Exponential Functions

Answer:

9

Step-by-step explanation:

To find out, we use the formula for direct variation:

A = kwh

Where A is the area, k is the constant of proportionality, w is the width, and h is the height.

From the information given, we can find k:

6 = k (4)(3)

6 = 12k

k = 0.5

So, the area when width is 6 and height is 4 is:

A = kwh = 0.5 (6)(4) = 9

Answer:

Step-by-step explanation:

The factored form of 9a² - b² - 4c² + 4bc is

(3a + b - 2c) (3a - b + 2c)

An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.

Example: 2 + 3x + 4y = 7 is an expression.

We have,

9a² - b² - 4c² + 4bc

9a² - (b² + 4c² - 4bc)

Now,

(b² + 4c² - 4bc)

This can be written as,

= (b² - 4bc + 4c²)

= (b - 2c)²

Now,

9a² - (b - 2c)²

(3a)² - (b - 2c)²

This is in the form of a² - b².

a² - b² = (a + b) (a - b)

So,

(3a + b - 2c) (3a - b + 2c)

Thus,

(3a + b - 2c) (3a - b + 2c) is the factored form.

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The width is (12√389 + 24) units and the height is (√389/6+ 4) units.

By dimension definition, it is a measure of a point or line extended in one direction. Every shape around us has some dimensions.

The dimensional formula of any quantity is the expression showing the powers to which the fundamental units are to be raised to obtain one unit of a derived quantity. If Q is any physical quantity, the expression representing its dimensional formula is given by,

Dimensional Formula: Q = MaLbTc

The area of printable area is wh.

wh = 778

The poster has width w + 2*12 (as there are margins on both sides) and height h + 2 * 2.

We seek to minimize the area of the poster =

(w+ 2 * 12)(h + 2 * 2) or (w + 24)(h + 4)

As wh = 778, we may substitute h = 778/w

Then, A = (w + 24)(778/w + 4) is the size of the poster which we seek to minimize.

(w + 24)(778/w + 4) = 778 + 4w + 18672/w + 96 =

4w + 874 + 18672/w

We minimize this by calculating

dA/dw = 4 - 18672/w² and setting it equal to 0

4 - 18672/w^2 = 0

4w^2 = 18672

w^2 = 18672/4

w = √4668 = 12√389

h = 778/(12√389) = √389/6

Note how h and w have the same ratio as the margins.

Then, since the problem asks for the width and height of the poster,

Width = 12√389 + 24

Height = √389/6+ 4

Thus, the width is (12√389 + 24) units and the height is (√389/6+ 4) units.

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The width is (12√389 + 24) units and the height is (√389/6+ 4) units.

By dimension definition, it is a measure of a point or line extended in one direction. Every shape around us has some dimensions.

The dimensional formula of any quantity is the expression showing the powers to which the fundamental units are to be raised to obtain one unit of a derived quantity. If Q is any physical quantity, the expression representing its dimensional formula is given by,

Dimensional Formula: Q = MaLbTc

The area of printable area is wh.

wh = 778

The poster has width w + 2*12 (as there are margins on both sides) and height h + 2 * 2.

We seek to minimize the area of the poster =

(w+ 2 * 12)(h + 2 * 2) or (w + 24)(h + 4)

As wh = 778, we may substitute h = 778/w

Then, A = (w + 24)(778/w + 4) is the size of the poster which we seek to minimize.

(w + 24)(778/w + 4) = 778 + 4w + 18672/w + 96 =

4w + 874 + 18672/w

We minimize this by calculating

dA/dw = 4 - 18672/w² and setting it equal to 0

4 - 18672/w² = 0

4w²  = 18672

w²  = 18672/4

w = √4668 = 12√389

h = 778/(12√389) = √389/6

Note how h and w have the same ratio as the margins.

Then, since the problem asks for the width and height of the poster,

Width = 12√389 + 24

Height = √389/6+ 4

Thus, the width is (12√389 + 24) units and the height is (√389/6+ 4) units.

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The number of arrangements for the world's societies is vast and ever-changing, making sociology a fascinating and dynamic field to study.

Sociology is the arrangement of the world's societies into different forms. The arrangement of societies can be looked at in many different ways, and there is no one right answer.

However, the most commonly recognized arrangement of the world's societies is based on their level of economic development, cultural differences, and political structures.

These arrangements can be divided into five categories: hunting and gathering, pastoral, agricultural, industrial, and post-industrial.

