Suppose that A is a 4 times 7 matrix that has an echelon form with two zero rows. Find the dimension of the row space of A, the dimension of the column space of A, and the dimension of the null space of A. The dimension of the row space of A is . The dimension of the column space of A is . The dimension of the null space of A is .

The approximate probability of obtaining an a-to-(1-a) split in the partition using the median-of-three method in Quicksort is about 0.6667 or 66.67%.

We'll first need to understand the median-of-three method in Quicksort and then approximate the probability of obtaining an a-to-(1-a) split in the partition.

1. Median-of-three method: This method involves selecting three random elements from the array, finding their median, and using it as the pivot. This helps improve the efficiency of Quicksort by reducing the chances of choosing a bad pivot.

2. Calculate the probability: Let's assume that a is a real number in the range 0 < a < 1. We want to find the probability of obtaining an a-to-(1-a) split in the partition. To get an a-to-(1-a) split, the pivot must be in the a-th percentile (or 1-a percentile, which is equivalent) of the sorted list.

Since there are three randomly selected elements, there are a total of 3! = 6 permutations for the order of these elements. The cases that result in an a-to-(1-a) split are when the median of the three elements is in the a-th percentile. There are two possible cases:

a) The first element is in the a-th percentile and the other two are larger.

b) The first element is larger, and the second element is in the a-th percentile.

Each of these two cases has 2! = 2 permutations (as the other two elements can be switched), giving us a total of 4 favorable permutations.

Now we can calculate the probability:

Probability = (Number of favorable permutations) / (Total permutations)

= 4/6 = 2/3 ≈ 0.6667

So, the approximate probability of obtaining an a-to-(1-a) split in the partition using the median-of-three method in Quicksort is about 0.6667 or 66.67%.

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Yes, the extreme value theorem guarantees the existence of an absolute maximum and minimum for f on the interval [−5,−1] since f is continuous on the closed and bounded interval.

The Extreme Value Theorem states that there is an existence of an absolute maximum and minimum for a function f(x) on a closed interval [a, b] if the function is continuous on that interval. In this case, f(x) = ln(x^2) is continuous on the interval [-5, -1], since x^2 is always non-negative and ln(x^2) is defined for all x in the interval.

Thus, the extreme value theorem applies on the function f(x)=ln(x2) over the interval [−5,−1]. And so, there must be the existence of an absolute maximum and minimum for the function f where f(x)=ln(x2) on the interval [-5, -1].

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The reason for each step to justify it is:

Division Property of Equality: This step involves dividing both sides of the equation by 4 to isolate the variable term on one side of the equation.

The step involved here is: (w+25)/4

Multiplication Property of Equality: This step involves multiplying both sides of the equation by 4 to simplify the equation and cancel out the denominator.
The step involved here is: 4*(w+25)

Addition Property of Equality: This step involves subtracting 25 from both sides of the equation to isolate the variable term on one side of the equation.

The step involved here is: w+25

Simplifying: This step involves simplifying the expression by combining like terms and performing arithmetic operations to simplify the equation.

The step involved here is: w+25-25=12-25

Subtraction Property of Equality: This step involves subtracting 25 from both sides of the equation to isolate the variable term and solve for the variable.

The step involved here is: w=-13

What is meant by division?

Division is a mathematical operation that involves splitting a quantity into equal parts or groups. It is represented by the symbol ÷ or / and is the inverse operation of multiplication. Division can be used to solve problems related to sharing, fractions, and ratios.

What is meant by multiplication?

Multiplication is a mathematical operation that involves adding a number to itself a certain number of times. It is represented by the symbol × or * and is the inverse operation of division. Multiplication can be used to solve problems related to scaling, area, volume, and rates.

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Answer:

The answer is trapezoid.

Trapezoids are quadrilaterals that have one set of parallel sides. Some authors claim it is exactly one set of parallel sides, other authors say at least one set of parallel sides, and by doing so allow parallelograms to be a special subgroup of trapezoids.

