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The lengths of the semimajor and semi-minor axes of an ellipse (or an ellipsoid in higher dimensions) are related to the singular values of the transformation matrix.

Hence, we can write the relationship between a, b, and det(A) as:

a^2 = λ1 = det(A) λ2^(-d+2)/2
b^2 = λ2 = det(A) λ1^(-1) λ3^(-d+2)/2^(d-2)

or equivalently,

a^2 b^2 = det(A)^2 / 2^(d-2).

The lengths of the semimajor and semi-minor axes of an ellipse (or an ellipsoid in higher dimensions) are related to the singular values of the transformation matrix. Specifically, if T is an invertible linear transformation with matrix A, then the lengths of the semi-axes of T(Ω) are given by the square roots of the eigenvalues of the matrix A^T A. Let λ1, λ2, λ3 be the eigenvalues of A^T A (or A^T A^T in 2D), arranged in decreasing order. Then the lengths of the semi-axes of T(Ω) are given by:

a = √(λ1)
b = √(λ2)  (in 2D) or b = √(λ3) (in 3D)

Moreover, the determinant of A is equal to the product of the singular values of A, which are the square roots of the eigenvalues of A^T A. Therefore, we have:

det(A) = λ1 λ2 λ3^(d-2)/2

where d is the dimension of the space (2 or 3 in the case of an ellipse in 2D or an ellipsoid in 3D, respectively).

Hence, we can write the relationship between a, b, and det(A) as:

a^2 = λ1 = det(A) λ2^(-d+2)/2
b^2 = λ2 = det(A) λ1^(-1) λ3^(-d+2)/2^(d-2)

or equivalently,

a^2 b^2 = det(A)^2 / 2^(d-2)

This relationship shows that the product of the semi-axes of T(Ω) is related to the determinant of A, but the individual semi-axes depend also on the singular values of A, which are related to the eigenvalues of A^T A.

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The power series of the function 3/((x-2)(x+1)) is:

-3/4 - 1/4(x-2) + 1/4(x-2)² - 1/4(x-2)³- 1/2 - 1/2(x+1) - 1/2(x+1)² - 1/2(x+1)³ + ...

To find the power series of the function 3/((x-2)(x+1)), we first need to find the partial fraction decomposition of the function:

3/((x-2)(x+1)) = A/(x-2) + B/(x+1)

To solve for A and B, we need to find a common denominator on the right-hand side:

3 = A(x+1) + B(x-2)

Setting x = 2, we get:

3 = A(3)

A = 1

Setting x = -1, we get:

3 = B(-3)

B = -1

Therefore, we have:

3/((x-2)(x+1)) = 1/(x-2) - 1/(x+1)

Now we can use the formula for the geometric series:

1/(1 - t) = 1 + t + t²+ t³ + ...

to write the power series of each term in the partial fraction decomposition. Substituting t = x-2 for the first term and t = -x-1 for the second term, we get:

1/(x-2) = -1/4 - 1/4(x-2) + 1/4(x-2)² - 1/4(x-2)³ + ...

1/(x+1) = -1/2 - 1/2(x+1) - 1/2(x+1)² - 1/2(x+1)³ - ...

Combining the two series, we have:

3/((x-2)(x+1)) = -3/4 - 1/4(x-2) + 1/4(x-2)² - 1/4(x-2)³ - 1/2 - 1/2(x+1) - 1/2(x+1)² - 1/2(x+1)³ + ...

Therefore, the power series of the function 3/((x-2)(x+1)) is:

-3/4 - 1/4(x-2) + 1/4(x-2)² - 1/4(x-2)³- 1/2 - 1/2(x+1) - 1/2(x+1)² - 1/2(x+1)³ + ...

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The algebraic expression for the width of rectangle X is:

w = (s - n - 5)/2

Since a square has four equal sides, each side of the square can be represented as s. The area of the square is s². When it is cut into three rectangles X, Y and Z, the area of the square is equal to the sum of the areas of the three rectangles.