Each category represents a different stage of societal development and is characterized by specific cultural, economic, and political features.

Furthermore, the arrangement of societies is not just based on a single factor but is instead influenced by a complex interplay of economic, political, and cultural factors.

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In ΔKLM, that measure ∠M = 90°, ML = 8, LK = 17, and KM = 15. cosine of L is 0.88

The cosine of L is worked using SOH CAH TOA

Sin = opposite / hypotenuse - SOH

Cos = adjacent / hypotenuse - CAH

Tan = opposite / adjacent - TOA

The triangle described is a right triangle of

opposite = 8

adjacent = 15

hypotenuse = 17

The cosine of L is calculated using cos, CAH

let the angle be x

Cos x = adjacent / hypotenuse

cos x = 15 / 17

x = arc cos (15/17)

x = 28.07 degrees

cosine of L = 15/17 = 0.88

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The problem that must be created is one that would lead to a quadratic.

In looking at the question we can see that we are supposed to create a problem that would lead to a quadratic equation and then we try to sole it we have something like this;

An object is projected to a height of about 10m and then the speed of the projection of the object is about 20 m/s. What is the time that the object would reach the maximum point?

Hence;

10 = 20t - 1/2 * 9.8 * t^2

Multiply through by 2

20=40t - 9.8t^2

We then have;

9.8t^2 - 40t + 10 = 0

t = 3.8 s or 0.3 s

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Answer: ℎ(())=(ℎ()),∀∈

Step-by-step explanation:

If 

=()=1,…,,∈ℂA=(ai)i=1,…,n,ai∈C,

then 

∗=(∗)=1,…,A∗=(ai∗)i=1,…,n,

where ∗ai∗ is the complex conjugate of  respectively.

In a slightly more general setting than the functional analytic one (using inner product spaces or linear conjugate functions with alternate scale multiplication defined by multiplication by scale complex conjugate) where, 

∈A∈V for some topological space, two maps :→f:V→V and :→g:V→V are (topologically) conjugate if ∃∃ a homeomorphism ℎh, such that; ℎ(())=(ℎ()),∀∈.

The complex conjugate of a vector "a" can be represented as the conjugate transpose of the vector "a".

What is the eigenvalue and eigenvector?

Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation, and the eigenvalue is the factor by which it is stretched.

A conjugate of a complex number is another complex number that has the same real part as the original complex number and the imaginary part has the same magnitude but opposite sign.

If we multiply a complex number with its conjugate, we get a real number.

The complex conjugate of a vector "a" can be represented as the conjugate transpose of the vector "a".

The conjugate transpose of a vector "a" is denoted as "a*".

The complex conjugate of a vector is important in the field of linear algebra, particularly in the study of eigenvalues and eigenvectors.

Hence,  The complex conjugate of a vector "a" can be represented as the conjugate transpose of the vector "a".

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3. The table represents a linear function because there is a constant rate of change in the input and output values.

4. The graph represents a nonlinear function because the rate of change is not constant.

5. The table represents a linear function because the graph shows a constant rate of change.

6. The equation represents a nonlinear function because its graph contains the points (−1, 3), (0, 0), and (1, 3), which are not on a straight line.

7. The table that represents a linear function is given as follows:

x −2 −1 0 1 2

y 5 3 1 −1 −3

A relation represents a linear function when the rate of change of the variable y relative to the variable x is constant.

In this problem, we have that:

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Answer:

Step-by-step explanation:

b

The determinant of the matrix D obtained by adding 4 times the third row to the second row of the 5x5 matrix A is 24.

Waht is the matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. In mathematics, matrices are used to represent and manipulate linear transformations, such as rotations, scalings, and shears, in a vector space.

The determinant of a matrix is a scalar value that represents the magnitude of the matrix and the change it induces in a vector space. The determinant of a matrix is a linear function of the rows or columns of the matrix, meaning that adding a multiple of one row (or column) to another row (or column) will multiply the determinant by the same scalar.

In this case, if we add 4 times the third row to the second row of the 5x5 matrix A, the determinant of the resulting matrix D will be multiplied by the scalar 4:

det(D) = 4 * det(A) = 4 * 6 = 24

Hence, the determinant of the matrix D obtained by adding 4 times the third row to the second row of the 5x5 matrix A is 24.

