X is a continuous uniform random variable defined over the interval [0, 4]. Y is an exponential random variable, independent from X, with a parameter λ = 2.
a) Compute the mean of 2X+3Y
b) Compute the variance of 2X+3Y
c) What is the joint density fXY(x, y)?
d) Find P(X > Y)
e) Find the characteristic function ψX(w) for variable X
f) Find the characteristic function ψY(w) for variable Y
g) Find the characteristic function ψz(w) for variable Z = X + Y

Answer:

incorrect

Step-by-step explanation:

Weight at 4 weeks old: 70 g

The weight increases 10 g per week.

Weight at 5 weeks old: 80 g

Percent change from 70 g to 80 g

percent change = (new amount - old amount)/(old amount) × 100%

percent change = (80 - 70)/70 × 100%

percent change = 14.3%

Weight at 5 week: 80 g

The weight increases 10 g per week.

Weight at 6 weeks: 90 g

percent change = (new amount - old amount)/(old amount) × 100%

percent change = (90 - 80)/80 × 100%

percent change = 12.5%

The percent change went from 14.3% to 12.5%.

He is incorrect. The percent change is smaller each week because the actual change is always the same, 10 g per week, but the starting weight each week is greater.

1. The image shows translation

2. It is a translated image because there is no change in the size from the pre-image

3. Point A from the pre-image corresponds with point D

In mathematics, there are four different types of transformation. They are listed as;

Now, it is important to note that for translation, we have that;

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The unit rate of granola bars per box is 6.

Given the data:

Granola Bars: 12, 42

Boxes: 2, 7

To determine the unit rate of granola bars per box, we need to find the ratio of the number of granola bars to the number of boxes.

For the first set, we have 12 granola bars and 2 boxes.

For the second set, we have 42 granola bars and 7 boxes.

To find the unit rate, we divide the number of granola bars by the number of boxes:

Unit rate = Number of granola bars / Number of boxes

For the first set:

Unit rate = 12 / 2 = 6 granola bars per box

For the second set:

Unit rate = 42 / 7 = 6 granola bars per box

Therefore, the unit rate of granola bars per box is 6.

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The complete question is as follows:

A factory packages breakfast items including granola bars. Use the table to determine the unit rate of granola bars per box.

Granola Bars: 12, 42

Boxes: 2, 7

The value of x₀ is approximately equal to 24.113.

The formula for the standard normal distribution is given by

=> z = (x - μ) / σ  [ where x is the random variable, μ is the mean, and σ is the standard deviation ]

It also involves using the standard normal distribution table or calculator to find the probability of a value being greater than a given z-score, and solving for the original random variable using the standardized value.

Here we have

Suppose x is a normal distribution random variable with mean 20 and standard deviation 2.5.

We can use the standard normal distribution to solve this problem.

First, we standardize the random variable x by subtracting the mean and dividing by the standard deviation:

=> z = (x - μ) / σ = (x - 20) / 2.5

Now we want to find the value x₀ such that P(x > x₀), which is equivalent to finding the value z such that P(z > (x₀ - 20) / 2.5).

Using a standard normal distribution table or calculator, we can find that the probability of z being greater than 1.645 is approximately 0.05.

So we have:

P(z > 1.645) = P((x - 20) / 2.5 > 1.645)

Simplifying and solving for x₀, we get:

=> (x - 20) / 2.5 > 1.645

=> x - 20 > 4.113

=> x > 24.113

Therefore,

The value of x₀ is approximately equal to 24.113.

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Answer: 3

Step-by-step explanation:

The type of graph that would be useful for determining if this is true is a scatter plot.

A scatter plot is a type of graph that displays the relationship between two variables. It is a collection of data points, where each point represents the value of two different variables for a single observation.

In a scatter plot, the two variables are plotted on the x-axis (horizontal axis) and y-axis (vertical axis).

A scatter plot would be a useful type of graph for determining if there is a relationship between students who can say all of their multiplication facts in under two minutes and their performance on three-digit by two-digit division.