So, we have:

s² = 10w + 5n + 5(s-n-5)/2

Simplifying this equation, we get:

s² = 10w + 5n + (5s - 5n - 25)/2

Multiplying both sides by 2, we get:

2s² = 20w + 10n + 5s - 5n - 25

Simplifying this equation, we get:

20w = 2s² - 10n - 5s + 5n + 25

Dividing both sides by 2 and rearranging the terms, we get the algebraic expression for the width of rectangle X:

w = (s - n - 5)/2

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Complete Question:

A square is cut into three rectangles X, Y and Z. Rectangle X has length 10cm. Rectangle Y has length n cm and width 5cm.  Write down an algebraic expression for the width of rectangle X.

Based on the definition of the the angles on a straight line, we have:

m<B = 40°; m<E = 50°; m<C = 90; m<D = 90°.

When two straight lines intersect, they form four angles. Angles that are formed on a straight line are called "angles on a straight line" or "linear pairs." These angles are also known as supplementary angles because they add up to 180 degrees. Therefore, if two angles are formed on a straight line, the measure of one angle added to the measure of the other angle equals 180 degrees.

m<B = 180 - 140 = 40° [straight angle]

m<E = 180 - 130 = 50°

m<C = 180 - m<B - m<E [triangle sum theorem]

Substitute:

m<C = 180 - 40 - 50

m<C = 90

m<D = 180 - m<C

m<D = 180 - 90 = 90°

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The formula to find the volume of attached figure: V = πr²h

The correct answer is an option (d)

In the attached image we can observe that the figure is of cylinder with radius 'r' and height 'h'

This means that we need to find the volume of cylinder.

We know that the formula for tha cylinder is:

Volume of cylinder = Base area × height of cylinder

As we know that the base of cylinder is circular in shape.

so, the base area of cylinder would be,

A = πr²

when we say height of the cylinder then it means the perpendicular distance between two parallel bases of cylinder. It is also known as length of the cylinder.

So, the formula for volume would be,

V = πr²h

Therefore, the correct answer is an option (d)

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The equation is separable, and the solution to the initial value problem is y(x) = Ce^(1/3sin(3x)), where C = ±e^(C1).

A differential equation of the form 3y'(x) = ycos(3x) is separable because we can write it as:

dy/dx = (1/3)ycos(3x)

We can separate the variables y and x and integrate both sides of the equation with respect to their respective variables:

∫(1/y)dy = ∫(1/3)cos(3x)dx

ln|y| = (1/3)sin(3x) + C1

where C1 is the constant of integration.

Solving for y, we get:

y(x) = Ce^(1/3sin(3x))

where C = ±e^(C1) is the constant of integration.

Therefore, the equation is separable, and the solution to the initial value problem is y(x) = Ce^(1/3sin(3x)), where C = ±e^(C1).

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The height of the tree in feet is 62 .

Two triangles are similar if their corresponding angles are congruent and corresponding sides are proportional.

From the diagram:

Leg 1 of the smaller triangle = 5ft 2in =  ( 5×12 + 12 )in = 62 in

Leg 2 of the smaller triangle = 10ft ( 10 × 12 ) = 120in

Leg 1 of the larger triangle = x

Leg 2 of the larger triangle = 120 ft = ( 120 × 12 )in = 1440 in

Since the corresponding sides of similar triangles are proportional.

We take equate their ratios

62/120 = x/1440

Solve for x

120x = 62 × 1440

120x = 89280

x = 89280/120

x = 744in

Convert back to feet

x = ( 744 ÷ 12 ) ft

x = 62 ft

Therefore, the value of x is 62 feets.

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The population mean is equal to 80 for the mean of a sampling distribution of sample means with n=49, and the standard deviation is equal to 1/√(n).

The standard deviation is a statistician's gauge of a group of values' degree of dispersion or variation. While a high standard deviation suggests that the values are dispersed over a wider range, a low standard deviation suggests that the values tend to be close to the mean (also known as the expected value) of the set.With a sample size of 49, the mean of a sampling distribution of sample means is equal to the population mean of 801.

With a sample size of n=49, the standard deviation of a sampling distribution of sample means is identical.

n=49 is equal to σ/√(n)

= 7/√(49) = 1¹

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Answer:

It's 84

Step-by-step explanation:

To calculate the area of a triangule you need to use this formula:

B * H / 2

So:

12 * 14 / 2 = 84

Hope this helps :)

Pls brainliest...

Answer:

41.6666%

Step-by-step explanation:

a) The value of zα for α = .0055 is -2.62.

b) The value of zα for α = .09 is 1.34.

c) The value of zα for α = .663 is .39 .