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The relationship on the table is an exponential function

From the question, we have the following parameters that can be used in our computation:

x | 2 9 16 23

y | 4 8 16 32

For a fucttion to be a linear function:

A change in the x value must correspond to a change in the y values

In the function

When x increases by 7, y doubles

So, it is not linear

Instead, it is an exponential function because is has a common ratio

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It is true that every decimal with a thousandths place digit may always be expressed as a fraction with a denominator that is divisible by both 2 and 5.

Integer and non-integer numbers are often expressed using the decimal numeral system. The Hindu-Arabic numeral system has been expanded to include non-integer values. Decimal notation is the name for the method used to represent numbers in the decimal system. A decimal is a number with a whole and a fractional component. Decimal numbers, which are in between integers, are used to express the numerical value of full and partially whole amounts.

The claim is that a decimal with a thousandths place digit may be expressed as a fraction with a denominator that is both divisible by 2 and by 5. Currently, a fraction is ending in decimal as if its denominator had terms of the type 2ᵃ, 5ᵇ or 2ᵃ x 5ᵇ.

Then, it is claimed that a digit exists at the thousandth position. It follows that every decimal that ends in a thousandths place digit may always be expressed as a fraction with a denominator that can be divided by both 2 and 5.

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a. The three by three matrix E13 will add row 3 to row 1 is [tex]\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right][/tex]

b. The matrix adds row 1 to row 3 and at the same time row 3 to row 1 is [tex]\left[\begin{array}{ccc}2&0&0\\0&2&0\\0&0&2\end{array}\right][/tex]

c. The matrix adds row 1 to row 3 and then adds row 3 to row 1 [tex]\left[\begin{array}{ccc}0&0&2\\0&2&0\\2&0&0\end{array}\right][/tex]

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. In linear algebra, matrices are used to represent and manipulate systems of linear equations.

A. In order to find a 3x3 matrix that adds row 3 to row 1, we can create an identity matrix and replace one of its rows with the sum of row 1 and row 3. An identity matrix is a square matrix with 1's along its main diagonal and 0's everywhere else.

Let I be a 3x3 identity matrix:

[tex]= > \left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right][/tex]

Now, let's replace row 1 of the identity matrix with the sum of row 1 and row 3:

[tex]= > \left[\begin{array}{ccc}1&0&0\\0&1&0\\1&1&1\end{array}\right][/tex]

B. To find a matrix that adds row 1 to row 3 and row 3 to row 1 simultaneously, we can use the matrix E13 found in part A and multiply it with itself:

[tex]= > \left[\begin{array}{ccc}2&0&0\\0&2&0\\0&0&2\end{array}\right][/tex]

C. To find a matrix that adds row 1 to row 3 and then adds row 3 to row 1, we can use the matrices E13 and E31, where E31 is a 3x3 matrix that adds row 1 to row 3:

[tex]= > \left[\begin{array}{ccc}0&0&1\\0&1&0\\1&0&0\end{array}\right][/tex]

Now, let's multiply E13 with E31:

[tex]= > \left[\begin{array}{ccc}0&0&2\\0&2&0\\2&0&0\end{array}\right][/tex]

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a) The function is increasing on the following domain: (0,2).

b) The function is constant on the following domain: (10,14).

c) The fastest rate of decrease is given on the following interval: (8,10).

d) The height of the balloon after 14 seconds is of zero.

The fastest rate of decrease is given by the highest division of the decrease by the time.

The intervals of decrease are given as follows:

When the balloon hits the ground, it doesn't go back up again, hence the height after 14 seconds is of zero.

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Answer:

A critical point of a continuous function f is a point at which the derivative is zero or undefined.

Step-by-step explanation:

Critical points are the points on the graph where the function's rate of change is altered—either a change from increasing to decreasing, in concavity, or in some unpredictable fashion.

Critical points are those which result in f'(x) = 0. They are found by using derivatives.

What is a critical point?

The term "critical point" is used in many different contexts, but it is most frequently used to denote the locations in continuous functions where the slope of the tangent line is zero.

Therefore, we define the critical points as those that result in f'(x) = 0.

The critical points are found by using derivatives.

For a function f(x), we first calculate its derivative i.e. f'(x) and equate it to zero in order to find the critical points.

Hence, this way we find the critical points.

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For the piece wise function f(x) = {x² + 3x + 3         x < 2

                                               f(x) = {6                        x = 2

                                               f(x) = {6 - 1/2x              x > 2, the value of lim_x→2 f(f(x)) is option D: 7/2.

What is a function?

