is the identity matrix.

4. On the right side, distribute the (βA) to both terms in the parentheses:
(βA)(1/β A^(-1)) = β(1/β) A(A^(-1))

5. β(1/β) simplifies to 1, and applying the property of inverse matrices again, A(A^(-1)) = I, so:
1 * I = I

Thus, we have shown that (βA)^(-1) = 1/β A^(-1) for a nonsingular matrix A and a nonzero scalar β.

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The expression for the nth term an, for the infinite series s=∑n=1[infinity]an is ∑n=4¹⁶an = 468

We know that the sum of the first n terms of the series is given by sn. Therefore, we can find an expression for the nth term an by taking the difference between successive values of sn:

sn - sn-1 = an

(4-4n²) - (4-4(n-1)²) = an

Simplifying this expression, we get:

an = 8n - 4

Now we can use this expression to find the values of ∑n=1¹⁰an and ∑n=4¹⁶an:

∑n=1¹⁰an = a1 + a2 + ... + a10

= (81 - 4) + (82 - 4) + ... + (8*10 - 4)

= 76

Therefore, ∑n=1¹⁰an = 76.

Similarly,

∑n=4¹⁶an = a4 + a5 + ... + a16

= (84 - 4) + (85 - 4) + ... + (8*16 - 4)

= 468

Therefore, ∑n=4¹⁶an = 468.

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To find the linear approximation of f(x) at x = 3, we need to calculate the derivative f'(x) and then use the formula for the linear approximation: L(x) = f(a) + f'(a)(x-a).

Step 1: Calculate the derivative f'(x) of the given function f(x).
As the function is not provided, I'll assume it's a general rational function, f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. To find the derivative, use the quotient rule: f'(x) = (P'(x)Q(x) - P(x)Q'(x))/Q(x)^2.

Step 2: Evaluate f(3) and f'(3).
Once you find f'(x), plug in x=3 to get f(3) and f'(3).

Step 3: Use the linear approximation formula.
L(x) = f(3) + f'(3)(x-3).

Now, estimate L(2.9):
L(2.9) = f(3) + f'(3)(2.9-3) = f(3) - 0.1f'(3).

To provide an exact answer in fractional form, compute the numerical values of f(3) and f'(3) and substitute them in the equation above.

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The system is solvable only if [tex]$-b_1 + 2b_2 - b_3 = 0$[/tex]. All solutions for the given condition are [tex]${(x, y, z) \mid x = \frac{1}{2}(2t - b_1 - b_2), y = \frac{1}{5}(4t + 2b_1 - 5b_2), z = t}$[/tex].

We can set up the augmented matrix as follows:

[tex]\begin{bmatrix}1 & 2 & -2 & b_1 \ 2 & 5 & -4 & b_2 \ 4 & 9 & -8 & b_3\end{bmatrix}[/tex]

We can row reduce this matrix to determine when the system is solvable and to find any solutions. Performing row operations, we get:

[tex]\begin{bmatrix}1 & 2 & -2 & b_1 \ 0 & 1 & 0 & 2b_1 - 5b_2 \ 0 & 0 & 0 & -b_1 + 2b_2 - b_3\end{bmatrix}[/tex]

So the system is solvable if and only if [tex]$-b_1 + 2b_2 - b_3 = 0$[/tex]. In this case, we can solve for z in terms of y and x by expressing z as a free variable and solving for x and y in terms of z. We get:

[tex]\begin{align*}z &= t \y &= \frac{1}{5}(4t + 2b_1 - 5b_2) \x &= \frac{1}{2}(2t - b_1 - b_2) \\end{align*}[/tex]\begin{align*}

z &= t \

y &= \frac{1}{5}(4t + 2b_1 - 5b_2) \

x &= \frac{1}{2}(2t - b_1 - b_2) \

\end{align*}

where t is any real number. So the solutions are given by the set:

[tex]${(x, y, z) \mid x = \frac{1}{2}(2t - b_1 - b_2), y = \frac{1}{5}(4t + 2b_1 - 5b_2), z = t}$[/tex]

where t is any real number, and [tex]$-b_1 + 2b_2 - b_3 = 0$[/tex].

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The exact length x of the diagonal of the rectangle is the exact length of the diagonal is 4sqrt(5).

We can use the Pythagorean theorem to solve for the diagonal:

The Pythagorean Theorem is a fundamental principle in mathematics that describes the relationship between the sides of a right triangle.

It states that the sum of the squares of the two shorter sides (the legs) of a right triangle is equal to the square of the length of the longest side (the hypotenuse).

x^2 = 4^2 + 8^2

x^2 = 16 + 64

x^2 = 80

x = sqrt(80)

x = 4sqrt(5)

Thus, the exact length of the diagonal is 4sqrt(5).

