Rewrite each expression using only positive exponents. Need this as soon as possible please :)

The origin traversed counterclockwise is ∮c xdx + ydy / x² + y² = 2πi

This is a classic example of a line integral in complex analysis.

To evaluate this integral, we need to use the Cauchy Integral Formula, which states that if f(z) is analytic inside and on a simple closed contour C, then:

∮C f(z) dz = 2πi Res(f, z)

Res(f, z) denotes the residue of f at z.

In this case, we have f(z) = x + iy / x² + y², and we want to integrate over a Jordan curve C that encloses the origin.

Since f(z) is analytic everywhere except at z = 0, we can apply the Cauchy Integral Formula to compute the value of the integral.

To do so, we need to find the residue of f(z) at z = 0.

We can do this by computing the Laurent series expansion of f(z) around z = 0:

f(z) = (x + iy) / (x² + y²) = (1 / z) [(x / z) + (iy / z)] = (1 / z) [1 - (1 / 2) z² + ...]

The coefficient of the z⁻¹ term is 1, which means that the residue of f(z) at z = 0 is 1.

The Cauchy Integral Formula to evaluate the integral:

∮C xdx + ydy / x² + y² = 2πi Res(f, z) = 2πi

The value of the integral is 2πi.

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The value of the line integral is zero for any Jordan curve c whose interior does not contain the origin, traversed counterclockwise.

This integral can be evaluated using Green's Theorem, which states that the line integral of a vector field around a simple closed curve is equal to the double integral of the curl of the vector field over the region enclosed by the curve.

Let F(x, y) = (x/(x^2 + y^2), y/(x^2 + y^2)) be the vector field in question. Then the curl of F is given by:

curl(F) = (∂y/∂x - ∂x/∂y) = (0 - 0)i - (0 - 0)j + (x^2 + y^2)^(-2) (1 - 1)k = 0i + 0j + 0k

Since the curl of F is zero, we know that F is a conservative vector field, which implies that the line integral of F around any closed curve is zero.

Therefore, we have:

∮c xd + yd / ^2 + ^2 = ∮c F · dr = 0

where the last step follows from the fact that F is conservative.

Hence, the value of the line integral is zero for any Jordan curve c whose interior does not contain the origin, traversed counterclockwise.

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The statistical values of the data given are :

Given the data : 6, 6, 7, 10, 10, 10, 11, 13, 13, 16, 16, 18, 18, 18 6,6,7,10,10,10,11,13,13,16,16,18,18,18

The statistical values in the data can be calculated thus:

Sort values in a sending order : 6,6,6,6,7,7,10,10,10,10,10,10,11,11,13,13,13,13,16,16,16,16,18,18,18,18,18,18

Minimum = 6 (least value)

Maximum= 18 (highest value)

Median = (N+1)/2 th term

Median = (11 + 13)/2 = 12

First quartile: 1/4(N+1)th term

First quartile = 10

Third quartile = 3/4(N+1)th term

Third quartile = 16

Interquartile Range: (Third Quartile - First quartile)

Interquartile range = 16-10 = 6

Therefore, the statistical values of a box and whisker plot are those calculated above .

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The surface area of the given pyramid, can be found to be A. 648 square units.

First find the area of the square base :

= 12 x 12

= 144 square units

Then find the area of a single triangular face of the regular pyramid :

= 1 / 2 x base  x height

= 1 / 2 x 12 x 21

= 126 square units

Seeing as there are 4 triangular faces, the total area would then be:

= 144 + ( 126 x 4 triangular faces )

= 648 square units

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The statement that best describes the size of the cross section is C. The height of the cross section is the same as the height of the prism, and the width of the cross section is the same as the width of the faces of the prism.

If a triangular prism is cut at a right angle to its base, the resulting section will resemble and have the same dimensions as the original triangular base of the prism.

The altitude of the cross-sectional triangle will equal the altitude of the triangular base of the prism. In a similar manner, the width of the triangular shape (its cross-section) will match the width of the base of the prism.