Trapezoids are not studied in US geometry to anywhere near the same depth as parallelograms. As a result in most US Geometry textbooks the only “special” trapezoids are isosceles trapezoids, where the two sides between the parallel sides are congruent to each other and the right trapezoid, where one of the non-parallel sides is perpendicular to the two parallel sides.

Types of trapezoids:

1. An isosceles trapezoid is a trapezoid where the sides (legs) have the same length, and the base angles have the same measure. It has reflection symmetry. This is possible for acute or right trapezoids. This is a cyclic quadrilateral. You can a draw a circumcircle around such a trapezoid.

2. It is called a right trapezoid (also called right-angled trapezoid) has two adjacent right angles. Right trapezoids are used in the trapezoidal rule for estimating areas under a curve.

3. An acute trapezoid has two adjacent acute angles on its longer base edge. The two adjacent angles on the shorter base edge are obtuse.

4. An obtuse trapezoid has one acute and one obtuse angle on each base.

5. A parallelogram is a trapezoid with two pairs of parallel sides. A parallelogram has central 2-fold rotational symmetry (or point reflection symmetry). It is possible for obtuse or right trapezoids.

6. A tangential trapezoid is a trapezoid that has an incircle.

The difference between the area of box and the pizza is 21.46 square inches .

First find the area of the square box to be :

= Length x Width

= 10 x 10

= 100 square inches

Then, find the area of the pizza inside the pizza box to be :

= π x radius x radius

= π x 5 x 5

= 78. 54 square inches

The difference between the areas of the box and pizza is :

=  100 - 78.54 = 21.46 square inches

In conclusion, the difference would be 21.46 square inches.

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The expected gestation period of this species with a 17-year life expectancy is approximately 430.2 days, rounded to one decimal place as needed.

A certain mammal has a life expectancy of about 17 years. To estimate the expected gestation period of this species, we can use the scatterplot provided, which shows Gestation Period (in days) vs. Life Expectancy (in years) for 18 species of mammals.

To determine if there is any evidence that an animal's gestation period is related to the animal's lifespan, we can look at the R-squared value provided, which is 86.5%. This indicates a strong positive correlation between gestation period and life expectancy.

To estimate the gestation period for the species with a 17-year life expectancy, we can use the provided linear regression equation:

Gestation = Constant + Coefficient * Life Expectancy

The given constant is 90.1808 and the coefficient is 20.

Gestation = 90.1808 + (20 * 17)

Gestation = 90.1808 + 340

Gestation ≈ 430.2 days

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The yearly appreciation of the value of Mike's coin collection over the 5-year period is $510 per year.

The given problem involves finding the yearly appreciation of value of Mike's coin collection over a period of 5 years, based on the values of the collection in 2000 and 2005.

To calculate the yearly appreciation of the value of the collection, we need to first find the total increase in value over the 5-year period. This can be calculated by subtracting the initial value of the collection in 2000 from its final value in 2005:

Total increase in value = 7675 - 5125 = 2550

Next, we can calculate the average annual increase in value by dividing the total increase by the number of years:

Average annual increase = Total increase in value / Number of years

Average annual increase = 2550 / 5 = 510

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Step-by-step explanation:

Let's represent the cost of one peach as "x" and the cost of one nectarine as "y". We can set up a system of equations using the given information:

4x + 12y = 2.28

2x + 14y = 2.10

We can write this system in matrix form as AX = B, where

A = |4 12|

|2 14|

X = |x|

|y|

B = |2.28|

|2.10|

To solve for X, we can use the formula X = A^(-1)B, where A^(-1) is the inverse of matrix A.