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The probability distribution used to model a random variable x that equals the number of events that occur within an interval or area of opportunity is the Poisson distribution. This is used to model a random variable representing the number of events occurring within a fixed interval or area of opportunity, given an average rate of occurrence.

The Poisson distribution is a discrete probability distribution that describes the probability of a given number of events occurring in a fixed interval of time or space, given the average rate at which events occur and the assumption that the events are independent of each other. It is commonly used in fields such as biology, physics, and engineering to model occurrences of rare events such as accidents, defects, or rare diseases.

The Poisson distribution has a single parameter λ, which represents the average rate of events occurring in the interval or area of opportunity. The probability of observing exactly k events in this interval is given by the Poisson probability mass function:

P(X=k) = (e^-λ * λ^k) / k!

where X is the random variable representing the number of events, e is the mathematical constant approximately equal to 2.71828, and k! is the factorial of k.

The Poisson distribution is similar to the binomial distribution but is used when the number of trials is very large and the probability of success is very small. In this case, the binomial distribution becomes impractical to use, and the Poisson distribution is a more appropriate model.

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The new equilibrium price is $12. The Option B is correct.

To find new equilibrium price, we need to compare the original supply and demand schedules to the new supply schedule after the technological advance.

The new supply schedule can be found by adding 60 units to the original quantity supplied at each price.

Price $10 11 12 13 14 15

Quantity Demanded 100 150 190 220 245 265

Original Quantity Supplied 295 275 250 220 180 135

New Quantity Supplied 355 335 310 280 240 195

The new equilibrium price is where the quantity demanded equals the quantity supplied. Looking at the new schedules, we can see that the new equilibrium price is $12.

Full question" Use the following table to answer the question below. Price $10 11 12 13 14 15 Quantity Supplied Quantity Demanded A) $13. 100 150 190 220 245 265 295 275 250 220 180 135 If a technological advance lowers production costs such that the quantity supplied increases by 60 units of this product at each price, the new equilibrium price would be Price $10 11 12 13 14 15 A) $13. B) $12. C. $14 D. $11

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The area of the shaded region in this problem is given as follows:

995.44 cm².

The area of a circle of radius r is given by the multiplication of π and the radius squared, as follows:

A = πr²

The radius of a circle represents the distance between the center of the circle and a point on the circumference of the circle, hence it's measure is given as follows:

r = 21 cm.

Then the area of the entire circle is given as follows:

A = π x 21²

A = 1385.44 cm².

The right triangle has two sides of length 39 cm and 20 cm, hence it's area is given as follows:

A = 0.5 x 39 x 10

A = 390 cm².

Then the area of the shaded region is given as follows:

1385.44 - 390 = 995.44 cm².

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The value of the chi-square test statistic and P-value are χ2 = 5.54 and between 0.10 and 0.15 respectively.

To determine the chi-square test statistic and P-value, we need to perform a chi-square test of independence using the observed frequencies and the expected frequencies based on the null hypothesis.

The null hypothesis states that the distribution of lunch location preference is as claimed in the article: 70% prefer the cafeteria, and the remaining options are preferred equally.

To calculate the expected frequencies, we need to assume that the null hypothesis is true. Since there are four lunch options, each option would be expected to have an equal probability of 0.10 (or 10%) if the null hypothesis is true.

Using the observed and expected frequencies, we can calculate the chi-square test statistic using the formula:

χ2 = Σ((O - E)² / E)

Substituting the values:

χ2 = ((118-105)²/105) + ((10-15)²/15) + ((12-15)²/15) + ((10-15)²/15)

= 5.54285714286

≈ 5.54

To determine the degrees of freedom, we subtract 1 from the number of categories (4 - 1 = 3).

Using the chi-square test statistic and degrees of freedom, we can find the P-value from a chi-square distribution table or using statistical software.

Based on the given answer choices, the correct option is:

χ2 = 5.54, P-value is between 0.10 and 0.15

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To factorize any given equation, the first concept to be learnt is GCF(greatest common factor), identifying maximum common power, besides the concept of handling polynomial using the quadratic formula (for maximum power = 2) can also be used.