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The factor or variable that is manipulated in an experiment is called the independent variable. Therefore, the correct answer is b. the independent variable.

The independent variable is a factor or variable that is manipulated or changed by the researcher in an experiment to observe its effect on the dependent variable. It is also called the predictor variable because it is used to predict changes in the dependent variable. The researcher controls the independent variable and can vary it as needed to test different hypotheses. In contrast, the dependent variable is the factor or variable that is being measured or observed in response to the changes made in the independent variable.

For example, in an experiment to test the effect of different doses of medication on blood pressure, the independent variable is the medication dosage, while the dependent variable is the blood pressure readings. The researcher can manipulate the dosage of the medication and measure the effect on the blood pressure to determine the optimal dosage for treating high blood pressure.

It is important to carefully choose and control the independent variable in an experiment to ensure accurate and reliable results. Any extraneous or confounding variables that could affect the dependent variable must be controlled or eliminated to isolate the effect of the independent variable.

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The statement "defects are additive in a multi-step manufacturing process" is generally false.

It depends on the type of defects and the specific manufacturing process. In general, defects are not necessarily additive in a multi-step manufacturing process.

Some defects may be additive, meaning that they can accumulate or worsen as the manufacturing process progresses. For example, if a part is slightly out of tolerance in one step of the process, subsequent steps may exacerbate the deviation, leading to a larger defect in the final product.

On the other hand, some defects may be independent or even compensatory, meaning that they do not accumulate or cancel each other out as the process progresses. For example, if one step of the process introduces a defect in one dimension of a part, another step may correct the defect in another dimension, resulting in a final product that meets specifications.

Therefore, the statement "defects are additive in a multi-step manufacturing process" is generally false, as it depends on the specific types of defects and the manufacturing process being considered.

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The statement "defects are additive in a multi-step manufacturing process" is generally false.

It depends on the type of defects and the specific manufacturing process. In general, defects are not necessarily additive in a multi-step manufacturing process.

Some defects may be additive, meaning that they can accumulate or worsen as the manufacturing process progresses. For example, if a part is slightly out of tolerance in one step of the process, subsequent steps may exacerbate the deviation, leading to a larger defect in the final product.

On the other hand, some defects may be independent or even compensatory, meaning that they do not accumulate or cancel each other out as the process progresses. For example, if one step of the process introduces a defect in one dimension of a part, another step may correct the defect in another dimension, resulting in a final product that meets specifications.

Therefore, the statement "defects are additive in a multi-step manufacturing process" is generally false, as it depends on the specific types of defects and the manufacturing process being considered.

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Using the concept of proportion, we have that:

1) Approximate number of students who prefer R & B is: 176 students

2) Approximate number of students who prefer R & B is:  464 students

Normally in table of values, we can easily tell if a table shows a proportional relationship by calculating the ratio of each pair of values. If those ratios are all the same, the table shows a proportional relationship.

We are told that there are a total of 1200 total students on the campus.

1) Total number of students = 1200

Total number surveyed = 150

Total who prefer R & B music = 22

Thus:

Approximate number of students who prefer R & B = (22/150) * 1200

= 176 students

2) Total number of students = 1200

Total number surveyed = 150

Total who prefer Pop or Rock music = 30 + 28 = 58

Thus:

Approximate number of students who prefer R & B = (58/150) * 1200

= 464 students

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From the test points, we find that f(x) is increasing on the interval (1, ∞), including the endpoint 1 for the function f(x) = x3+ 3x2 - 9x + 14.

To determine on what interval f(x) is increasing, we need to find the derivative of f(x) and solve for when it is greater than zero.
f'(x) = 3x^2 + 6x - 9

Setting f'(x) > 0, we can solve for x: 3x^2 + 6x - 9 > 0

Dividing by 3, we get: x^2 + 2x - 3 > 0

Factoring, we have: (x + 3)(x - 1) > 0

This expression is greater than zero when both factors are either both positive or both negative.

Thus, we have two intervals: x < -3 and x > 1

Testing values in each interval, we can see that f(x) is increasing on:
(-infinity, -3) and (1, infinity)

Therefore, the interval on which f(x) is increasing (including the endpoints) is: [-3, 1]

To determine the interval on which the function f(x) = x^3 + 3x^2 - 9x + 14 is increasing, we first need to find its critical points by taking the derivative and setting it equal to 0.

f'(x) = 3x^2 + 6x - 9

Now, set f'(x) to 0 and solve for x:

0 = 3x^2 + 6x - 9

We can factor out a 3:

0 = 3(x^2 + 2x - 3)

Now, factor the quadratic equation:

0 = 3(x - 1)(x + 3)

So, the critical points are x = 1 and x = -3.

