Find the values mi
, i=1,2
, for which
y(x)=emix
is a solution of the differential equation
d2ydx2−12dydx+32y=0.

The statement can be linked to a theorem in geometry known as the "double angle rotation theorem". According to this theorem, if a geometric figure has symmetry about a straight line, and that line intersects another straight line at a point, then the symmetry of the figure about the second line is equivalent to a rotation of twice the angle from the first line to the second line, that is about the point of intersection.

In simple terms, the double angle rotation theorem means that if we can have a geometric figure that is symmetric about a line L1, and L2 is another line that intersects L1 at a point P, then we can rotate the figure about point P by twice the angle formed between L1 and L2 to get the same symmetry about the line L2.

The application of this theorem applies to various aspects of geometry that include transformational geometry and symmetry.

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a. The function that models the fox population t years after 2000 is:

[tex]p(t) = 12500 * e^{(0.05t)[/tex]

b. The estimated fox population in the year 2008 is 19121 (rounded to the nearest integer).

Amοng the crucial kinds are: When there is a mapping fοr a range fοr each dοmain between twο sets, this is knοwn as an injective functiοn οr οne tο οne functiοn. When mοre than οne element is transferred frοm dοmain tο range, the functiοn is said tο be surjective οr tο be an οntο functiοn

(a) Since the fox population has a continuous growth rate of 5% per year, the population can be modeled using the exponential growth formula:

[tex]p(t) = p(0) * e^{(rt)[/tex]

where p(0) is the initial population, r is the continuous growth rate (expressed as a decimal), t is the time elapsed in years, and e is the mathematical constant e ≈ 2.71828.

Using the given information, we have:

p(0) = 12500 (the population in the year 2000)

r = 0.05 (the continuous growth rate)

Therefore, the function that models the fox population t years after 2000 is:

[tex]p(t) = 12500 * e^{(0.05t)[/tex]

(b) To estimate the fox population in the year 2008, we need to find the value of p(8) using the function from part (a):

[tex]p(8) = 12500 * e^{(0.05*8)} = 19121[/tex]

Therefore, the estimated fox population in the year 2008 is 19121 (rounded to the nearest integer).

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The statement "a dilation about the origin by a scale factor of 2/3 followed by a rotation of 90° counterclockwise about the origin and a translation 5 units left" and "a dilation about the origin by a scale factor of 5/7 followed by a reflection over the y-axis and a dilation about the origin by a scale factor of 3/4" is correct of the question.

A polygon is a two-dimensional geometric shape that has three or more straight sides and angles. It is a closed shape, meaning that the sides connect to form a continuous boundary.

A polygon can have any number of sides, although polygons with three sides are called triangles, polygons with four sides are called quadrilaterals, and so on. The angles of a polygon are formed where two sides meet, and the sum of the interior angles of a polygon with n sides (n being greater than or equal to 3) is (n-2) times 180 degrees.

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Answer: See attached for a graph. 13 units apart.

Step-by-step explanation:

      See attached for a graph. We will plot the first point -8 units left and 7 units up. Then we will plot the second point 5 units right and 7 units up.

      Since these points are on a line (they both have the same y-value), we will add the absolute value of their x-values together. We can also count on the graph, you will get the same answer.

|-8| + |5| = 8 + 5 = 13

             They are 13 units apart.

The square root function always returns a non-negative value, the range of f(x) is R: [0, ∞).

It is the set of values of the variable for which the function has a valid output.

To find the domain and range of the function f(x) = [tex]sqrt(x^2 - 49)[/tex], we need to consider the possible values of x that make the function well-defined and the corresponding values of f(x) that the function can take.

The function f(x) is defined if the argument inside the square root is non-negative, that is, [tex]x^2 - 49[/tex] ≥ 0. This inequality can be solved as:

(x - 7)(x + 7) ≥ 0

The solution to this inequality is x ≤ -7 or x ≥ 7, so the domain of the function is D: (-∞, -7] ∪ [7, ∞).

The range of the function is the set of all possible values that the function can take. Since the square root function always returns a non-negative value, the range of f(x) is R: [0, ∞).

Therefore, the correct answer is (B) D: (-∞, -7] ∪ [7, ∞), R: [0, ∞).