Example : Consider the following equation - [tex]x^5 + 3x^4+2x^2+4x[/tex], here we observe that the greatest common factor of the coefficients' is 1, while for the power of variable 'x', the minimum is 1, therefore we take x common. One way to factorize :

[tex]x(x^4+3x^3+2x+1)[/tex] . Another way to write the above mentioned is :

[tex]x^4(x+3)+2x(x+2)[/tex]

Hope, the steps are clear and easy to understand. You can also use quadratic formula to find roots and then put them in (x+a)(x+b) form

For the set {u1, u2, u3} spans R3 and A = [u1 u2 u3], the nullity of A is 0.

If the set {u1, u2, u3} spans R3, it means that any vector in R3 can be expressed as a linear combination of the three vectors. Therefore, the three vectors are linearly independent and form a basis for R3.

If we construct a matrix A whose columns are the three vectors, we can find the nullity of A by determining the dimension of the null space of A, which is the set of all vectors x that satisfy the equation Ax = 0.

Since the three vectors span R3, the matrix A is a 3x3 matrix with rank 3. By the rank-nullity theorem, the nullity of A is given by:

nullity(A) = n - rank(A)

where n is the number of columns of A. In this case, n = 3, so:

nullity(A) = 3 - rank(A) = 3 - 3 = 0

Therefore, the nullity of A is 0.

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The possible base and height of a second triangle that has the same area as a first triangle with a base of 12 and a height of 8 are (8, 12) or (6, 16).

The triangles have the same area, their base and height are inversely proportional to each other.

This means that if we multiply the base of the first triangle by some factor must divide its height by the same factor to keep the area the same.

Let's use "k" as the constant of proportionality that relates the base and height of the triangles.

Then, we can write:

base₁ × height₁ = base₂ × height₂

The subscripts 1 and 2 refer to the first and second triangles respectively.

The base and height of the first triangle so we can substitute those values:

12 × 8 = base₂ × height₂

Simplifying, we get:

96 = base₂ × height₂

Now, since the base and height are inversely proportional can choose any pair of values that multiply to 96.

The base of the second triangle is 8 then its height must be:

height₂ = 96 / base₂

= 96 / 8

= 12

Alternatively, if the base of the second triangle is 6, then its height must be:

height₂ = 96 / base₂

= 96 / 6

= 16

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The distance between the spheres defined by x^2 + y^2 + z^2 = 4 and x^2 + y^2 + z^2 - 4x - 4y - 4z + 11 = 0 will be determined.



The first sphere equation can be written as:

x^2 + y^2 + z^2 = 4 ............. (1)

The second sphere equation can be written as:

x^2 + y^2 + z^2 - 4x - 4y - 4z + 11 = 0 ............. (2)

To find the distance between the spheres, we need to find the distance between their centers. The centers of the spheres can be determined by completing the square for each equation.

For Equation (1):

x^2 + y^2 + z^2 = 4

We have a sphere centered at the origin (0, 0, 0) with a radius of 2.

For Equation (2):

x^2 + y^2 + z^2 - 4x - 4y - 4z + 11 = 0

Rearranging terms:

x^2 - 4x + y^2 - 4y + z^2 - 4z = -11

To complete the square, we need to add and subtract appropriate constants:

x^2 - 4x + 4 + y^2 - 4y + 4 + z^2 - 4z + 4 = -11 + 4 + 4 + 4

(x^2 - 4x + 4) + (y^2 - 4y + 4) + (z^2 - 4z + 4) = 1

Simplifying:

(x - 2)^2 + (y - 2)^2 + (z - 2)^2 = 1

We have a sphere centered at (2, 2, 2) with a radius of 1.

Now that we have the centers of the two spheres, the distance between them can be found using the distance formula:

Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

Using the coordinates of the centers, we have:

Distance = sqrt((2 - 0)^2 + (2 - 0)^2 + (2 - 0)^2)

Distance = sqrt(4 + 4 + 4)

Distance = sqrt(12)

Distance ≈ 3.464

Therefore, the distance between the spheres x^2 y^2 z^2 = 4 and x^2 y^2 z^2 = 4x + 4y + 4z - 11 is approximately 3.464 units.

