In Problems 21 through 30, set up the appropriate form of a
particular solution yp, but do not determine the values of the
coefficients.y" – 2y' + 2y = et sin x = . =

The two requirements for a discrete probability distribution are that the probabilities assigned to each possible outcome must be between 0 and 1 inclusive, and the sum of probabilities of all possible outcomes must be equal to 1.

There are two main requirements for a discrete probability distribution,

The probabilities assigned to each possible outcome must be between 0 and 1 inclusive. That is, the probability of each outcome must be a non-negative number, and the sum of probabilities of all possible outcomes must be equal to 1.

Each possible outcome must be mutually exclusive. That is, only one of the possible outcomes can occur on any given trial of the random experiment.

These two requirements ensure that the probabilities assigned to each outcome are valid and that the total probability space is complete, allowing us to make valid inferences and predictions about the outcomes of random events.

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I have solved the question in general, as the given question is incomplete.

The complete question is:

What are the two requirements for a discrete probability distribution?

The equation which correctly relates the measure of diameter and radius of a circle is (c) r = d/2.

The Diameter (d) of a circle is defined as the distance across the circle through its center. The radius (r) of a circle is defined as the distance from the center of the circle to any point on the circle.

We know that the radius of the circle is half of diameter, because it extends from the center to the edge of the circle, while the diameter extends all the way across the circle.

So, we can express the relationship between d and r as:

⇒ d = 2r

To solve for r, we can divide both sides by 2:

We get,

⇒ r = d/2

Therefore, The correct equation is Option (c) r = d/2.

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The given question is incomplete, the complete question is

Which of the following correctly relates the measures of the diameter (d) and radius (r) of a circle?

(a) d = r/2

(b) r = 2d

(c) r = d/2

(d) d = 2/r

We can estimate that the middle 95% of all miniature Tootsie rolls will fall within the range of 3.04 grams to 3.56 grams for standard deviation of 0.13 gram.

A normal distribution is a symmetric, bell-shaped continuous probability distribution that is defined by its mean and standard deviation. The majority of the data in a normal distribution is located close to the mean, and the number of data points decreases as you deviate from the mean in either direction. Because many real-world events, like human height or test scores, have a tendency to follow a normal distribution, the normal distribution is frequently utilised in statistics. A helpful technique for determining the range of values within a normal distribution based on the mean and standard deviation is the empirical rule, commonly known as the 68-95-99.7 rule.

Given that, the mean of 3.30 grams and standard deviation of 0.13 gram.

Using the empirical formula the range that falls in 95% is associated to two standard deviations.

Mean + 2 standard deviations = 3.30 + 2(0.13) = 3.56 grams

Mean - 2 standard deviations = 3.30 - 2(0.13) = 3.04 grams

Hence, we can estimate that the middle 95% of all miniature Tootsie rolls will fall within the range of 3.04 grams to 3.56 grams.

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Answer:

86.24

Step-by-step explanation:

If you divide 12 ÷ 98 you will get 11.76

Now with that information you will subtract 11.76 to 98 and will get a total of  

86.24

Hope this helped :)

Answer:

Step-by-step explanation:

Let's use the Pythagorean theorem to solve this problem.

Let x be the length of the larger leg of the triangle. Then, the length of the smaller leg is x - 14 cm. The length of the hypotenuse is x + 2 cm.

Using the Pythagorean theorem, we have:

(x - 14)^2 + x^2 = (x + 2)^2

Simplifying and solving for x:

x^2 - 28x + 196 + x^2 = x^2 + 4x + 4

2x^2 - 32x + 192 = 0

Dividing both sides by 2:

x^2 - 16x + 96 = 0

Factorizing:

(x - 8)(x - 12) = 0

Therefore, x = 8 or x = 12.

If x = 8, then the smaller leg is 8 - 14 = -6 cm, which is not possible.

If x = 12, then the smaller leg is 12 - 14 = -2 cm, which is also not possible.

Therefore, there are no real solutions to this problem.

(1) 4y-7z is a binomial.

(2) 8-xy² is a binomial.

(3) ab-a-b can be written as ab - (a + b) which is a binomial.

(4) z²-3z+8 is a trinomial.

In algebra, monomials, binomials, and trinomials are expressions that contain one, two, and three terms, respectively.

A monomial is an algebraic expression with only one term. A monomial can be a number, a variable, or a product of numbers and variables.

A binomial is an algebraic expression with two terms that are connected by a plus or minus sign. For example, 2x + 3y and 4a - 5b are both binomials.

A trinomial is an algebraic expression with three terms that are connected by plus or minus signs.