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(a) The standard deviation of the sample mean is 1.94 (rounded to two decimal places).

(b) How large should the sample be to achieve a halved standard deviation of the sample mean?

To find the standard deviation of the sample mean (also known as the standard error), we divide the population standard deviation by the square root of the sample size. Given that the population has a variance of 25, the standard deviation is √25 = 5. Since we are working with a sample size of 54, we divide the population standard deviation by the square root of 54 to obtain the standard deviation of the sample mean, which is approximately 1.94 when rounded to two decimal places.

To halve the standard deviation of the sample mean, we need to increase the sample size. The standard deviation of the sample mean decreases as the square root of the sample size increases. In other words, if we want to halve the standard deviation, we need to quadruple the sample size. Therefore, the sample size should be increased to 216 (54 * 4) in order to achieve this reduction.

In conclusion, the standard deviation of the sample mean for a random sample of size 54 is approximately 1.94. To halve the standard deviation, the sample size should be increased to 216.

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, f'(x) is increasing on the intervals (-infinity, -2sqrt(14)) and (2sqrt(14), infinity), and decreasing on the interval (-2sqrt(14), 2sqrt(14)).

To find the intervals on which f'(x) is increasing or decreasing, we need to first find the critical points of f(x), i.e., the values of x where f'(x) = 0 or where f'(x) does not exist. Then, we can use the first derivative test to determine the intervals of increase and decrease.

We have:

f'(x) = x^2 - 56

Setting f'(x) = 0, we get:

x^2 - 56 = 0

Solving for x, we obtain:

x = ±sqrt(56) = ±2sqrt(14)

So, the critical points of f(x) are x = -2sqrt(14) and x = 2sqrt(14).

Now, we can use the first derivative test to find the intervals of increase and decrease. We construct a sign chart for f'(x) as follows:

|       -    2sqrt(14)   +    2sqrt(14)   +

f'(x) | - 0 + 0 +

From the sign chart, we see that f'(x) is negative on the interval (-infinity, -2sqrt(14)), and positive on the interval (-2sqrt(14), 2sqrt(14)) and (2sqrt(14), infinity).

Therefore, f'(x) is increasing on the intervals (-infinity, -2sqrt(14)) and (2sqrt(14), infinity), and decreasing on the interval (-2sqrt(14), 2sqrt(14)).

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Here is the rule

y = (x - h)² + k

Vertex = (h, k)

The vertex represents the lowest point on the graph, or the minimum value of the quadratic function.

y= (x - 1)² + 2

Vertex = (1, 2)

Second graph or the middle picture one

The half-angle x/2 lies in the range 90° < x/2 < 135° (Quadrant II) and has a sine value of √(25/34).

Based on the given information, cos(x) = -8/17, and the angle x lies in the range 180° < x < 270°, which places it in Quadrant III. In this quadrant, cosine is negative, which confirms the value of cos(x). Now, we need to find the half-angle (x/2) and determine its range.

Since x is in Quadrant III, the angle x/2 will lie in the range 90° < x/2 < 135°, placing it in Quadrant II. In this quadrant, sine and cosine have opposite signs, so while cos(x) is negative, sin(x/2) will be positive. To find the value of sin(x/2), we can use the half-angle identity:

sin(x/2) = ±√[(1 - cos(x))/2] = √[(1 - (-8/17))/2] = √(25/34)

Since x/2 is in Quadrant II, sin(x/2) must be positive, so we take the positive square root:

sin(x/2) = √(25/34)

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Based on the Year 2 balance sheet, the largest cash dividend that Dakota could pay is $16,500.

Cash dividends refers to the payments that companies make to their shareholders which is usually on the strength of earnings. They often represent opportunity for companies to share the benefit of business profits.

Based on the balance sheet, the largest cash dividend that Dakota could pay in Year 2 is:

= $ 31,500 + $ 5,000 - $ 20,000

= $ 16,500.

Missing questions:Dakota Company experienced the following events during Year 2:

Acquired $20,000 cash from the issue of common stock.