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The distance between the spheres [tex]\(x^2 + y^2 + z^2 = 4\) and \(x^2 + y^2 + z^2 = 4x + 4y + 4z - 11\)[/tex] is [tex]\(\sqrt{153}\)[/tex] units.

To find the distance between the spheres [tex]\(x^2 + y^2 + z^2 = 4\) and \(x^2 + y^2 + z^2 = 4x + 4y + 4z - 11\)[/tex], you can use the formula for the distance between two points in three-dimensional space.

The general formula for the distance between two points [tex]\((x_1, y_1, z_1)\)[/tex] and [tex]\((x_2, y_2, z_2)\)[/tex] is given by:

[tex]\[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\][/tex]

In this case, you can consider one point on the first sphere as the center of the sphere, which is at the origin (0, 0, 0), and the point on the second sphere as another point (4, 4, 11), as it satisfies the equation [tex]\(x^2 + y^2 + z^2 = 4x + 4y + 4z - 11\).[/tex]

Now, you can plug these values into the distance formula:

[tex]\text{Distance} &= \sqrt{(4 - 0)^2 + (4 - 0)^2 + (11 - 0)^2} \\\\&= \sqrt{16 + 16 + 121} \\\\&= \sqrt{153}[/tex]

So, the distance between the two spheres is [tex]\(\sqrt{153}\)[/tex] units.

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Greg drove approximately 199.1 miles in the truck.

We can subtract the base fee from the total amount he paid and then divide the remaining amount by the additional charge per mile.

Total amount paid - Base fee = Additional charge for miles

$185.12 - $19.95 = $165.17 (additional charge for miles)

To calculate the number of miles travelled, divide the additional fee by the fee per mile:

$165.17 / $0.83 per mile = 199.1 miles

Therefore, Greg drove approximately 199.1 miles in the truck.

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Answer:

Step-by-step explanation:

The value represented by point A is less than -2.5.

Their sum Cn = Cn-2 + Cn-1 is also divisible by 3.

Thus, by strong induction, we have proved that Cn is divisible by 3 for all integers n > 1.

What is recurrence relations?

In mathematics, a recurrence relation is a mathematical equation that recursively defines a sequence of values. Recurrence relations are used to describe sequences of numbers or other mathematical objects that depend on previous terms in the sequence.

We will use strong induction to prove that Cn is divisible by 3 for all integers n > 1.

Base case: n = 2

C2 is given to be -9, which is divisible by 3.

Base case: n = 3

C3 = C1 + C2 = 0 - 9 = -9, which is not divisible by 3. However, we will show that the statement holds for all integers up to n - 1, and then use that to prove the statement for n.

Inductive step:

Assume that Ck is divisible by 3 for all integers k such that 2 < k < n. We want to prove that Cn is divisible by 3.

From the recursive definition of the sequence, we have:

Cn = Cn-2 + Cn-1

By our assumption, Cn-2 and Cn-1 are both divisible by 3.

Therefore, their sum Cn = Cn-2 + Cn-1 is also divisible by 3.

Thus, by strong induction, we have proved that Cn is divisible by 3 for all integers n > 1.

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Answer:

280 degrees.

Step-by-step explanation:

The way you work this out is by:

1) Work out 240-360

2) Subtract the answer from 360

3) Divide the answer of step 2 into however many angles you have (in this instance 4).

4) Then use one of these angles and subtract it from 360. This will give you a reflex angle.

5) That's it.

Answer:

----------------------

We know that the sum of interior angles of a quadrilateral is 360°.

Show this as a sum using angle measures in the diagram:

Answer:3/4

Step-by-step explanation:

The probability of throwing at most two 6s is (5/6)^8 + 8*(1/6)(5/6)^7 + (28/2)(1/6)^2*(5/6)^6, which is approximately equal to 0.983.

We want to find the probability of throwing at most two 6s, which means we want to find the probability of throwing zero, one, or two 6s. The probability of throwing zero 6s is (5/6)^8, since we need to roll a non-6 on all 8 rolls.