In statistics, the letter "z" usually refers to the z-score, which is a measure of how many standard deviations a particular value is from the mean. In order to determine the value of zα, we need to use the standard normal distribution table or a calculator that has this function built-in.

a. To find zα for α = .0055, we need to locate the area of .0055 in the body of the standard normal distribution table. This area is between z-scores of -2.61 and -2.62. Therefore, zα = -2.62.

b. For α = .09, we need to locate the area of .09 in the body of the standard normal distribution table. This area is between z-scores of 1.34 and 1.35. Therefore, zα = 1.34.

c. Finally, for α = .663, we need to locate the area of .663 in the body of the standard normal distribution table. This area is between z-scores of .38 and .39. Therefore, zα = .39.

In summary, zα is the z-score that corresponds to a given value of α (the level of significance). We can find this value using a standard normal distribution table or a calculator.

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The complement of A has a probability of 0.044.

The occurrence that A does not occur is the complement of event A. A', often known as "not A," is used to indicate it. One less than the probability of A gives you the probability of A'.

P(A') = 1 - P(A)

In probability and statistics, the concept of the complement of an event is helpful because it enables us to calculate the probability of an event by calculating the probability of its complement and deducting it from 1. The probability of the complement may also, in some circumstances, be more easily determined than the probability of the initial occurrence.

To find the complement we subtract the probability with 1 as follows:

P(A') = 1 - P(A)

Thus,

P(A') = 1 - P(A) = 1 - 0.956 = 0.044

Hence, the complement of A has a probability of 0.044.

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The circumference of the shaded face is 62.8 mm while the width of the rectangle is 62.8 mm

An equation is an expression that can be used to show the relationship between two or more numbers and variables using mathematical operators.

The area of a figure is the amount of space it occupies in its two dimensional state.

The radius (r) of the circle is 10mm. Hence:

Circumference = 2π * radius = 2π * 10 = 62.8 mm

The width of the rectangle = Circumference of circle = 62.8 mm

The circumference of the shaded face is 62.8 mm

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Answer:

2(3w+4)

Step-by-step explanation:

Given:

(W) is a variable with an unknown number

you are multiplying 5 by (w) variable

you are adding the 8 to (w) and 5w

1. Combine like terms:

8+w+5w

8+6w

2. Rearrange terms:

8+6w

6w+8

3. Common factor:

6w+8

2(3w+4)

Answer:

96/120 = 4/5

4/5 × 425 = 340 students

A sequence is a collection of numbers in which a pattern between consecutive numbers is established.

One example is a geometric sequence with first term of 2 and common ratio of 3, which is:

2, 6, 18, 54, ...

A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed number called the common ratio q.

Considering a geometric sequence with first term of 2 and common ratio of 3, the other terms are given as follows:

And the pattern continues for however many terms are there in the sequence.

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The width of rectangle Y is 8 units.

The width of rectangle Y is obtained applying the proportions in the context of the problem.

The dimensions of rectangle X are given as follows:

Height of 3 units.

Width of 2 units.

The dimensions of rectangle Y are given as follows:

Height of 12 units.

Width of x units.

The height was enlarged by a scale factor of 4, hence the width is given as follows:

2 * 4 = 8 units.

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Complete Question

Check attached image

Answer:

[tex]2[/tex] × [tex]5^{2}[/tex]

Step-by-step explanation:

2 X 5 X 5 = 50

Hope this helps

Answer:

see explanation

Step-by-step explanation:

(a)

x² - 11x + 24

consider the factors of the constant term (+ 24) which sum to give the coefficient of the x- term (- 11)

the factors are - 3 and - 8 , since

- 3 × - 8 = + 24 and - 3 - 8 = - 11, then

x² - 11x + 24 = (x - 3)(x - 8) ← in factored form

(b)

([tex]x^{4}[/tex] - 8x²y² + 16[tex]y^{4}[/tex] ) - 289 ← is a difference of squares and factors in general as

a² - b² = (a - b)(a + b)

= (x² - 4y² )² - 17²

= (x² - 4y² - 17)(x² - 4y² + 17) ← in factored form

The probability that for one such phone number is approximately 0.344

There are 10 possible digits (0-9) for each of the last four digits of the telephone number. Therefore, there are [tex]10^4[/tex]= 10,000 possible telephone numbers that can be generated using this method.