Paid $20,000 cash to purchase land.

Borrowed $2,500 cash.

Provided services for $40,000 cash.

Paid $1,000 cash for utilities expense.

Paid $20,000 cash for other operating expenses.

Paid a $5,000 cash dividend to the stockholders.

Determined that the market value of the land purchased in Event 2 is now $25,000.

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The required radius of the water bottle is approximately 2.88 cm.

To find the radius of the cylindrical water bottle, we can use the formula for the volume of a cylinder:

Volume = π * radius² * height

Given that the volume of the bottle is 207 cm³ and the height is 7.9 cm, we can rearrange the formula to solve for the radius:

207 = 3.14 * radius² * 7.9

Dividing both sides of the equation by (3.14 * 7.9), we get:

radius² = 207 / (3.14 * 7.9)

radius² = 8.308

radius = √(8.308)

radius ≈ 2.88 cm (rounded to two decimal places)

Therefore, the radius of the water bottle is approximately 2.88 cm.

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We are given that the equation

x^2 y'' - 2xy'^2 y = 0

has a solution y = x, which satisfies the homogeneous equation. To find the general solution of the nonhomogeneous equation

x^2 y'' - 2xy'^2 y = x^2,

we can use the method of undetermined coefficients.

Assume a particular solution of the form y_p(x) = Ax^2 + Bx. Then, we have

y_p'(x) = 2Ax + B,

y_p''(x) = 2A.

Substituting these into the nonhomogeneous equation, we get

x^2 (2A) - 2x(2Ax + B)^2 (Ax^2 + Bx) = x^2.

Simplifying and collecting terms, we get

2A - 2B^2 = 1.

We can choose A = 1/2 and B = -1/2 to satisfy this equation. Therefore, a particular solution of the nonhomogeneous equation is

y_p(x) = (1/2)x^2 - (1/2)x.

The general solution of the nonhomogeneous equation is then

y(x) = c1 x + c2 - (1/2)x + (1/2)x^2,

where c1 and c2 are constants determined by the initial or boundary conditions of the problem.

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All values of b that prove Stella's claim is not correct by making the rate of change of function B greater than the rate of change of function A are:

D. 15

E. 17

In Mathematics and Geometry, the rate of change (slope) of any straight line can be determined by using this mathematical equation;

Rate of change = (Change in y-axis, Δy)/(Change in x-axis, Δx)

Rate of change = rise/run

Rate of change = (y₂ - y₁)/(x₂ - x₁)

When b = 6, the rate of change of function B is given by:

Rate of change = (6 - 3)/(4 - 1)

Rate of change = 3/3

Rate of change = 1 (not greater than 3).

When b = 12, the rate of change of function B is given by:

Rate of change = (12 - 3)/(4 - 1)

Rate of change = 9/3

Rate of change = 3 (not greater than 3).

When b = 15, the rate of change of function B is given by:

Rate of change = (15 - 3)/(4 - 1)

Rate of change = 12/3

Rate of change = 4 (greater than 3).

When b = 15, the rate of change of function B is given by:

Rate of change = (17 - 3)/(4 - 1)

Rate of change = 14/3

Rate of change = 4.7 (greater than 3).

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Missing information:

Select all values of b that prove Stella's claim is not correct by making the rate of change of function B greater than the rate of change of function A.

6

8

12

15

17

The critical t-values are approximately ±4.032.

To determine the statistically significant values of the t statistic at the α = 0.001 level, with a sample size of 6 and a two-tailed test, refer to a t-distribution table.

To test H0: μ = 0 against Ha: μ ≠ 0 with an SRS of 6 observations from a normal population, follow these steps:

1. Determine the degrees of freedom (df): Since n = 6, the df = n - 1 = 5.
2. Identify the significance level (α): In this case, α = 0.001.
3. Determine the type of test: As Ha: μ ≠ 0, this is a two-tailed test.
4. Refer to a t-distribution table: Look up the critical t-values for a two-tailed test with df = 5 and α = 0.001.
5. Find the critical t-values: The table will show that the critical t-values are approximately ±4.032.