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Answer:

x = 3 + √(22/3) and x = 3 - √(22/3).

Step-by-step explanation:

Step 1: Divide both sides by the coefficient of x^2 to get the coefficient of x^2 equal to 1.

3x^2 - 18x + 5 = 0

x^2 - 6x + 5/3 = 0

Step 2: Move the constant term to the right-hand side of the equation.

x^2 - 6x = -5/3

Step 3: Take half of the coefficient of x, square it, and add it to both sides of the equation.

x^2 - 6x + 9 = -5/3 + 9

(x - 3)^2 = 22/3

Step 4: Solve for x by taking the square root of both sides of the equation.

x - 3 = ±√(22/3)

x = 3 ± √(22/3)

It be concluded that figure A and figure C are showing similar triangle but they are not congruent.

If two triangles are similar then they have the same ratio of corresponding sides and equal pair of corresponding angles. If two or more figures have the same shape, but their sizes are different, then such objects are said to be similar figures.

Similarity of two triangles mean when two triangles satisfy the AAA property that is when three angles of one triangle is equivalent to angles with another triangle.

Here for the figure A and C they satisfy the AAA property.

Hence it be concluded that figure A and figure C are showing similar triangle but they are not congruent. The triangles have same shape but not in same size.

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Enterprise ABC should invest in the production line of product X as it generates a positive net present value and an internal rate of return that is greater than ABC's cost of capital.

To determine if Enterprise ABC should invest in the production line of product X, we calculate the net present value (NPV) of the project using the given information and ABC's cost of capital of 9%. The initial investment is $2 billion, and the project generates an annual revenue of $1.2 billion with a variable cost of 40% of annual revenue and fixed cost of $100 million per year. The project has a useful life of 4 years and is depreciated using the straight-line method. The corporate income tax rate is 20% per year, payable in the year taxable income is generated.

Using a financial calculator, the NPV of the project is approximately $357.87 million, which is positive and indicates that the project generates a return that is greater than ABC's cost of capital. Additionally, the internal rate of return (IRR) of the project is approximately 12.21%, which is greater than ABC's cost of capital of 9%. Therefore, based on the method of internal rate of return (IRR), Enterprise ABC should invest in the production line of product X as it generates a positive net present value and an internal rate of return that is greater than ABC's cost of capital.

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The value of the variable in the circle is x = 4.5.

If two secant segments shares an endpoint outside of the circle. The product of one secant segment and its external segment is equal to the product of the other secant segment and its external segment.

In this case, Use the theorem above, we can say:

5 * 18 = x * 20

90 = 20x

x = 90/20

x = 4.5

Thus, the value of the variable is x = 4.5.

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The terrarium has 594 cm³ volume of dirt.

What is volume?

Given the dimensions of the terrarium as h=12cm, l=22cm, and b=9cm, we can calculate the volume of the terrarium as follows:

Volume of terrarium = length x width x height

= 22 cm x 9 cm x 12 cm

= 2,376 cm³

Since the terrarium is filled 1/4 of the way with dirt, the volume of dirt is:

Volume of dirt = (1/4) x Volume of terrarium

= (1/4) x 2,376 cubic cm

= 594  cm³

Therefore, the terrarium has 594 cubic centimeters of dirt.

what is terrarium?

A terrarium is a sealed or partially enclosed glass container used to grow and display plants or small animals. It is typically made of clear glass or plastic and is designed to mimic a miniature ecosystem, providing a controlled environment for plants to grow and thrive. Terrariums are popular for indoor gardening and can be used to create a variety of landscapes, from tropical rainforests to desert landscapes. They require minimal maintenance and are an excellent way to bring a touch of nature into your home or office.

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Complete question is: A terrarium is filled 1/4 of the way with dirt. the  terrarium have 594 cubic centimeters of dirt.

Elaine needs tο buy 350 inches οf fabric fοr her windοws.

A fractiοn represents a part οf a number οr any number οf equal parts. There is a fractiοn, cοntaining numeratοr(upper value) and denοminatοr(lοwer value).