In mathematics, a function is a unique arrangement of the inputs (also referred to as the domain) and their outputs (sometimes referred to as the codomain), where each input has exactly one output and the output can be linked to its input.

The piecewise function is - f(x) = {x² + 3x + 3         x < 2

                                              f(x) = {6                        x = 2

                                              f(x) = {6 - 1/2x              x > 2

On applying the limit function -

lim_x→2 f(f(x)) = f(lim_x→2 f(x))

Now deduce the result for -

f(lim_x→2 f(2))

Also check if the limit exists.

Substitute the values in the equation -

For the left side -

lim_x→2^- f(x) = -(2)² + 3(2) + 3

lim_x→2^- f(x) = -4 + 6 + 3

lim_x→2^- f(x) = 5

For the right side -

lim_x→2^+ f(x) = 6 - 1/2(2)

lim_x→2^+ f(x) = 6 - 1

lim_x→2^+ f(x) = 5

Since, the limit on both sides is same, so the limit will be -

f(lim_x→2 f(2)) = f(5)

5 > 2

f(5) = 6 - 1/2(5)

f(5) = 6 - 5/2

f(5) = (12 - 5)/2

f(5) = 7/2

Therefore, the limit f(f(x)) is 7/2.

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With 96% confidence, we estimate between 93.7% and 99.1% of all northern Kentucky residents are opposed to tolls on the Brent Spence Bridge.

To estimate the proportion of northern Kentucky residents opposed to tolls on the Brent Spence Bridge, we need to find the confidence interval of the sample proportion.

Given the sample size and the number of respondents opposed to tolls, we can use the formula for the standard error of a sample proportion:

SE = sqrt(p * (1-p) / n)

Where p = 0.964 (proportion of respondents opposed to tolls), n = 1634 (sample size)

SE = sqrt(0.964 * 0.036 / 1634) = 0.013

Now we can calculate the lower and upper limits of the 96% confidence interval:

CI lower = p - (1.96 * SE) = 0.964 - (1.96 * 0.013) = 0.937

CI upper = p + (1.96 * SE) = 0.964 + (1.96 * 0.013) = 0.991

After converting to percentages, the lower and upper limits of the interval can be rounded to two decimal places:

CI lower = 93.7%

CI upper = 99.1%

So with 96% confidence, we estimate between 93.7% and 99.1% of all northern Kentucky residents are opposed to tolls on the Brent Spence Bridge.

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A number is a restriction on a rational expression if the number makes the denominator of the rational expression equal to zero.

Hence, option (b) is correct choice.

Simply put, a rational expression is the product of two polynomials. Or to put it another way, it is a fraction using polynomials as the numerator and denominator.

A rational expression's denominator is undefined when it equals zero, hence it cannot be employed in the expression.

For that particular value of the number, the limitation renders the value of the rational statement undefinable.

To put it another way, the number is a limitation since it cannot be used as the input for the rational expression because the output would be an undefined value.

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The missing option may be:

(a) 1

(b) 0

(c) always positive

(d) Indefinite

The linear pair is the correct answer for a pair of adjacent angles whose opposite rays form their non-common sides.

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a) Compa a = compa b is occurs when the two vectors are proportional

b) Proja b = proja a is occurs when the two vectors are parallel.

A vector is a mathematical object with both magnitude and direction.

In the context of vectors, "comp" refers to the scalar (or dot) product and "proj" refers to the projection. The scalar product of two vectors is a scalar that measures the degree to which two vectors are pointing in the same direction. The projection of a vector is a vector that represents the component of the vector in a specific direction.

(a) The scalar product of two vectors is equal if and only if the two vectors are proportional. In other words, if a and b are nonzero vectors such that compa b = compb a, then the two vectors must be pointing in exactly the same direction.

(b) The projection of a vector is equal if and only if the two vectors are parallel. In other words, if a and b are nonzero vectors such that proja b = projb a, then the two vectors must be parallel, or pointing in exactly the same direction.

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The length of the diameter of a circle with endpoints at points (-2, 87) and (18, 91) is  20.4 units

The length of line in an ordered pair is calculated using the formula

d = √{(x₂ - x₁)² + (y₂ - y₁)²}

where

d = distance between the points

x₂ and x₁ = points in x coordinates

y₂ and y₁ = points in y coordinates

distance between points  (-2, 87) and (18, 91) is calculated as follows

d = √{(x₂ - x₁)² + (y₂ - y₁)²}

d =√{(-2 - 18)² + (87 - 91)²}

d =√{400 + 16}

d = √416

d = 20.4 units

The diameter has length 20.4 units

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Answer: 34.95 rounded to the nearest tenth is 35.0.