Therefore, t statistic values less than -4.032 or greater than 4.032 are statistically significant at the α = 0.001 level.

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Asymptotes are typically found in rational functions (ratios of polynomials) or logarithmic or exponential functions.

An asymptote is a straight line that a curve approaches but never touches.

In other words, it is a line that the function gets closer and closer to but never actually intersects.
Now, let's take a look at the function f(x) = [tex]x^2[/tex]/x.

This function can be simplified to f(x) = x.

Therefore,

The graph of this function is simply a straight line passing through the origin with a slope of 1.
Since the graph of this function is a straight line, it does not have any vertical or horizontal asymptotes.

However, it is worth noting that the function has a hole at x = 0, since the value of the function is undefined at that point.
In summary, the equation of each asymptote for the function f(x) = x^2/x is as follows:
- There is no vertical asymptote.
- There is no horizontal asymptote.

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The equations of the asymptotes for the given function are:

Vertical asymptotes: x1 = 0 and x = 0

Horizontal asymptotes: None

Slant asymptote: y = x.

Since the given function is a rational function, it may have vertical, horizontal, and slant asymptotes.

Vertical Asymptotes:

The vertical asymptotes occur at those values of x where the denominator of the function becomes zero. Here, the denominator is x1x, so the vertical asymptotes are given by x1 = 0 and x = 0.

Horizontal Asymptotes:

To find the horizontal asymptotes, we can compare the degrees of the numerator and the denominator of the function. Here, the degree of the numerator is 2 and the degree of the denominator is 3, so there is no horizontal asymptote.

Slant Asymptotes:

Since the degree of the numerator is less than the degree of the denominator by exactly 1, we can find the slant asymptote by long division of the numerator by the denominator. Performing long division, we get:

x2 x1 x = (x + 0)x1 x - 0

So the slant asymptote is y = x.

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Answer:

2160/13.

Step-by-step explanation:

Exterior angle = 360/number of sides

Interior angle = 180 – exterior angle

exterior angle = 360/26

= 180/13.

interior angle = 180 - (180/13)

= 2160/13.

False, the number of true arithmetical statements involving positive integers, +, x,(,) and = is countable, i.e. "(17+31) x 2 = 96".

The number of true arithmetical statements involving positive integers, +, x,(,) and = is uncountable. There are infinitely many true arithmetical statements involving positive integers and the other specified symbols. For any given set of positive integers, there are infinitely many arithmetic statements that can be formed using those integers and the symbols. Additionally, there are infinitely many possible sets of positive integers that could be used to form arithmetic statements. Therefore, the total number of true arithmetical statements involving positive integers, +, x,(,) and = is uncountable. It's worth noting that the set of possible arithmetical statements involving positive integers, +, x,(,) and = is a subset of the set of all possible mathematical statements involving those symbols, which is itself uncountable.

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The child who received more land if the land was divided evenly among them compared to the way the mother divided it is Sofia. Option (d) is correct.

A mother divided her land of 141/4 acres among her 5 adopted adult children such that Samuel received 11 acres, Sophia received 9 acres, Anthony received 8 acres, Jasmine received 5 acres, and Juan received 4 1/4 acres.

Among the given children, the child who received more land if the land was divided evenly among them compared to the way the mother divided it is Sofia.

To determine the land received by each child if the land was divided evenly among them, divide the total area by the number of children.

Therefore, each child would receive 141/4 ÷ 5 acres.

141/4 ÷ 5 = 113/20 ≈ 5.65 acres.

(to two decimal places)

Approximately, each child would receive 5.65 acres of land.

Which is not how the mother divided her land among the children.

Now, let's compare the area of land that each child received with the 5.65 acres they would have received if the land was divided evenly among them.Samuel received 11 acres.

11 > 5.65

So, Samuel received more land than they would have received if the land was divided evenly among them.Sophia received 9 acres.