To determine if f(x) is increasing or decreasing in each interval, we can use a number line with the critical points:

-∞ < x < -3,  -3 < x < 1,  1 < x < ∞

Choose a test point in each interval and evaluate f'(x):

For x = -4: f'(-4) = -16 (negative)
For x = 0: f'(0) = -9 (negative)
For x = 2: f'(2) = 15 (positive)

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Answer:

67.5%

Step-by-step explanation:

To calculate the percentage decrease in the share price of Bitcoin, we can use the following formula:

Percentage decrease = ((original value - new value) / original value) x 100%

Here, the original value is the share price before the crash, which is $60,000, and the new value is the share price after the crash, which is $19,500.

Substituting these values into the formula, we get:

Percentage decrease = ((60,000 - 19,500) / 60,000) x 100%

= 40,500 / 60,000 x 100%

= 67.5%

Therefore, the percentage decrease in the share price of Bitcoin is 67.5%.

The Extreme Value Theorem does not guarantee the existence of an absolute maximum and minimum for f(x) on this interval.

The extreme value theorem states that if a function is continuous over a closed interval, then it must have at least one absolute maximum and one absolute minimum on that interval. In the case of f(x) = cot(x) 10 over the interval [−π,π3], this function is continuous over the interval since it is the composition of two continuous functions (cot(x) and 10). Therefore, the extreme value theorem guarantees that there must be at least one absolute maximum and one absolute minimum for f(x) on this interval.
The Extreme Value Theorem states that if a function is continuous on a closed interval, then it has an absolute maximum and minimum on that interval. The function f(x) = cot(x) is not continuous over the interval [-π, π/3] due to the presence of vertical asymptotes, where the function is undefined. Therefore, the Extreme Value Theorem does not guarantee the existence of an absolute maximum and minimum for f(x) on this interval.

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Step-by-step explanation:

26 is 355 plus the product of 265 and p

26 = 355 + 265p

Answer:

[tex]y= (x+\frac{1}{2})^2-8.25[/tex]

Step-by-step explanation:

First, we move the c in [tex]ax^2+bx+c[/tex] to the other side of the equation by adding 8 onto both sides:

[tex]x^2+x=8[/tex].

Then, since [tex]x^2\\[/tex] has no coefficient, we make the left side a perfect square trinomial by adding [tex](\frac{b}{2})^2[/tex] on both sides of the equation. We do this because adding this to the equation will make the left side equal to [tex](x\pm\frac{b}{2})^2[/tex] (plus-minus because the sign depends on if b is negative or positive):

[tex]x^2+x+\frac{1}{4}=8+\frac{1}{4}[/tex].

Then, simplify the left side:

[tex](x+\frac{1}{2})^2=8.25[/tex]

Finally, subtract 8.25 on both sides to make it vertex form:

[tex]y= (x+\frac{1}{2})^2-8.25[/tex]

The missing angles in the right angle triangle are as follows:

∠1 = 53.1°

∠2 = 36.9°

A right angle triangle is a triangle that has one of its angles as 90 degrees. The angles of the right angle triangle can be found using trigonometric ratios.

The sum of angles in a triangle is 180 degrees.

Therefore,

cos ∠1 = adjacent / hypotenuse

Hence,

cos ∠1 = 18 / 30

∠1 = cos⁻¹ 0.6

∠1 = 53.1301023542

∠1 = 53.1 degrees

Therefore,

∠2 = 180 - 90 - 53.1

∠2 = 36.9 degrees

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9. A - Each person is assigned a three-digit number (from 000 to 999). On a Random Digit Table, numbers are read, three at a time. The first hundred three-digit numbers read will represent the people in the sample.

10. C - Every man in the sampling frame will be assigned 8 sequential 4-digit numbers and every woman in the sampling frame will be assigned 12 sequential 4-digit numbers. From a Random Digit Table, groupings of 4 numbers will be read and the first 100 subjects with their number read will be in the sample (duplicate selections will be ignored).

9. Method A is probability sampling because each individual in the sampling frame has an equal chance of being selected, and the selection is based on random digits.

10. Method C is stratified sampling because the sampling frame is divided into two strata based on gender, and each stratum is sampled separately using a random selection method. This allows for a more representative sample by ensuring that both men and women are adequately represented in the sample.

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9. A - Each person is assigned a three-digit number (from 000 to 999). On a Random Digit Table, numbers are read, three at a time. The first hundred three-digit numbers read will represent the people in the sample.

10. C - Every man in the sampling frame will be assigned 8 sequential 4-digit numbers and every woman in the sampling frame will be assigned 12 sequential 4-digit numbers. From a Random Digit Table, groupings of 4 numbers will be read and the first 100 subjects with their number read will be in the sample (duplicate selections will be ignored).

9. Method A is probability sampling because each individual in the sampling frame has an equal chance of being selected, and the selection is based on random digits.

10. Method C is stratified sampling because the sampling frame is divided into two strata based on gender, and each stratum is sampled separately using a random selection method. This allows for a more representative sample by ensuring that both men and women are adequately represented in the sample.