The probability that the last four digits do not include any 0s is:

P(no 0s) = [tex](9/10)^4[/tex] = 0.6561

So, the probability that the last four digits include at least one 0 is:

P(at least one 0) = 1 - P(no 0s) = 1 - 0.6561 = 0.3439

Therefore, the probability that for one such phone number, the last four digits include at least one 0 is approximately 0.344 (rounded to three decimal places).

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[tex]\textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \hspace{5em}\stackrel{center}{(\underset{}{h}~~,~~\underset{}{k})}\qquad \stackrel{radius}{\underset{}{r}} \\\\[-0.35em] ~\dotfill\\\\ (x+5)^2+(y-3)^2=4^2\implies ( ~~ x-(\stackrel{ h }{-5}) ~~ )^2+(y-\stackrel{ k }{3})^2=\stackrel{ r }{4^2}\qquad \begin{cases} \stackrel{ center }{(-5,3)}\\ \stackrel{radius}{4} \end{cases}[/tex]

The solution to the transversal problem is determined using the alternate exterior angles theorem, therefore, x = 15.

The diagram shows two given alternate exterior angles, to solve this transversal problem, apply the alternate exterior angles theorem, which states that alternate exterior angles are congruent.

Thus, we have:

6x + 8 = 7x - 7

Combine like terms:

6x - 7x = -8 - 7

-x = -15

x = 15

The value of x is: 15.

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Answer: you travel 80 miles in 2 hours while driving at a speed of 40 miles per hour.

Answer: 80 mi/hr

Step-by-step explanation: If you are traveling at a speed of 40 mi/hr for 2 hours, you need to do 40x2 which is equal to 80.

The approximate probability that John saves less than $400 is 0.166 or 16.6%.

First, we need to determine the mean and standard deviation of the amount of money John receives each day:

Mean = (0.50 * $5) + (0.35 * $10) + (0.15 * $20) = $6.25

Standard deviation = sqrt[(0.50 * ($5 - $6.25)^2) + (0.35 * ($10 - $6.25)^2) + (0.15 * ($20 - $6.25)^2)] = $5.54

Next, we need to calculate the mean and standard deviation of the amount of money John saves each day:

Mean savings = $6.25 - $4 = $2.25

Standard deviation of savings = $0.75

To calculate the total amount of money John saves over a certain period of time, we need to use the following formula:

Total savings = (Number of days) * (Mean savings)

We can estimate the probability distribution of John's total savings using the Central Limit Theorem, which states that the sum of a large number of independent, identically distributed random variables approaches a normal distribution.

Since we don't know the exact number of days, we can assume it is large enough for the Central Limit Theorem to apply.

Using the formula for the z-score, we can calculate the z-score for a total savings of $400:

z = (400 - (Number of days) * (Mean savings)) / (Standard deviation of savings * sqrt(Number of days))

To simplify this calculation, we can use a continuity correction and assume that John saves between $399.50 and $400.50:

z = (399.5 - (Number of days) * (Mean savings)) / (Standard deviation of savings * sqrt(Number of days))

z = (400.5 - (Number of days) * (Mean savings)) / (Standard deviation of savings * sqrt(Number of days))

We can use a standard normal distribution table or calculator to find the probability that z is less than or equal to the calculated z-score.

The resulting probability is approximately 0.166 or 16.6%, which is the approximate probability that John saves less than $400 over the given period of time.

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The shape of the distribution of the box plot is described as: B: It is Skewed

The box plot shape is used to indicate if a statistical particular set is either normally distributed or skewed.

There is a property of the box plot i.e. When the median is in the center of the box, and the whiskers are about the same on both flanks of the box, then the distribution is symmetric. However, when the median is anywhere to the box except the center, and if the whisker is more concise on the left or right end of the box, then the distribution is skewed.

In this question, we can clearly see that  the whisker is more concise on the right end of the box, and as such we can say that the distribution is skewed.

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The number of numbers that Bill wrote (v-8) numbers.

How to form inequalities?

A mathematical phrase that states the order relationship between two integers or algebraic expressions as greater than, greater than or equal to, less than, or less than or equal to.