The probability of throwing one 6 is 8*(1/6)(5/6)^7, since there are 8 ways to choose which roll will be the 6, and we need to roll a 6 on that one roll and a non-6 on the other 7 rolls.

The probability of throwing two 6s is (28/2)(1/6)^2*(5/6)^6, since there are 28 ways to choose which 2 rolls will be the 6s, and we need to roll a 6 on both of those rolls and a non-6 on the other 6 rolls.

Therefore, the probability of throwing at most two 6s is (5/6)^8 + 8*(1/6)(5/6)^7 + (28/2)(1/6)^2*(5/6)^6, which is approximately equal to 0.983.

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The eigenvalues of the matrix A are λ₁ = 1, λ₂ = 2, and λ₃ = 12. For each eigenvalue, the rank of the matrix A - λI is 2, 2, and 3, respectively. The matrix A is diagonalizable.

a. To find the eigenvalues of the matrix A = [ [4 0 1][ 2 3 2][ 1 0 4]], we need to solve the characteristic equation det(A - λI) = 0, where I is the identity matrix and λ is the eigenvalue.

The characteristic equation is:

det([ [4 0 1][ 2 3 2][ 1 0 4]] - λ[ [1 0 0][ 0 1 0][ 0 0 1]]) = 0

Simplifying, we get:

det([ [4 - λ 0 1][ 2 3 - λ 2][ 1 0 4 - λ]]) = 0

Expanding the determinant, we get:

(4 - λ) * (3 - λ) * (4 - λ) - 2 * (4 - λ) - 2 * (3 - λ) + 2 * (1 - λ) = 0

Simplifying, we get:

-λ^3 + 11λ^2 - 32λ + 24 = 0

Factoring, we get:

-(λ - 1) * (λ - 2) * (λ - 12) = 0

Therefore, the eigenvalues of the matrix A are λ₁ = 1, λ₂ = 2, and λ₃ = 12.

b. For each eigenvalue, we need to find the rank of the matrix A - λI, where I is the identity matrix and λ is the eigenvalue.

For λ₁ = 1, we have:

A - λ₁I = [ [3 0 1][ 2 2 2][ 1 0 3]]

The rank of A - λ₁I is 2.

For λ₂ = 2, we have:

A - λ₂I = [ [2 0 1][ 2 1 2][ 1 0 2]]

The rank of A - λ₂I is 2.

For λ₃ = 12, we have:

A - λ₃I = [ [-8 0 1][ 2 -9 2][ 1 0 -8]]

The rank of A - λ₃I is 3.

c. To determine if matrix A is diagonalizable, we need to check if it has n linearly independent eigenvectors, where n is the size of the matrix.

Since matrix A is a 3x3 matrix, we need to find three linearly independent eigenvectors. We can find the eigenvectors by solving the system of equations (A - λI)x = 0 for each eigenvalue.

For λ₁ = 1, we have:

(A - λ₁I)x = [ [3 0 1][ 2 2 2][ 1 0 3]]x = 0

Solving the system of equations, we get:

x1 = -1/3 * x3

x2 = 1/2 * x3

Therefore, the eigenvector corresponding to λ₁ is [x1, x2, x3] = [-1, 3, 6].

For λ₂ = 2, we have:

(A - λ₂I)x = [ [2 0 1][ 2 1 2][ 1 0 2]]x = 0

Solving the system of equations, we get:

x1 = -1/2 * x3

x2 = x3

Therefore, the eigenvector corresponding to λ₂ is [x1, x2, x3] = [-1, 1, 2].

For λ₃ = 12, we have:

(A - λ₃I)x = [ [-8 0 1][ 2 -9 2][ 1 0 -8]]x = 0

Solving the system of equations, we get:

x1 = -1/8 * x3

x2 = -2/9 * x3

Therefore, the eigenvector corresponding to λ₃ is [x1, x2, x3] = [-1, -16/9, 8].

Since we have found three linearly independent eigenvectors, the matrix A is diagonalizable.