First, we need to find the inverse of matrix A:

|4 12| |7/30 -2/15|

|2 14| = |-1/30 2/15|

Now we can use the formula X = A^(-1)B:

|x| |7/30 -2/15||2.28| |0.24|

|y| = |-1/30 2/15||2.10| = |0.12|

Therefore, the cost for a peach is $0.24 (rounded to two decimal places) and the cost for a nectarine is $0.12 (rounded to two decimal places).

The area of the triangle from the vertices of the triangle is B. 32 square units

From the question, we have the following parameters that can be used in our computation:

The vertices of a triangle are

L(2, 2), M(6, -2), N(2, -6)

The area of the triangle is calculated using

Area = 1/2 * |Lx * (My - Ny) + Mx * (Lx - Ny) + Nx * (Lx - My)|

Substitute the known values in the above equation, so, we have the following representation

Area = 1/2 * |2 * (-2 + 6) + 6 * (2 + 6) + 2 * (2 + 2)|

Evaluate

Area = 32

Hence, the area is B. 32 square units

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The missing values of the polynomial are listed as below

The Polynomial x² - 8x + 15 is factored to (x - 3)(x - 5)

comparing with (ax + b)(cx + d)

a = 1, b  = -3, c = 1 d = -5.

the Polynomial 2x² - 24x is factored to 2x(ax + b)(cx + d)

2x² - 24x = 2x(x + 0)(2x - 6) = 2x(x)(2x - 6)

comparing with 2x(ax + b)(cx + d)

a = 1, b  = 0, c = 2 d = -6.

2x² - 24x = 2x(x)(2x - 6)

the Polynomial 6x² + 14x + 4 is factored to (2x + 2) (3x + 2)

comparing with  (ax + b)(cx + d)

a = 2, b = 2, c = 3, and d = 2.

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The missing values of the polynomial are listed as below

The Polynomial x² - 8x + 15 is factored to (x - 3)(x - 5)

comparing with (ax + b)(cx + d)

a = 1, b  = -3, c = 1 d = -5.

the Polynomial 2x² - 24x is factored to 2x(ax + b)(cx + d)

2x² - 24x = 2x(x + 0)(2x - 6) = 2x(x)(2x - 6)

comparing with 2x(ax + b)(cx + d)

a = 1, b  = 0, c = 2 d = -6.

2x² - 24x = 2x(x)(2x - 6)

the Polynomial 6x² + 14x + 4 is factored to (2x + 2) (3x + 2)

comparing with  (ax + b)(cx + d)

a = 2, b = 2, c = 3, and d = 2.

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a= 1

h= 1

k= 0

The graph moves to the right 1 unit.

a) The first negative term of the sequence is given as follows: -1.

b) -30 is not a term on the sequence, as it has a difference of 29 from -1, and 29 is not divisible from the common difference of -4.

An arithmetic sequence is a sequence of values in which the difference between consecutive terms is constant and is called common difference d.

The common difference for this problem is defined as follows:

d = 27 - 31

d = -4.

Hence the next terms on the sequence are given as follows:

15, 11, 7, 3, -1.

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To determine where g has a local maximum, we need to find the critical points of g'(x) and analyze the sign changes around those points.

Here's a step-by-step explanation:
1. Identify the critical points of g'(x) by setting it equal to 0:
g'(x) = x(x - 5)(x + 1)^4 = 0
The critical points are x = -1, x = 0, and x = 5.

2. Determine the sign of g'(x) around the critical points:
For x < -1, g'(x) > 0 (since there are three negative factors)
For -1 < x < 0, g'(x) > 0 (since there are two negative factors)
For 0 < x < 5, g'(x) < 0 (since there is one negative factor)
For x > 5, g'(x) > 0 (since all factors are positive)

3. Analyze the sign changes and apply the First Derivative Test:
- At x = -1, there is no sign change, so it is not a local maximum.
- At x = 0, g'(x) changes from positive to negative, indicating a local maximum.
- At x = 5, there is no sign change, so it is not a local maximum.

So, g has a local maximum at x = 0 only. The correct answer is (c).