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Classify into monomials, binomials and trinomials.

(1) 4y-7z

(1) 8-xy²

(v) ab-a-b

(ix) z2-3z+8

Answer:

7^1/3 * y^1/3

Step-by-step explanation:

(7y) ^ 1/3

We know that (ab) ^c = a^c * b^c

7^1/3 * y^1/3

3 individuals believe Elvis is still alive but do not believe that they have been abducted by rebels.

A Venn diagram is a graphical representation of set theory that illustrates the relationships between sets. It consists of a rectangle or a circle representing the universal set and circles or ovals representing the subsets.

a) Here's a Venn diagram to model the information:

Label the left circle "Elvis is still alive" and the right circle "Abducted by rebels." The overlapping region of the circles represents the individuals who believe both.

b) To find out how many believe neither of these things, we need to subtract the number of individuals in the overlapping region from the total number of individuals surveyed.

Total surveyed = 64

Number who believe both = 42

Therefore, the number who believe neither is:

64 - 42 = 22

So, 22 individuals believe neither of these things.

c) To find out how many believe Elvis is still alive but do not believe that they have been abducted by rebels, we need to subtract the number of individuals in the overlapping region from the number who believe Elvis is still alive.

Number who believe Elvis is still alive = 45

Number who believe both = 42

Therefore, the number who believe Elvis is still alive but do not believe that they have been abducted by rebels is:

45 - 42 = 3

So, 3 individuals believe Elvis is still alive but do not believe that they have been abducted by rebels.

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The directional derivative of f(x,y) at point P(1,5) in the direction of PQ is -2√2.

Find the directional derivative of the function f(x,y) = 3x² - y² + 4 at point P(1,5) in the direction of PQ, where P(1,5) as well as Q(4,2), we need to first calculate the gradient of f(x,y) at point P.

The gradient of f(x,y) at P is:

∇f(x,y) = [∂f/∂x, ∂f/∂y] = [6x, -2y]

Evaluating this at point P(1,5), we get:

∇f(1,5) = [6(1), -2(5)] = [6, -10]

Now, we need to find the unit vector in the direction of PQ. This can be calculated as follows:

u = PQ/|PQ|

where PQ = Q - P = [4 - 1, 2 - 5] = [3, -3] and |PQ| = √(3² + (-3)²) = √18 = 3√2

So, u = PQ/|PQ| = [3/3√2, -3/3√2] = [1/√2, -1/√2]

The directional derivative of f(x,y) at P in the direction of PQ is then given by:

D_u f(P) = ∇f(P) · u

where · represents the dot product.

Substituting the values we obtained earlier, wehave:

D_u f(P) = [6, -10] · [1/√2, -1/√2]

D_u f(P) = (6/√2) + (-10/√2)

D_u f(P) = -2√2

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So, out of 75 pulls, we would anticipate choosing a yellow pebble 25 times.

The measurement and study of random events are the focus of the mathematic branch known as probability. It entails calculating the probability of an event happening, with a scale from 0 (impossible) to 1. (certain). The fundamental meaning of chance is: Number of favourable outcomes minus the total number of outcomes equals the probability of an occurrence.

given

There are a total of the following painted stones in the box:

Total painted stones equals Claire's painted stones plus Laura's painted stones.

Total number of decorated stones: 35 + 40 = 75

Number of yellow stones equals the sum of Claire's and Laura's yellow pebbles.

5 Plus 20 = 25 yellow pebbles total.

As a result, the likelihood of getting a yellow pebble on any given draw is:

Number of yellow pebbles / total painted pebbles represents the likelihood of painting a yellow pebble.

25/75 is the likelihood of getting a yellow pebble.

The likelihood of drawing a yellow pebble on each draw is the same because we are drawing with substitution.

In 75 pulls, there should be an average of:

Number of draws times the likelihood of getting a yellow pebble yields the expected number of yellow pebbles.

Number of yellow pebbles anticipated Equals 75 * (1/3)

Expected quantity of golden stones: 25

So, out of 75 pulls, we would anticipate choosing a yellow pebble 25 times.

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With regard to p-charts, the general recommendation for the number of samples to be taken when estimating p is  to take at least 20-25 samples.

The number of samples needed when estimating p using p-charts depends on the desired level of accuracy and confidence.

A general recommendation is to take at least 20-25 samples to obtain a reasonably accurate estimate of p. This recommendation is based on the central limit theorem, which states that the distribution of sample proportions approaches a normal distribution as the sample size increases.

With a sample size of 20-25, the estimate of p is likely to be within a reasonable margin of error and have a sufficient level of confidence.