9 < 5.65

So, Sophia received less land than they would have received if the land was divided evenly among them.

Anthony received 8 acres.

8 < 5.65

So,

Anthony received less land than they would have received if the land was divided evenly among them.

Jasmine received 5 acres.

5 = 5.65

So, Jasmine received the same land as they would have received if the land was divided evenly among them.

Juan received 4 1/4 acres.

21/4 = 5.25 and

5.25 < 5.65

So, Juan received less land than they would have received if the land was divided evenly among them.

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The mean number of boxes ordered per month is approximately 29.75 boxes.

Mean number of boxes = (Total number of boxes ordered in the first three months + Total number of boxes ordered for the rest of the year) / Total number of months

The total number of boxes ordered in the first three months would be 35 boxes/month * 3 months = 105 boxes.

For the rest of the year, the number of boxes ordered is reduced to 28 per month. Since there are 12 months in a year, the total number of boxes ordered for the rest of the year would be 28 boxes/month * 9 months = 252 boxes.

Mean number of boxes = (105 boxes + 252 boxes) / 12 months

Mean number of boxes = 357 boxes / 12 months

Mean number of boxes = 29.75 boxes/month

Therefore, the mean number of boxes ordered per month is approximately 29.75 boxes.

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To express 1 as a power with the base 6, we can use logarithms.

We have the equation:

[tex]36 = 6^2[/tex]

Taking the logarithm base 6 of both sides:

[tex]\log_6(36) = \log_6(6^2)[/tex]

Applying the logarithmic property, we can bring down the exponent:

[tex]\log_6(36) = 2\log_6(6)[/tex]

Since [tex]\log_b(b) = 1[/tex], where b is the base of the logarithm, we have:

[tex]\log_6(36) = 2 \times 1[/tex]

Simplifying the expression:

[tex]\log_6(36) = 2[/tex]

Therefore, 1 expressed as a power with the base 6 is [tex]6^0[/tex].

After enlarging triangle ABC with a scale factor of 3 about the origin (0, 0), the coordinates of A' and B' are (12, 0).

To find the coordinates of A' and B' after the enlargement, we can use the formula for enlarging a point (x, y) by a scale factor of k about a center point (h, k):

A' = (k * (A - O)) + O

B' = (k * (B - O)) + O

Given that AB = 4 cm and the scale factor is 3, we can assume that point O is the origin (0, 0).

Let's calculate A'

A' = (3 * (A - O)) + O

= 3 * (A - O) + O

= 3 * (4, 0) + (0, 0)

= (12, 0) + (0, 0)

= (12, 0)

Therefore, A' has the coordinates (12, 0).

Now let's calculate B'

B' = (3 * (B - O)) + O

= 3 * (B - O) + O

= 3 * (4, 0) + (0, 0)

= (12, 0) + (0, 0)

= (12, 0)

Therefore, B' also has the coordinates (12, 0).

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--The given question is incomplete, the complete question is given below " A triangle ABC is enlarged, about the point O as centre of enlargement, and the scale factor is 3. Find A'B', if AB = 4cm."--

The limaçon has a characteristic loop is evident in the Cartesian equation. e.

The given equation is in polar form and it represents a limaçon.

We can convert this equation into Cartesian form using the following equations:

x = r cosΘ()

y = r sin(Θ)

Substituting r = 3 cos(Θ) get:

x = 3 cos(Θ) cos(Θ)

= 3 cos²(Θ)

y = 3 cos(Θ) sin(Θ)

= 3 sin(Θ) cos(Θ)

= (3/2) sin(2Θ)

The Cartesian equation of the curve is:

x²/9 + y²/27 = cos²(Θ) + (1/4)sin²(2Θ)

This equation represents a limaçon is a type of curve formed by a point moving around a fixed point while a second point moves around the first.

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The power booster can be operated by engine vacuum or through hydraulic pressure, which is usually generated by the power steering pump or an electric-driven pump.