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The charge enclosed by the cube is ±96 times the permittivity of free space ε0.

Gauss's Law states that the electric flux through any closed surface is proportional to the total charge enclosed by the surface. We can use this law to find the charge enclosed by the given cube by constructing a closed surface that encloses the cube and calculating the electric flux through that surface.

The cube has vertices at (±2, ±2, ±2), so we can choose a cube of edge length 4 centered at the origin as our closed surface. The surface is then defined by the six faces of the cube, each of which is a square of area 4² = 16.

The electric field is given by E(x, y, z) = xi + yj + zk, so the flux through each face is the dot product of the field and the outward normal vector to the face, multiplied by the area of the face:

Φ = ∫ E · dA = ∫ (xi + yj + zk) · (±1, 0, 0) dA

= ± ∫ x dA

= ± 16 ∫₂⁻² x dx = ± 16 [x²/2]₂⁻² = ±16

Here we used the fact that the x-component of the field is constant over each face and equal to ±1, depending on the orientation of the face. The other two components of the field do not contribute to the flux through the x-oriented faces.

Since there are six faces, the total flux through the closed surface is Φ = 6(±16) = ±96. By Gauss's Law, this flux is proportional to the charge enclosed by the surface:

Φ = ∫ E · dA = Q/ε0

Solving for Q, we get Q = Φ ε0 = ±96 ε0.

Therefore, the charge enclosed by the cube is ±96 times the permittivity of free space ε0.

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Answer:.40 or .4 would be the correct answer

Answer:

0.4

Step-by-step explanation:

40% can be written as 0.4. Recall that to get a decimal to a percent or vice versa, you can move the decimal place 2 places to the right or left to get the desired outcome. In this case, the decimal needs to move 2 places to the left leaving you with the decimal 0.4 or 4 parts out of 10 (40 parts out of 100= 40%)

The bias of mu1 is 0, the variance of mu1 is 1/2, and the mean squared error of mu1 is 1/2.

We need to find the bias, variance, and mean squared error of mu1, given that x1 and x2 are independent with an unknown mean (mu) and a known variance (sigma^2 = 1), and mu1 = (x1 + x2)/2.

Step 1: Compute the expected value of mu1.
E(mu1) = E((x1 + x2)/2) = (E(x1) + E(x2))/2 = (mu + mu)/2 = mu

Step 2: Calculate the bias of mu1.
Bias (mu1) = E (mu1) - mu = mu - mu = 0

Step 3: Calculate the variance of mu1.
Var(mu1) = Var((x1 + x2)/2) = (1/4) * (Var(x1) + Var(x2)) = (1/4) * (1 + 1) = 1/2

Step 4: Calculate the mean squared error of mu1.
MSE(mu1) = Bias(mu1)^2 + Var(mu1) = 0^2 + 1/2 = 1/2

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The statement that alone is enough to prove that a parallelogram is a rectangle is Opposite sides of the quadrilateral must be congruent and one angle must be 90 degrees

Option A is the correct answer.

We have,

A rectangle is a type of parallelogram with four right angles.

So, if a quadrilateral has opposite sides that are congruent and one angle is 90 degrees, then it must be a parallelogram with four right angles, i.e., a rectangle.

Statement 2 describes the properties of a rhombus, which is a special type of parallelogram with all sides congruent, not a rectangle.

Statement 3 describes the properties of a parallelogram, but it does not guarantee that the angles are right angles.

Therefore, it does not prove that the parallelogram is a rectangle.

Statement 4 describes the properties of a square, which is a special type of rectangle with all sides congruent, not a general rectangle.

Thus,

Opposite sides of the quadrilateral must be congruent and one angle must be 90 degrees is enough to prove that a parallelogram is a rectangle.

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The distribution of (M,N) is a bivariate Poisson distribution with parameters λ_1 = 9 (rate of local calls) and λ_2 = 3 (rate of long distance calls) since the rates of the two processes are independent Poisson processes.  and the number of people on line are 60.