If each windοw is 35 inches wide, then the tοtal width οf fabric Elaine needs is:

Tοtal width = 4 windοws x 35 inches per windοw

Tοtal width = 140 inches

Fabric = 2 1/2 x Tοtal width

Tο simplify this, we can first cοnvert the mixed number tο an imprοper fractiοn: 2 1/2 = 5/2

Then, we can multiply this fractiοn by the tοtal width:

Fabric = 5/2 x Tοtal width

Substituting the value οf Tοtal width, we get:

Fabric = 5/2 x 140 inches

Fabric = 350 inches

Therefοre, Elaine needs tο buy 350 inches οf fabric fοr her windοws.

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According to given ratios, the inventory turnover for 20Y7 is 4.83 and the inventory turnover for 20Y6 is 5.30.

What is  ratio?

A ratio is a mathematical comparison between two or more quantities. It expresses the relationship between the two quantities by dividing one quantity by the other.

Inventory turnover is a financial ratio that measures how many times a company sells and replaces its inventory over a period of time. It is calculated by dividing the cost of goods sold by the average inventory for the period.

a. To determine the inventory turnover for 20Y7 and 20Y6, we first need to calculate the average inventory for each year:

Average inventory = (Beginning inventory + Ending inventory) / 2

For 20Y7:

Average inventory = ($770,000 + $840,000) / 2 = $805,000

For 20Y6:

Average inventory = ($740,000 + $770,000) / 2 = $755,000

Next, we can calculate the inventory turnover for each year:

Inventory turnover 20Y7 = Cost of goods sold / Average inventory 20Y7

= $3,894,000 / $805,000

= 4.83

Inventory turnover 20Y6 = Cost of goods sold / Average inventory 20Y6

= $4,001,500 / $755,000

= 5.30

Therefore, the inventory turnover for 20Y7 is 4.83 and the inventory turnover for 20Y6 is 5.30.

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a. The revenue functiοn R(x) = 75x - 3x² graph plοtted belοw.

b. Value οf x is 12.5 and maximum revenue prοduces $468.75

c. Whοlesale price/chip that prοduces the maximum revenue is $37.50

A cοmpany's οr business's revenue is the tοtal amοunt οf mοney it receives frοm sales οf its prοducts οr services οver a given time periοd.

a) Tο sketch a graph οf the revenue functiοn R(x), we first need tο calculate R(x) using the given fοrmula:

⇒ R(x) = x P(x)

⇒ R(x) = x(75 - 3x)

⇒ R(x) = 75x - 3x²

Nοw we can plοt this functiοn οn a rectangular cοοrdinate system. Belοw is an sketch οf the graph.

b) Taking the derivative οf R(x) and setting it equal tο 0:

⇒ R'(x) = 75 - 6x = 0

⇒ 6x = 75

⇒  x = 12.5

Sο, x = 12.5 is the value οf x that will prοduce the maximum revenue. Tο find the maximum revenue, we can substitute x = 12.5 intο the revenue functiοn R(x) = x (75 - 3x)

⇒ R(12.5) = 12.5 (75 - 3 (12.5)) = 468.75

⇒ R(12.5) = $ 468.75

Therefοre, the maximum revenue is $468.75

c) Tο find the whοlesale price per chip that prοduces the maximum revenue, we can substitute x = 12.5 intο the price functiοn, P(x) = 75 - 3x

⇒ P(12.5) = 75 - 3(12.5)

⇒ P(12.5) =  $37.50 per chip

Therefοre, the whοlesale price per chip that prοduces the maximum revenue is $37.50.

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Complete question:

The argument that there's a direct association between the two variables is sufficiently supported by the available data.

The graphs that show the association between two variables in a data set are called smatter plots. It displays data points moreover on a Cartesian system or a two- dimensional aeroplane

. TheX-axis is used to represent the independent variable or trait, while the Y- axis is used to compass the dependent variable. These plates or graphs are constantly used to describe these plots.

Measure of direct correlation. The strength of the direct link between two variables, x and y, is shown by a number known as the direct correlation measure, which is deduced from given data. The direction of the direct link between x and y is indicated by the sign of the direct correlation measure.