Therefore, the terms "hydraulic pressure" and "power steering pump" are relevant to the operation of the power booster.The power booster, also known as the brake booster, is a device that helps in applying more force to the brakes with less pressure on the brake pedal. This results in an enhanced braking performance. The power booster can be operated using either of two methods:

Engine vacuum, or Hydraulic pressure, which is produced by the power steering pump or an electric-driven pump.

In both methods, the power booster serves to augment the force that is applied to the brake master cylinder.

This increases the hydraulic pressure that is applied to the brakes, resulting in an enhanced braking performance.

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The value of (f ₀ f⁻¹) (5) of the function f(x) = x + 9 is 5.

Given is a function f(x) = x + 9,

Setting x + 9 = y and making x the subject we have

y = x + 9

x = y - 9

replacing x with f⁻¹(x) and y with x we have

f⁻¹(x) = x - 9

so, the (f ₀ f⁻¹) (x) can be gotten by substituting x in f(x) with x - 9 so that we have,

(f ₀ f⁻¹) (x) = (x - 9) + 9

(f ₀ f⁻¹) (x) = x

Therefore,

(f ₀ f⁻¹) (5) = 5

Hence the value of (f ₀ f⁻¹) (5) of the function f(x) = x + 9 is 5.

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From the given equation the series diverge is  iii only. The correct answer is B.

First, note that the series in option I is not an alternating series, so we cannot apply the Alternating Series Test to check for convergence.

For option II, we can use the Limit Comparison Test. We compare it to the harmonic series, which is known to diverge:

lim(n→∞) (1 + 1/n) / (1/n) = lim(n→∞) (n + 1) / n = 1

Since the limit is positive and finite, the series in option II diverges.

For option III, we can use the Divergence Test, which states that if the limit of the terms of the series does not approach zero, then the series must diverge.

lim(n→∞) (n + 1/n^2) = ∞

Since the limit is infinite, the series in option III also diverges.

Therefore, the answer is (B) iii only.

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the eigenvalues of A are λ = 2 and μ = -2/3, with algebraic multiplicities 1 and 2, respectively.

We know that the trace of a matrix is the sum of its eigenvalues and the determinant is the product of its eigenvalues. Let the two distinct eigenvalues of A be λ and μ. Then, we have:

tr(A) = λ + μ + λ or μ (since the eigenvalues are distinct)

-3 = 2λ + μ ...(1)

det(A) = λμ(λ + μ)

-28 = λμ(λ + μ) ...(2)

We can solve this system of equations to find λ and μ.

From equation (1), we can write μ = -3 - 2λ. Substituting this into equation (2), we get:

-28 = λ(-3 - 2λ)(λ - 3)

-28 = -λ(2λ^2 - 9λ + 9)

2λ^3 - 9λ^2 + 9λ - 28 = 0

We can use polynomial long division or synthetic division to find that λ = 2 and λ = -2/3 are roots of this polynomial. Therefore, the eigenvalues of A are 2 and -2/3, and their algebraic multiplicities can be found by considering the dimensions of the eigenspaces.

Let's find the algebraic multiplicity of λ = 2. Since tr(A) = -3, we know that the sum of the eigenvalues is -3, which means that the other eigenvalue must be -5. We can find the eigenvector corresponding to λ = 2 by solving the system of equations (A - 2I)x = 0, where I is the 3 x 3 identity matrix. This gives:

|1-2 2 1| |x1| |0|

|2 1-2 1| |x2| = |0|

|1 1 1-2| |x3| |0|

Solving this system, we get x1 = -x2 - x3, which means that the eigenspace corresponding to λ = 2 is one-dimensional. Therefore, the algebraic multiplicity of λ = 2 is 1.

Similarly, we can find the algebraic multiplicity of λ = -2/3 by considering the eigenvector corresponding to μ = -3 - 2λ = 4/3. This gives:

|-1/3 2 1| |x1| |0|

| 2 -5/3 1| |x2| = |0|

| 1 1 5/3| |x3| |0|

Solving this system, we get x1 = -7x2/6 - x3/6, which means that the eigenspace corresponding to λ = -2/3 is two-dimensional. Therefore, the algebraic multiplicity of λ = -2/3 is 2.