If Bill wrote all the natural numbers between 9 and v, then the number of numbers he wrote would be equal to the difference between v and 9 plus one (since he included both 9 and v).

v > 9

(where v is greater than 9 i.e., (v-9)+1. )

=(v-9) + 1

= v - 9 + 1

= v - 8

Therefore, the number of numbers he wrote would be (v - 8) numbers.

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The population of the country in 2003 was 979 million.

We are given that;

a=979e 0.0008t

Now,

To find the population of the country in 2003, we need to plug in t = 0 into the model, since 2003 is the starting year.

a = 979e^(0.0008t)

a = 979e^(0.0008(0))

a = 979e^0

a = 979(1)

a = 979

Therefore, by the exponential mode the answer will be 979 million.

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The population of the country in 2003 was 979 million.

We are given that;

a=979e 0.0008t

Now,

To find the population of the country in 2003, we need to plug in t = 0 into the model, since 2003 is the starting year.

a = 979e^(0.0008t)

a = 979e^(0.0008(0))

a = 979e^0

a = 979(1)

a = 979

Therefore, by the exponential mode the answer will be 979 million.

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Answer:

There are two possible coordinates for A:  Either A (-5, 10) or A (-5, -6)

Step-by-step explanation:

To find the possible coordinates of A, we will need to find the possible y-coordinate.

We can do this using the distance (d) formula, which is

[tex]d=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }[/tex], where (x1, x2) are one set of coordinates and (y1, y2) are the other set of coordinates

We can allow point B (1, 2) to be our x1 and x2 coordinates and A (-5, y2) to be our x2 and y2.  Thus, we must plug into the formula 10 for d and solve for y2:

[tex]10=\sqrt{(-5-1)^2+(y_{2}-2)^2}\\ 10=\sqrt{(-6)^2+(y_{2}-2)^2}\\ 10=\sqrt{36+(y_{2}-2)^2}\\ 100=36+(y_{2}-2)^2\\64=(y_{2}-2)^2\\[/tex]

To finish solving, we must take the square root of both sides.  Whenever you take a square root, there is both a positive answer and a negative answer, since squaring a negative number also yields a positive number (e.g., 5 * 5 = 25 and -5 * -5 = 25):

Positive answer:

[tex]8=y_{2}-2\\10=y_{2}[/tex]

Negative answer:

[tex]-8=y_{2}-2\\-6=y_{2}[/tex]

To check our answers, we can plug in both 10 for y2 into the formula and -6 for y2 into the formula and check that we get 10 each time:

Plugging in 10 for y2:

[tex]10=\sqrt{(-5-1)^2+(10-2)^2}\\ 10=\sqrt{(-6)^2+(8)^2}\\ 10=\sqrt{36+64}\\ 10=\sqrt{100}\\ 10=10[/tex]

Plugging in -6 for y2:

[tex]10=\sqrt{(-5-1)^2+(-6-2)^2}\\ 10=\sqrt{(-6)^2+(-8)^2}\\ 10=\sqrt{36+64}\\ 10=\sqrt{100}\\ 10=10[/tex]

Thus, the two possible coordinates of A are (-5, 10), where 10 is the y-coordinate or (-5, -6), where - 6 is the y-coordinate.

A. z < -0.65. 25.78% of observations are less than -0.65. B. z > -0.65. 74.22% of observations are greater than -0.65. C. z < 1.32. 90.66% of observations are less than 1.32. D. For z = -0.65, It represents the proportion of observations that have a z-score of less than -0.65.

A. To find the proportion of observations from a standard normal distribution that satisfies the statement z < -0.65, we can use a standard normal table or calculator to find the area under the curve to the left of -0.65. This area is equal to approximately 0.2578, or 0.2579 when rounded to four decimal places.

B. To find the proportion of observations that satisfy the statement z > -0.65, we can find the area under the curve to the right of -0.65. This is equal to 1 - P(z < -0.65), or 1 - 0.2578, which equals approximately 0.7422, or 0.7421 when rounded to four decimal places.

C. To find the proportion of observations that satisfy the statement z < 1.32, we can find the area under the curve to the left of 1.32. This is equal to approximately 0.9066, or 0.9065 when rounded to four decimal places.

D. The statement "-0.65" is not actually a statement, so there is no proportion of observations to calculate. If this was meant to be a typo and the statement was meant to be "z = -0.65", then the proportion of observations that satisfy this statement would be extremely small, as the probability of getting a single specific value from a continuous distribution is zero.


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