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(a) 12 cm 40° : Area of shaded segments = 301.44 sq. cm.

(b) 58° 16 cm ​: Area of shaded segments = 777.04 sq. cm.

Two radii that meet at the center to form a sector define a circle. The sector is the portion of the circle created by these two radii. Knowing a circle's central angle calculation and radius measurement are both crucial for solving circle-related difficulties.

Area of sector of circle = Ф/360 * πr²

π = 3.14

r  is the radius

Ф is the angle subtended.

(a) 12 cm 40°

Area of shaded segments = 40/60 * 3.14* 12²

Area of shaded segments = 40/60 * 452.16

Area of shaded segments = 301.44 sq. cm.

(b) 58° 16 cm ​

Area of shaded segments = 58/60 * 3.14* 16²

Area of shaded segments = 58/60 * 803.84

Area of shaded segments = 777.04 sq. cm.

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The diagram for the question is attached.

therefore, we have shown that: [tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})[/tex] assuming that x2 ≠ x1.

Slope refers to the measure of steepness of a line or a curve. In mathematics, slope is usually denoted by the letter "m" and is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between two points on a line.

The formula for calculating the slope between two points (x1, y1) and (x2, y2) on a line is:

[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})[/tex]

by the question.

To finds the slope of a line passing through two points (x1, y1) and (x2, y2), we use the slope formula:

[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})[/tex]

This formula represents the change in y divided by the change in x between the two points.

Now, assuming that x2 ≠ x1, we can simplify the formula as follows:

[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})*(1/1)[/tex]

Multiplying the numerator and denominator by 1, which in this case is (x2 - x1) / (x2 - x1), we get:

[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})*(x_{2}-x_{1})/(x_{2}-x_{1})[/tex]

Simplifying the numerator, we have:

[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})/[(x_{2}-x_{1})*1][/tex]

The term (x2 - x1) cancels out, leaving us with:

[tex]m=(y_{2}-y_{1} /1[/tex]

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Answer:

the value of r that makes the slope of the line passing through (3, 5) and (-3, r) equal to 3/4 is r = 1/2

Step-by-step explanation:

We can use the formula for the slope of a line passing through two points, which is:

m = (y2 - y1)/(x2 - x1)

where (x1, y1) and (x2, y2) are the coordinates of the two points.

In this case, we have two points (3, 5) and (-3, r), and we know that the slope m is 3/4. So we can write:

3/4 = (r - 5)/(-3 - 3)

Simplifying this equation, we get:

3/4 = (r - 5)/(-6)

Multiplying both sides by -6, we get:

-9/2 = r - 5

Adding 5 to both sides, we get:

r = -9/2 + 5

Simplifying, we get:

r = 1/2

The answer is letter A and letter F.

The question is asking if any of these choices are less than 300 centimeters. To find this out, first, you have to convert the meters to centimeters, to make it easier. (1 centimeter is equal to 0.01 meters) Multiply the meters by 100. For example 1.8m x 100 = 180cm. Do the rest for the other meters and then add them to the centimeters. Example: 180cm + 110cm = 290cm, which is less than 300cm.

Answer:

when y=3.2, the value of x is 0.4.

Step-by-step explanation:

If y is directly proportional to x, it means that y = kx, where k is the constant of proportionality. To find the value of k, we can use the given information that when x=20, y=160:

y = kx

160 = k(20)

k = 8

Now that we have the value of k, we can use it to find the value of x when y=3.2:

y = kx

3.2 = 8x

x = 0.4

Therefore, when y=3.2, the value of x is 0.4.

The probability of -20 < x < 15, where x be a normal random variable with a mean of 5 and a standard deviation of 10 is approximately 0.8351.

To solve this problem, we need to find the area under the normal distribution curve between -20 and 15, with a mean of 5 and a standard deviation of 10.

We can standardize the normal distribution by subtracting the mean and dividing by the standard deviation, which gives us the standard normal distribution with a mean of 0 and a standard deviation of 1.

So, we can first calculate the z-scores for -20 and 15:

z1 = (-20 - 5) / 10 = -2.5

z2 = (15 - 5) / 10 = 1

Next, we use a standard normal distribution table or calculator to find the probabilities associated with these z-scores:

P(z < -2.5) = 0.0062

P(z < 1) = 0.8413

To find the probability of -20 < x < 15, we subtract the probability associated with the lower z-score from the probability associated with the higher z-score:

P(-20 < x < 15) = P(-2.5 < z < 1) = P(z < 1) - P(z < -2.5)

P(-20 < x < 15) = 0.8413 - 0.0062 = 0.8351

Therefore, the probability of -20 < x < 15 is approximately 0.8351.