The joint probability mass function of (M,N) is given by:

[tex]P(M=m, N=n) = e^{-(\lambda_1 +\lambda_2) (lambda_1^m/ m!) (\lambda_2^n/ n!)[/tex]

The number of people on the line is the expected value of the total number of minutes of all calls, which can be calculated as E[10M + 5N] = 10E[M] + 5E[N].

Since M and N are independent Poisson random variables, we have E[M] = λ_1/(1-p_1) and E[N] = λ_2/(1-p_2), where p_1 and p_2 are the probabilities of a call being local or long distance, respectively.

Substituting in the given values, we get:

E[10M + 5N] = 10(9/3) + 5(3/1) = 45 + 15 = 60

Therefore, the expected number of people on the line is 60.

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Mr. David Miller, a statistician, is interested in selecting a sample. He want to give equal probability to each population unit to be selected in the sample. This method is called option (a) simple random sampling

Simple random sampling is a method of selecting a sample from a population where each unit has an equal chance of being selected. This means that every member of the population has the same probability of being chosen for the sample, making it a fair and unbiased method of sampling.

The process involves selecting units at random, without any specific order or pattern. Simple random sampling is commonly used in research studies to obtain representative samples from a population, as it ensures that the sample is a true reflection of the population and reduces the risk of bias or error in the findings.

Therefore, the correct option is (a) simple random sampling

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To the left of z₁ = -1.74 and to the right of z₂ = 1.74, the total area under the standard normal curve is roughly 0.0818.

The standard normal distribution table calculates the probability that a regularly distributed random variable Z, with a mean of 0 and a difference of 1, is not exactly or equal to z. A persistent probability distribution is the normal distribution. It is also known as the Gaussian distribution. It only applies to positive z estimations.

The total area under the standard normal curve to the left of z₁ and to the right of z₂ is the sum of the area to the left of z₁ and the area to the right of z₂.

Using a standard normal distribution table or calculator, we can find:

Therefore, the total area under the standard normal curve to the left of z₁ and to the right of z₂ is:

0.0409 + 0.0409 = 0.0818

Rounding this answer to four decimal places, we get:

0.0818

Therefore, the total area under the standard normal curve to the left of z₁ = -1.74 and to the right of z₂ = 1.74 is approximately 0.0818.

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To simulate a coin tossing environment using a random function, you can follow these steps:

1. Import the random module: `import random`
2. Initialize variables to count the number of heads and tails: `heads = 0` and `tails = 0`
3. Use a for loop to toss the coin 100 times: `for i in range(100):`
4. In the loop, generate a random number between 0 and 1: `toss = random.randint(0, 1)`
5. If the toss is 0, increment the heads counter, otherwise increment the tails counter: `heads += toss == 0` and `tails += toss == 1`
6. After the loop, print the number of heads and tails.

For Part 2, you can use the `randint()` function from the random module to generate random integers within specific ranges:

a) `n = random.randint(1, 3)`
b) `n = random.randint(1, 150)`
c) `n = random.randint(0, 11)`
d) `n = random.randint(1000, 3112)`
e) `n = random.randint(-1, 3)`
f) `n = random.randint(-4, 15)`

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The chances of winning a rare item at least once in 20 attempts is approximately 33.3%.

If you have a 2% chance of winning a rare item, and you want to know the chances after being multiplied by 20

To calculate the probability of winning the rare item at least once in 20 attempts, you can use the complement probability method.

First, find the probability of not winning the item in a single attempt, which is 1 - 0.02 = 0.98.

Then, find the probability of not winning the item in all 20 attempts by multiplying the probability of not winning in a single attempt (0.98) by itself 20 times, which is[tex]0.98^20.[/tex]

Finally, subtract that result from 1 to find the probability of winning at least once in the 20 attempts.