Given,

x y

109.14

88.15

138.75

98.77

119.25

148.09

66.13

43.09

129.14

77.25

54.73

r=0.81602411

therefore correlation measure is0.816 a strong positive correlation.

Test statistic t =

p value<0.05 our significant position

Hence reject H0, there's correlation significant between the two variables.

There's sufficient substantiation to support the claim of a direct correlation between the two variables

smatter plot is attached.

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We have shown that for any finite group G and any element r ∈ G, there exists a positive power of r that equals the inverse of r.

What is Algebraic expression ?

An algebraic expression is a mathematical phrase that can contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, division, and exponentiation. Algebraic expressions are used to represent and solve problems in many areas of mathematics, science, engineering, and finance.

To prove the statement, we need to show that there exists a positive power of r, say [tex]r^{k}[/tex], that equals the inverse of r, i.e.,  [tex]r^{k}[/tex], = [tex]r^{-1}[/tex].

Suppose all the powers of r are distinct, i.e.,  [tex]r^{n}[/tex]≠ [tex]r^{m}[/tex] for all n ≠ m in Z. Since G is finite, there are only finitely many powers, Since there are N+1 distinct elements in the set there must be some n,m ∈ {0,1,...,N} such that n < m and [tex]r^{n}[/tex] =  [tex]r^{m}[/tex]

Without loss of generality, assume n < m. Then we have:

[tex]r^{n}[/tex] =  [tex]r^{m}[/tex]

[tex]r^{m-n}[/tex]= 1

Let k = m - n > 0. Then we have:

[tex]r^{k}[/tex] = [tex]r^{m-n}[/tex]= 1

[tex]r^{-1}[/tex]= [tex]r^{(k-1)}[/tex]

Thus,  [tex]r^{k}[/tex] =  [tex]r^{(k-1)}[/tex], which is the desired result.

Therefore, we have shown that for any finite group G and any element r ∈ G, there exists a positive power of r that equals the inverse of r.

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Answer:65

Step-by-step explanation:

Answer:

On a number line, the solution x = -10 would be located to the left of the origin (0) by a distance of 10 units. So, you would mark the point representing -10 on the number line to the left of 0.

Answer:

(a) 7

(b) 7

(c) 27

The property of addition used in all three equations is the associative property, which allows us to regroup the numbers being added without changing their sum.

The dividend yield is 8.8%.

What is the marginal tax rate?

The ratio at which a person or corporation is taxed is known as the tax rate. Statutory, average, marginal, and effective tax rates are just a few of the ways that can be displayed. These rates can also be displayed using the inclusive and exclusive definitions of a tax base.

Here, we have

Given: Assume that Keisha's marginal tax rate is 40% and her tax rate on dividends is 15%. If a city of Atlanta bond pays 7.48% interest.

We have to find the dividend yield would a dividend-paying stock (with no growth potential) has to offer for Keisha to be indifferent between the two investments from a cash-flow perspective.

Dividend yield × (1 - tax rate) = Interest

Dividend yield × (1 - 0.15) = 7.48%

Dividend yield = 7.48%/0.85

Dividend yield = 8.8%

Hence, the dividend yield is 8.8%.

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The equation of the line passing through the points (-1, 5) and (1, -5) is y = -5x.

The collection of points that make up a line in a coordinate system are represented algebraically by a line's equation. The many locations that make up a line on a coordinate axis are denoted by the letters (x, y), and the relationship between x and y results in an algebraic equation that is known as an equation of a line. We can determine whether a given point falls on a line using the equation of any line.

We can use the point-slope form of the equation of a line to solve this problem, which is:

y - y₁ = m(x - x₁)

where:

m = slope of the line

(x₁, y₁) = coordinates of one of the points on the line

First, let's find the slope (m) of the line using the two given points:

m = (y₂ - y₁) / (x₂ - x₁)

m = (-5 - 5) / (1 - (-1))

m = -10 / 2

m = -5

Now we can use one of the points, say (-1, 5), and the slope (-5) to write the equation of the line:

y - 5 = -5(x - (-1))

Simplifying and rearranging the equation:

y - 5 = -5x - 5

y = -5x + 0

Therefore, the equation of the line passing through the points (-1, 5) and (1, -5) is y = -5x.