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1. The total yards of each fabric that Karina will buy to make a quilt is as follows:

a) Green Fabric = 12 square yards

b) Burgundy Fabric = 19 square yards

2. The total yards of fabric she will buy is 31 square yards.

The total yards of fabric can be determined by unit conversion using division operation.

Given that 36 inches = 1 yard, the square inches of fabric are converted to square yards by dividing the total by 36.

The total number of green fabric Karina requires = 420 square inches

= 12 square yards (420/36)

The total number of burgundy fabric Karina requires = 688 square inches

= 19 square yards (688/36)

The total number of fabric (green and burgundy) = 1,108 square inches (420 + 688)

36 inches = 1 yard

1,108 inches = 30.78 square yards (1,108/36)

= 31 square yards or (12 + 19)

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If a data set has an even number of observations, the median is the middle value of the two middle values when all values are placed in ascending order. Thus, option d is correct.

The median is the central value in a set of groups of data. The data should be organized and ordered from smallest value to biggest value. To find the middle value, we should divide the number of observations by two.

To determine the median, we must determine the Mean. Then, we need to calculate out how many observations there are in the data set. If there are an odd number of observations, the median cannot be calculated as a regular calculation.

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Answer:341.97 N

Step-by-step explanation:Answer and Explanation: The gravitational force on the 35 kg mass at the surface of the Earth is 341.97 N while at the surface of the Moon the 35 kg mass feels only 57.56 N. The gravitational force on the surface of the Earth is 5.94 times greater than that of the surface of the moon.

We can see that 36 is a composite number.

We also know that [tex]36=1\times36[/tex], [tex]3\times15[/tex], or [tex]4\times9[/tex]. All of these numbers are prime numbers so they are all factors of 36.

Thus we see that the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Answer:

$85.47

Step-by-step explanation:

Before tax and before tip: $74

Tax is 5%.

5% of $74 = 0.05 × $74 = $3.70

Cost including tax (but not tip): $74 + $3.70 = $77.70

After tax but before tip: $77.70

Tip is 10%.

10% of $77.70 = 0.1 × $77.70 = $7.77

Total price after tax & tip:

$77.70 + $7.77 = $85.47

The breadth, B which the soap carvings should be equal for the top of the bookcase to have the correct area is 5 inches.

In Mathematics, the area of a rectangle can be calculated by using the following mathematical equation:

A = LB

Where:

Substituting the given parameters into the area of rectangle formula, we have the following;

Area of soap carvings = length × width

300 = 60B

Width, B = 300/60

Width, B = 5 inches.

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Answer:

f(x) = (x^2 - 25) / (x - 5)

Step-by-step explanation:

Note that this function is undefined at x=5, which satisfies condition (a). Also, the function is defined for all other real numbers, which satisfies the domain and range requirement of (a).

For condition (b), note that the function is defined for all positive numbers greater than 1, including 1, since the denominator (x-5) will be positive for these values of x.

For condition (c), note that the function is undefined at x=1, since the denominator (x-5) will be negative for x slightly less than 1. Therefore, the function is defined for all positive numbers greater than 1, but not including 1.

Multiple linear regression is a statistical technique used to model the relationship between a dependent variable and multiple independent variables.

In multiple linear regression,

The dependent variable is modeled as a linear function of several independent variables.

Regression coefficient that quantifies the strength and direction of the relationship between independent and dependent variable.

Coefficient standard deviation refers to the standard deviation of the estimated regression coefficients in multiple linear regression.

Provides a measure of the variability of the estimated coefficients and can be used to assess the precision of the estimates.

New multiple regression model,

Collect data on the dependent variable and multiple independent variables of interest.

Perform a multiple linear regression analysis, which involves fitting a linear equation to the data using the method of least squares.

Estimates the regression coefficients and their standard deviations.

Used to assess the significance of each independent variable and the overall fit of the model.

Therefore,  multiple regression model consider adding or removing independent variables, transforming the data, or machine learning algorithms.

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The above question is incomplete , the complete question is:

What is multiple linear regression with coefficient standard deviation and mean to get new multiple regression ?

Answer: B (3,333 yards, 1 foot)

Step-by-step explanation: 10,000 feet / 3yards= 3,333.33 yards

3 feet in every yard so divide 10,000 by 3 and get 3,333.33 yards. since there are 3 feet in every yard the .33 is 1 foot.

Answer: 3 cm

Step-by-step explanation:

The formula for the area of a circle is A = πr², where A is the area and r is the radius. We are given that the area of the circle is 28½ cm².