Calculate the probability of not winning in a single attempt.

1 - 0.02 = 0.98

Calculate the probability of not winning in all 20 attempts.

[tex]0.98^20.[/tex] ≈ 0.667

Calculate the probability of winning at least once in 20 attempts.

1 - 0.667 ≈ 0.333

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The answer is (a) 0.50.

To find the percent of observations that fall between 1 and 4, we need to find the area under the curve between 1 and 4 and divide by the total area under the curve.

The total area under the curve is equal to 1, since this is a probability density function.

The area under the curve between 1 and 4 can be found by calculating the integral of the density function from x = 1 to x = 4:

∫1^4 (1/6) dx = (1/6)(4-1) = 1/2

So the area between 1 and 4 is 1/2, and the percent of observations that fall in this range is:

(1/2) * 100% = 50%

Therefore, the answer is (a) 0.50.

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The series Σ^[infinity]_n=1 (3 / (4^n + 5)) converges.

To determine whether the series converges or diverges, consider the series Σ^[infinity]_n=1 (3 / (4^n + 5)).

Step 1: Apply the Comparison Test. We will compare our given series with a known convergent or divergent series. In this case, let's compare it to the geometric series Σ^[infinity]_n=1 (3 / 4^n).

Step 2: Since 4^n + 5 > 4^n, we have (3 / (4^n + 5)) < (3 / 4^n). Now we need to determine if the geometric series converges.

Step 3: Check the geometric series. A geometric series converges if the absolute value of the common ratio is less than 1. The common ratio in our case is 1/4 (since each term is multiplied by 1/4 to get the next term). The absolute value of the common ratio |1/4| = 1/4, which is less than 1.

Step 4: Conclude the convergence of the geometric series. Since the geometric series Σ^[infinity]_n=1 (3 / 4^n) converges, and our given series is term-by-term smaller, it also converges by the Comparison Test.

Thus, the series Σ^[infinity]_n=1 (3 / (4^n + 5)) converges.

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a. The total number of possible passwords is 45,697,014,176.

b. The total number of possible passwords is 44,146,368,000

(a) If digits and letters can be repeated, then there are 10 possible choices for each of the 3 digits as well as 26 possible choices for each of the 4 letter. Therefore, the total number of possible passwords is:

10^3 * 26^4 = 45,697,014,176

(b) If digits and letters cannot be repeated, then there are 10 choices for the first digit, 9 choices for the second digit (since one digit has already been used), 8. Therefore, the total number of possible passwords is:

10 * 9 * 8 * 26 * 25 * 24 * 23 = 44,146,368,000

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a. The total number of possible passwords is 45,697,014,176.

b. The total number of possible passwords is 44,146,368,000

(a) If digits and letters can be repeated, then there are 10 possible choices for each of the 3 digits as well as 26 possible choices for each of the 4 letter. Therefore, the total number of possible passwords is:

10^3 * 26^4 = 45,697,014,176

(b) If digits and letters cannot be repeated, then there are 10 choices for the first digit, 9 choices for the second digit (since one digit has already been used), 8. Therefore, the total number of possible passwords is:

10 * 9 * 8 * 26 * 25 * 24 * 23 = 44,146,368,000

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For 2 sec the snowboarder in the air.

We have,

The height (in feet) of a snowboarder above the bottom of the half-pipe can be modeled by h=-16t²+24t + 16.4

So, 3.2 = -16t² + 24t + 16.4

-16t² + 24t + 12.4 = 0

Now, solving the above quadratic equation we get

t = -0.40650, 1.90650

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The solutions of the equation sec² x - 4 = 0 are x = π/3 + 2kπ, 2π/3 + 2kπ, 4π/3 + 2kπ, 5π/3 + 2kπ. Therefore, option d. is correct.