Therefore, the eigenvalues of the matrix A are λ₁ = 1, λ₂ = 2, and λ₃ = 12. For each eigenvalue, the rank of the matrix A - λI is 2, 2, and 3, respectively. The matrix A is diagonalizable.

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Answer:

8

Step-by-step explanation:

put the numbers in order smallest to largest

3 5 5 7 8 8 9 9 9 10

you need to find the middle number but because there is an even amount of numbers you need to find the two middle numbers and add them then divide by 2.

In this case it's the 8+8 =16

16/2=8

The value of r=-0.854 indicates a stronger correlation than r=0.835.

To determine which value of r indicates a stronger correlation, r=0.835 or r=-0.854, we need to compare their absolute values.

Step 1: Find the absolute values of both correlation coefficients.
|r=0.835| = 0.835
|r=-0.854| = 0.854

Step 2: Compare the absolute values.
0.835 < 0.854

The value of r=-0.854 indicates a stronger correlation than r=0.835.

This is because the absolute value of -0.854 (0.854) is greater than the absolute value of 0.835 (0.835), meaning that the correlation is stronger, regardless of the negative sign.

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Step-by-step explanation:

The amplitude is the value that the cosine is being multiplied by.

The general equation of a sinusoid is

[tex] a \cos(b(x + c) ) + d[/tex]

where a is the amplitude

[tex] \frac{2\pi}{ |b| } [/tex]

is the period

-c is the phase shift

d is the midline(vertical shift)

Here the amplitude is -1/2 so b is the correct answer.

Answer:

the amplitude of the function that is y= -1/2 cos x, is 1/2.

Step-by-step explanation:

To find the equations of the tangent and normal lines to the hyperbola x^2/4 − y^2/1 = 1 at the point where x = 4, we need to first find the y-coordinate of the point of tangency. We can do this by substituting x = 4 into the equation of the hyperbola and solving for y:

x^2/4 - y^2/1 = 1

(4)^2/4 - y^2/1 = 1

16/4 - y^2/1 = 1

4 - y^2 = 1

y^2 = 3

y = ±√3

So, the point of tangency is (4, √3).

Now, to find the equation of the tangent line at this point, we need to take the derivative of the equation of the hyperbola implicitly with respect to x:

x^2/4 - y^2/1 = 1

Differentiating both sides with respect to x:

x/2 - 2y(dy/dx) = 0

dy/dx = x/(4y)

At the point (4, √3), we have:

dy/dx = 4/(4√3) = √3/3

So the slope of the tangent line at this point is √3/3. Using the point-slope form of the equation of a line, we can write the equation of the tangent line as:

y - √3 = (√3/3)(x - 4)

Simplifying, we get:

y = (√3/3)x - (√3/3)∙4 + √3

y = (√3/3)x - (√3/3) + √3

y = (√3/3)x + 2√3/3

To find the equation of the normal line, we first need to find its slope, which is the negative reciprocal of the slope of the tangent line. So:

m(normal) = -1/m(tangent) = -1/(√3/3) = -√3

Using the point-slope form again, the equation of the normal line is:

y - √3 = (-√3)(x - 4)

Simplifying, we get:

y = -√3x + 4√3 + √3

y = -√3x + 5√3

So the equations of the tangent and normal lines to the hyperbola x^2/4 − y^2/1 = 1 at the point where x = 4 are:

Tangent line: y = (√3/3)x + 2√3/3

Normal line: y = -√3x + 5√3

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Angle facts refer to the relationships between angles in a triangle or other shapes.

These relationships include the fact that the sum of all angles in a triangle is 180 degrees, that angles opposite each other in a parallelogram are equal, and that angles on a straight line add up to 180 degrees.
In Susan's case, she is trying to find angle b, and she first finds angle a before using that information to find angle b.

So let's break down each step:
a) To find angle a, Susan must have used an angle fact that relates to the triangle she is working with.

Since she did not provide any information about the triangle, we cannot be sure which angle fact she used.

However,

We do know that the sum of all angles in a triangle is 180 degrees, so it is likely that she used this fact in some way to find angle a.
b) Once Susan has found angle a, she uses another angle fact to find angle b. Again, we do not have enough information to know exactly which angle fact she used.