So, 28½ = πr²

We need to solve for r. Dividing both sides by π, we get:

r² = 28½/π

r² = 9

Taking the square root of both sides, we get:

r = 3√1 = 3 cm

Therefore, the radius of the circle is 3 cm.

Check the explanation.

Step-by-step explanation:

As the graph of a linear function f passes through the point (-2,-10) and has a slope of 5/2.

As the slop-intercept form is given by:

where m is the slope and b is the y-intercept.

substituting the values (-2, -10) and m = 5/2 in the slop-intercept form to determine y-intercept.

And the equation of the line in the slope-intercept form will be:

putting b = -5 and slope = m = 5/2

Determining the zero of function.

As we know that the real zero of a function is the x‐intercept(s) of the graph of the function.

so let us determine the value of x (zero of function) by setting y = 0.

Therefore, the zeros of the function will be:

x = 2

Answer:

To find the slope, m, of a linear function that passes through two points, (x1, y1) and (x2, y2), we can use the following formula:

m = (y2 - y1) / (x2 - x1)

In this case, the two points are (3, 10) and (6, 8), so we have:

x1 = 3

y1 = 10

x2 = 6

y2 = 8

Substituting these values into the formula, we get:

m = (8 - 10) / (6 - 3) = -2 / 3

Therefore, the slope of the linear function is -2/3.

V is not a vector space, since at least one of the vector space axioms fails. Specifically, the axiom of closure under addition fails, since for any positive real numbers x and y, their sum xy may not be positive.

V is not a vector space, since some of the vector space axioms fail with the given operations.

Closure under addition fails For example, taking x=2 and y=3, we have xy = 6, but xy is not a positive real number, so xy is not in V.

Distributivity of scalar multiplication over vector addition fails For example, taking x=2, y=3, and c=-1, we have c(x+y) = cxy = -6, but cx + cy = -2 -3 = -5, which is not in V.

Other vector space axioms, such as associativity of addition, existence of additive identity and inverse, and compatibility of scalar multiplication with field multiplication, are still satisfied with the given operations.

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_____The given question is incomplete, the complete question is given below:

Let V be the set of all positive real numbers. Determine whether V is a vector space with the following operations. x + y = xy Addition cx = x Scalar multiplication If it is, then verify each vector space axiom; if it is not, then state all vector space axioms that fail. STEP 1: Check each of the 10 axioms. (1) U + v is in V. This axiom holds. This axiom fails. (2) U + V = V + u This axiom holds. This axiom fails. (3) U+ (v + w) = (u + v) + w This axiom holds. This axiom fails. STEP 2: Use your results from Step 1 to decide whether V is a vector space. O V is a vector space. O Vis not a vector space.

The solution using matrix inversion is x = -1 and y = 0.

To solve this system of equations using matrix inversion, we first need to write the system in matrix form:

[tex]\left[\begin{array}{cc}4&1\\4&-3\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{c}-4\\-4\end{array}\right][/tex]

Next, we need to invert the coefficient matrix on the left-hand side of the equation by finding its inverse. The inverse of a 2x2 matrix is given by:

[tex]\left[\begin{array}{cc}a&b\\c&d\end{array}\right]^{-1} =\frac{1}{(ad - bc)} \left[\begin{array}{cc}-d&b\\c&-a\end{array}\right][/tex]

Using this formula, we can find the inverse of the coefficient matrix [4 1; 4 -3]:

[tex]\left[\begin{array}{cc}4&1\\4&-3\end{array}\right]^{-1} =\frac{1}{(-12 - 4)} \left[\begin{array}{cc}3&1\\4&-4\end{array}\right]=\frac{-1}{16} \left[\begin{array}{cc}3&1\\4&-4\end{array}\right][/tex]

Now we can solve for [x y] by multiplying both sides of the equation by the inverse of the coefficient matrix:

[tex]\left[\begin{array}{cc}4&1\\4&-3\end{array}\right]^{-1}\left[\begin{array}{cc}4&1\\4&-3\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \frac{-1}{16} \left[\begin{array}{cc}3&1\\4&-4\end{array}\right]\left[\begin{array}{c}-4\\-4\end{array}\right][/tex]

[tex]\left[\begin{array}{cc}1&0\\0&1\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \frac{-1}{16} \left[\begin{array}{c}-12-4\\-16+16\end{array}\right] = \left[\begin{array}{c}-1\\0\end{array}\right][/tex]

Therefore, x = -1 and y = 0, which is the same solution we obtained using row reduction.

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