We start by solving for x in the equation sec² x - 4 = 0:

sec² x - 4 = 0
sec² x = 4
sec x = ±2

Recall that sec x = 1/cos x, so we can rewrite the equation as:

1/cos² x = 4
cos² x = 1/4
cos x = ±1/2

Now we need to find all values of x that satisfy cos x = ±1/2. These values are:

x = π/3 + 2kπ or x = 5π/3 + 2kπ (for cos x = 1/2)
x = 2π/3 + 2kπ or x = 4π/3 + 2kπ (for cos x = -1/2)

Combining these solutions, we get:
x = π/3 + 2kπ, 2π/3 + 2kπ, 4π/3 + 2kπ, 5π/3 + 2kπ

Therefore, the correct answer is d. pi/3 + 2k pi, 2pi/3 + 2k pi, 4pi/3 + 2k pi, 5pi/3 + 2k pi.

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a) The probability that daily sales will fall between 2,500 and 3,000 gallons is 0.2.

b) The probability that the service station will sell at least 4,000 gallons is 0.3.

c) The probability that the station will sell exactly 2,500 gallons is 0.1.

Gasoline sold daily is uniformly distributed between 2,000 and 5,000 gallons.

a) To find the probability that daily sales will fall between 2,500 and 3,000 gallons, we need to find the area under the uniform distribution curve between 2,500 and 3,000.

Since the distribution is uniform, the probability density function (PDF) is a constant:

f(x) = 1 / (5000 - 2000) = 1 / 3000, for 2000 <= x <= 5000.

The probability of the daily sales being between 2,500 and 3,000 gallons is equal to the area under the PDF curve between 2,500 and 3,000:

Therefore, the probability that daily sales will fall between 2,500 and 3,000 gallons is 0.1667 or 16.67%.

b) To find the probability that the service station will sell at least 4,000 gallons,

we need to find the area under the uniform distribution curve from 4,000 to 5,000 gallons:

Therefore, the probability that the service station will sell at least 4,000 gallons is 0.3333 or 33.33%.

c) To find the probability that the station will sell exactly 2,500 gallons, we need to find the area under the uniform distribution curve at 2,500 gallons:

P(X = 2500) = 0 (since the probability of a single point is zero for a continuous distribution)

Therefore, the probability that the station will sell exactly 2,500 gallons is zero.

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The possible rational zeros of the polynomial are of the form:

±1, ±3, ±7, ±21, ±1/3, ±7/3.

The Rational Root Theorem proclaims that if any polynomial is composed of integer coefficients, then its potential rational zeros will have a numerator which is a divisor of the constant term and a denominator that can be factored into the leading coefficient.

Considering the case at hand, with the leading coefficient as 3 and the constant term being -21, the reasonable zeros conceived are in this format:

±1, ±3, ±7, ±21, ± 1/3, ± 7/3.

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The following can be answered by the concept of Trigonometry.

The rearranged functions for each case, using appropriate mathematical techniques and identities:

(a) For 1 - cos(x) / x², we can use the identity sin²(x) + cos²(x) = 1 and rearrange as follows:
1 - cos(x) = sin²(x) / (1 + cos(x))
Then, divide by x² to get the rearranged function:
(sin²(x) / (1 + cos(x))) / x²

(b) For log(x + 1) - log(x) with x being large, we can use the logarithmic property of subtraction to combine the logs:
log((x + 1) / x)
As x is large, the expression simplifies to:
log(1 + (1/x))

(c) For sin(a + x) - sin(a), we can use the angle sum identity for sine:
sin(a + x) - sin(a) = (sin(a)cos(x) + cos(a)sin(x)) - sin(a)
Rearrange to get:
cos(a)sin(x) + sin(a)(cos(x) - 1)

(d) For √(1 + x - 1), we can simplify by removing the 1 and -1 inside the square root:
√(x)

(e) For √(4 + x - 2) / x, we can first simplify inside the square root:
√(2 + x) / x

These rearranged functions help avoid loss-of-significance errors in each of the given cases.

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