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β is an angle with cos(β) = -5/6 and β is not in the third quadrant, we need to compute the exact value of csc(β), without rationalizing the denominator.

Step 1: Determine the quadrant of angle β.
Since cos(β) = -5/6 and β is not in the third quadrant, then β must be in the second quadrant, as cosine values are negative in the second quadrant.

Step 2: Compute the sine of angle β.
We know that sin^{2}(β) + cos^{2}(β) = 1 (Pythagorean identity). So,
sin^{2}(β) + (-5/6)^{2} = 1
sin^{2}(β) + 25/36 = 1
sin^{2}(β) = 11/36
sin(β) = √(11/36) (since sin is positive in the second quadrant)

Step 3: Compute the exact value of csc(β).
Since csc(β) is the reciprocal of sin(β), then csc(β) = 1/sin(β).
csc(β) = 1/(√(11/36))

Therefore, the exact value of csc(β) is 1/(√(11/36)).

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β is an angle with cos(β) = -5/6 and β is not in the third quadrant, we need to compute the exact value of csc(β), without rationalizing the denominator.

Step 1: Determine the quadrant of angle β.
Since cos(β) = -5/6 and β is not in the third quadrant, then β must be in the second quadrant, as cosine values are negative in the second quadrant.

Step 2: Compute the sine of angle β.
We know that sin^{2}(β) + cos^{2}(β) = 1 (Pythagorean identity). So,
sin^{2}(β) + (-5/6)^{2} = 1
sin^{2}(β) + 25/36 = 1
sin^{2}(β) = 11/36
sin(β) = √(11/36) (since sin is positive in the second quadrant)

Step 3: Compute the exact value of csc(β).
Since csc(β) is the reciprocal of sin(β), then csc(β) = 1/sin(β).
csc(β) = 1/(√(11/36))

Therefore, the exact value of csc(β) is 1/(√(11/36)).

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The Cartesian equation of the curve is  Y = 1/X.

To the parameter and find a Cartesian equation of the curve using the given terms. Consider the following:

X = sin(t), Y = csc(t), and 0 ≤ t ≤ 2π

Step 1: Rewrite Y in terms of sin(t)
Since Y = csc(t), we know that csc(t) = 1/sin(t). Therefore, Y = 1/sin(t).

Step 2: Eliminate the parameter t
We already have X = sin(t), so we can substitute this into the equation for Y:
Y = 1/X

Step 3: Write the Cartesian equation of the curve
Now that we have eliminated the parameter t, the Cartesian equation of the curve is simply:

Y = 1/X

Therefore, the Cartesian equation of the curve is Y = 1/X.

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The simplified form of the surd is 1 - 1/3√5

Here are the general steps to follow when rationalizing a surd:

Identify the surd in the denominator of the fraction.

Multiply the numerator and denominator of the fraction by the conjugate of the denominator. The conjugate is obtained by changing the sign of the surd term in the denominator.

Simplify the resulting expression by expanding the brackets and collecting like terms.

If there is still a surd in the denominator, repeat the process until no surds remain in the denominator.

Given that;

√2 - √10/√2 + √10

Then;

√2 - √10/ √2 + √10  * √2 - √10/√2 - √10

2 -√20 - √20  + 10/2 -√20 + √20 + 10

2 - 2√20 + 10/2 + 10

12 - 2√20/12

1 - 1/3√5

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She will need to purchase 3 boxes of gloves.

He exceeded his goal by 10 devices.

There are 28 phalange bones in each foot.

There will be 2 smoothies left in stock after 4 days from one case.

Frank needs to consume no more than 15 grams of fat for the rest of the day.

Since there are 100 gloves in each box, she will need 222/100 = 2.22 boxes of gloves. Since she cannot purchase a partial box, she will need to purchase 3 boxes of gloves.

The medical sales rep sold devices for a total of 17 x 30 = 510 devices in November. Since his goal was to sell 500 devices, he exceeded his goal by 510 - 500 = 10 devices.

Since there are 56 phalange bones in the body and 14 phalange bones in each hand, there are 56 - (14 x 2) = <<56-(14*2)= 28 phalange bones in each foot.

Frank needs to consume no more than 56 - 41 = 15 grams of fat for the rest of the day.

The rec center sells 12 smoothies per day for 4 days, for a total of 12 x 4 = 48 smoothies. Therefore, there will be 50 - 48 = 2 smoothies left in stock after 4 days from one case.

Since Ashton drank a 24 oz bottle of water, he still needs to drink 64 - 24 = 40 ounces of water for the rest of the day.

Kathy walks a total of 3 x 2.5 =7.5 miles with her friend each week. Therefore, she still needs to walk 10 - 7.5 = 2.5 more miles to meet her goal for the week.

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