However, we do know that angle b is not directly opposite angle a, since they are both named angles in the same triangle.

Therefore, she must have used some other relationship between angles in the triangle to find angle b.
Without more information about the triangle and the specific angle facts Susan used, we cannot say for sure how she found angle a and angle b. However, we can say that angle facts are a useful tool for finding missing angles.

in a variety of shapes, and it is always a good idea to review your work and double-check your answers to ensure accuracy.

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The slant height of the roof x is 8.6 ft.

The right circular cone is the cone in which the line joining the peak of the cone to the center of the base of the circle is perpendicular to the surface of its base.

Let consider the dimensions of the given cone:

c = hypotenuse = slant height

a = base = radius = 5 ft

b = height = 7 ft

SO,

[tex]\sf x^2=5^2+7^2[/tex]

[tex]\sf x^2=25+49[/tex]

[tex]\sf x^2=74[/tex]

[tex]\sf x^2=\sqrt{74}[/tex]

[tex]\sf x^2=8.602\thickapprox\bold{8.6 \ ft}[/tex]

Hence, The slant height of the roof is 8.6 ft.

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The value of the diameter of the base of the cylinder is,

⇒ d = 8

We have to given that;

The volume of a cylinder is 96 π cubic meters

And, the height is 6 meters.

Since, We know that;

Volume of cylinder is,

V = πr²h

Substitute all the values we get;

96π = π × r² × 6

16 = r²

r = √16

r = 4

Thus, The value of the diameter of the base of the cylinder is,

⇒ d = 4 × 2

⇒ d = 8

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The distribution of x^2 y^2 z^2 is not uniform. To find the distribution, we need to use the transformation method.

Let g(x,y,z) = x^2 y^2 z^2. Then, we need to find the Jacobian of the transformation. J = | ∂(x,y,z)/∂(u,v,w) |, where (u,v,w) = (x^2 y^2 z^2, θ, φ)

∂(x,y,z)/∂u = 2xy^2z^2
∂(x,y,z)/∂v = -x^2y^2zsin(φ)
∂(x,y,z)/∂w = -x^2y^2z^2cos(φ)

Therefore, J = 2x^2y^3z^3sin(φ)cos(φ)

The joint distribution of (u,v,w) is given by:

f(u,v,w) = f(x,y,z) |J|, where (x,y,z) is uniform on the unit sphere.

Since (x,y,z) is uniform on the unit sphere, we know that:

f(x,y,z) = 1/(4π)

Substituting the Jacobian, we get:

f(u,v,w) = 1/(4π) * 2x^2y^3z^3sin(φ)cos(φ)

To find the marginal distribution of u, we integrate out v and w:

f(u) = ∫∫ f(u,v,w) dv dw
     = ∫∫ 1/(4π) * 2x^2y^3z^3sin(φ)cos(φ) dv dw

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Therefore, The Taylor series for f(x)=sin(x) at x=π/2 is ∑n=0[infinity](-1)^n(x−π/2)^{2n+1}/(2n+1)! and can be found by evaluating the derivatives of sin(x) at x=π/2.

The Taylor series for f(x)=sin(x) at x=π/2 can be found by taking the derivative of sin(x) and evaluating it at x=π/2. We get f(π/2) = sin(π/2) = 1 and f'(x) = cos(x). Evaluating f'(π/2) gives us cos(π/2) = 0. We can then find the second derivative f''(x) = -sin(x) and evaluate it at x=π/2 to get f''(π/2) = -1. This pattern continues, with each derivative evaluated at x=π/2 giving us a coefficient for our Taylor series. Therefore, the Taylor series for f(x)=sin(x) at x=π/2 is ∑n=0[infinity](-1)^n(x−π/2)^{2n+1}/(2n+1)!.

Therefore, The Taylor series for f(x)=sin(x) at x=π/2 is ∑n=0[infinity](-1)^n(x−π/2)^{2n+1}/(2n+1)! and can be found by evaluating the derivatives of sin(x) at x=π/2.

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