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1. False   2. False    3. False

4. The distance between the shark and the bird is |1834 – 145| = 12634 feet.  False

To determine the truth value of each statement, we need to calculate the absolute differences between the given coordinates.

1: The distance between the fish and the dolphin is |–3812 – (–8414)| = |3812 + 8414| = 12226 feet.

Since the calculated distance is 12226 feet, the statement "The distance between the fish and the dolphin is 4534 feet" is false.

2: The distance between the shark and the dolphin is |–145 – 8414| = |-145 - 8414| = 8559 feet.

Since the calculated distance is 8559 feet, the statement "The distance between the shark and the dolphin is 22934 feet" is false.

3: The distance between the fish and the bird is |1834 – (–3812)| = |1834 + 3812| = 5646 feet.

Since the calculated distance is 5646 feet, the statement "The distance between the fish and the bird is 5714 feet" is false.

4: The distance between the shark and the bird is |1834 – 145| = |1834 - 145| = 1689 feet.

Since the calculated distance is 1689 feet, the statement "The distance between the shark and the bird is 12634 feet" is false.

Therefore:

False

False

False

False

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Number of significant digits is -169.

In the given expression, 320.72 is subtracted by the result of dividing 6109.7 by 3.4.

To express the answer with the appropriate number of significant digits, we must consider the significant digits in each component of the calculation.

Starting with the division, 6109.7 divided by 3.4 yields approximately 1799.9117647058824.

Since 3.4 has two significant digits, the division result should also have two significant digits. Therefore, we round it to 1800.

Subtracting 1800 from 320.72 gives us -1479.28. However, since 320.72 has five significant digits, the final answer should have the same number of significant digits.

Rounding to the appropriate number of significant digits, we obtain -169.

Therefore, the result of the given expression, rounded to the appropriate number of significant digits, is -169.

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14 vertices in this polyhedron.

Euler's formula, which states that for any polyhedron (a three-dimensional solid object with flat polygonal faces), the number of vertices (V), edges (E), and faces (F) are related by the equation:

V - E + F = 2

This formula is named after the mathematician Leonhard Euler, who first discovered it in the 18th century.

It's a fundamental result in geometry and has many important applications.

The polyhedron has 9 faces and 21 edges.

We can plug these values into Euler's formula and solve for the number of vertices:

V - 21 + 9 = 2

Simplifying the left-hand side, we get:

V - 12 = 2

Adding 12 to both sides, we get:

V = 14

So the polyhedron has 14 vertices.

Euler's formula is a powerful tool for analyzing polyhedra, and can be used to derive many other results in geometry.

It's also related to other important mathematical concepts, such as topology and graph theory.

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The length on the blueprint is approximately 0.2667 units.

To find the length on the blueprint, we can set up a proportion using the given information.

Let's denote the length on the blueprint as "x".

According to the blueprint, the width is 4 inches, and the actual length is 20 feet. We can set up the following proportion:

Width on Blueprint / Actual Width = Length on Blueprint / Actual Length

Plugging in the values:

4 inches / 20 feet = x / 16 feet

Now, we need to convert the units to be consistent. Since we have feet in the denominator on both sides, we can convert inches to feet by dividing by 12:

(4 inches / 12 feet) / 20 feet = x / 16 feet

Simplifying:

1/3 / 20 = x / 16

Now, we can cross multiply:

(1/3) × 16 = 20 ×x

Simplifying further:

16/3 = 20x

To solve for x, we can divide both sides by 20:

(16/3) / 20 = x

Simplifying:

16 / (3 × 20) = x

16 / 60 = x

Now, we can simplify the fraction:

4/15 = x

So, the length on the blueprint is 4/15 of a unit.

Alternatively, if you want to convert the fraction to decimal form, you can divide 4 by 15:

4 ÷ 15 ≈ 0.2667

Therefore, the length on the blueprint is approximately 0.2667 units.

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