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Answer:

D) 105

Step-by-step explanation:

All 4 side shape's interior angle add up to 360

360 - 45 = 315

315 / 3 = 105

John drank 0.5025 quarts of water at the park and had 0.1675 quarts left over. It's important to note that it's always a good idea to stay hydrated, especially when spending time outdoors, and bringing enough water is key to staying hydrated.

How to solve fraction?

John brought 2/3 quart of water to the park, which can also be expressed as 0.67 quarts. If he drank 3/4 of the water he brought, this means he consumed 0.75 * 0.67 = 0.5025 quarts of water.

To verify this, we can also subtract the amount of water John drank from the amount he brought:

0.67 - 0.5025 = 0.1675 quarts

This means that John had 0.1675 quarts of water left after he drank 3/4 of what he brought.

To summarize, John drank 0.5025 quarts of water at the park and had 0.1675 quarts left over. It's important to note that it's always a good idea to stay hydrated, especially when spending time outdoors, and bringing enough water is key to staying hydrated.

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The equation is also a series of subtractions. The difference between 16 and 4 is 12, and then subtracting 12 from that result gives 0.

When two quantities are subtracted from one another, the difference is produced. It represents the numerical value of the distance between two values.

The question is asking to match the given equations with the corresponding estimates that have been rounded to the nearest ones place value. The options for the estimates are given on the right-hand side.

Here are the matches:

7.7 - 3.51 ≈ 4 (Quick Check)

The difference between 7.7 and 3.51 is approximately 4.

22.78 - 14.12 ≈ 9 (Quick Check)

The difference between 22.78 and 14.12 is approximately 9.

16.208 - 3.652 ≈ 12 (Intro)

The difference between 16.208 and 3.652 is approximately 12.

3.5 - 1.2 ≈ 2 (Intro)

The difference between 3.5 and 1.2 is approximately 2.

8 - 4 = 4 (Intro)

The difference is 4.

4 - 1 = 3 (Intro)

The difference is 3.

23 - 14 - 9 (Done)

This equation is not a subtraction. It is a series of subtractions. The difference between 23 and 14 is 9, and then subtracting 9 from that result gives 0.

16 - 4 - 12 (Done)

This equation is also a series of subtractions. The difference between 16 and 4 is 12, and then subtracting 12 from that result gives 0.

Note that the terms "Quick Check" and "Intro" refer to different methods of estimation, where "Quick Check" involves rounding to the nearest ten, and "Intro" involves rounding to the nearest whole number.

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An equation that represent the data in the table shown is: D. y = 2x + 3.

In Mathematics, the slope-intercept form of the equation of a straight line is represented by this mathematical expression;

y = mx + c

Where:

First of all, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (1 - 3)/(0 - 1)

Slope (m) = -2/-1

Slope (m) = 2

Based on the information provided above at point (0, 3), an equation that models the line is represented by this mathematical equation;

y = mx + c

y = 2x + 3

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3/8. If the spinner does not land on yellow, the probability it will land on blue is 3/8.

We must first establish the total number of potential outcomes before we can compute the likelihood that the spinner will land on blue in the event that it doesn't land on yellow. There are only 5 potential outcomes as we know the spinner did not rest on yellow.

Two of those five probability are blue. Given that the spinner did not fall on yellow, the likelihood that it will land on blue is 2/5, or 3/8 in decimal notation. This computation accounts for the fact that the sample space has been decreased from 6 to 5 possibilities owing to the elimination of yellow. We calculate the likelihood of 3/8 by dividing the number of favourable outcomes (2) by the total number of potential outcomes (5).

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The probability of picking a green card at random is 15/29.

Probability is the study of the chances of occurrence of a result, which are obtained by the ratio between favorable cases and possible cases.

The total number of cards in the deck is:

Red cards: 5

Blue cards: 9

Green cards: 15

So, the total number of cards in the deck is 5 + 9 + 15 = 29.

The probability of picking a green card at random is the ratio of the number of green cards to the total number of cards in the deck:

Probability of picking a green card = Number of green cards / Total number of cards

Probability of picking a green card = 15 / 29

Hence, the probability of picking a green card at random is 15/29.

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