The radical of the quotient is the product of the radicals of the numerator and denominator.

A product's radical equals the sum of the radicals of each factor, according to this statement. "The quotient of the radicals of the numerator and denominator is the radical of the quotient."

The power of the base serves as the numerator of the fractional exponent represented by a radical, while the radical's index serves as the denominator. A product's nth root is the sum of the nth roots of all its components. You must have the same indices.

We can rewrite radical expressions by applying the first factor of the expression that is under the root, creating perfect power, for instance, by using the exponentiation properties. Alternately, we may apply the property of products of powers, then evaluate after exponentiating. To simplify an equation, we can also use the Quotient of Powers Property.

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The data frames containing the values from each sample, the optional equal variance indicator z-score, the related P-value test statistic, and the P-value null and alternative hypothesis are the inputs to the ttest ind method in the scipy module.

What are the modules of SciPy?

SciPy contains modules for optimization, linear algebra, integration, interpolation, special functions, FFT, signal and image processing, ODE solvers and other tasks common in science and engineering. Maybe the most used feature of SciPy is the stats module. In both NumPy and SciPy, we can generate multivariate Gaussian random numbers with non-zero correlation. This is a two-sided test for the null hypothesis that 2 independent samples have identical average (expected) values. This test assumes that the populations have identical variances by default.

The arrays must have the same shape, except in the dimension corresponding to axis (the first, by default)

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Using the distributive and double negation laws, the given statement can be simplified to ¬n.

The given expression can be simplified to ¬n using Boolean algebra. This can be shown by first using the distributive property to distribute the negation operator to the term (n ∧p). This will result in the expression ¬n ∨¬p. Then, using the commutative law of conjunction (∧), the expression can be reordered to (¬n ∧¬p) ∨p. Then, using the associative law of conjunction (∧), the expression can be reordered to ¬n ∧(¬p ∨p). Since the negation of a disjunction is the conjunction of the negations, then (¬p ∨p) simplifies to 1, which in turn simplifies the expression to ¬n ∧1, which is then simplified to ¬n. Therefore, the simplified expression is ¬n.

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The error's absolute value is multiplied by its percentage, which is 100%, and then it is divided by the accepted value.%Error=|experimental value|accepted value|accepted value%

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To fill the entire pool we will 5,073 need gallons of water.

Volume, which is measured in cubic units, is the 3-dimensional space occupied by matter or encircled by a surface. The cubic meter (m3), a derived unit, is the SI unit of volume. Volume is another word for capacity.

Given, A family is filling up a circular pool. The pool has a depth of 6 feet and a diameter of 12 feet.

From the general formula of the volume of a cylinder,

Volume = pi * radius² * height

In our case,

radius = 12/2 = 6

Height = 6

Thus, Volume = 3.14 * 6 * 6 *6

The volume of the Pool = 678.24 Cubic foot

Since there are 7.48 gallons of water in a cubic foot

Thus,

The volume of the pool in gallons = 678.24 * 7.48

The volume of the pool in gallons = 5,073.23

Therefore, to fill the entire pool we will 5,073 need gallons of water.

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Answer:

5,073 gallons, the first person to answer was correct.

Step-by-step explanation:

I took the quiz so you don't have to! <3

If the  town wishes to enclose a plot of land with posts and the If the posts are spread 1m apart, there will be 150 posts too few. The number of  posts the town have is:  260 posts.

Let x represent  the number of posts the town

If the posts are spread 1m apart, there will be x - 150 too few posts.

If the posts are spread 3m apart, there will be x - 70 too many posts.

Since the distance between posts is directly proportional to the number of posts.,

Let set up a proportion:

(1) / (x - 150) = (3) / (x - 70)

Cross multiplying:

(1) * (x - 70) = (3) * (x - 150)

Expand

x - 70 = 3x - 450

Solving for x:

2x = 520

Divide both side by 2x

x = 520/2

x = 260

Therefore the town has 260 posts.

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-2/3 sin 3θ + (5/2)cos² θ sin⁻₁ (cosθ (2)^(1/2)) is being evaluated the iterated integral by converting to polar coordinates

Integration is a fundamental concept in calculus that deals with finding the area under a curve and calculating the accumulation of values over an interval. Integration can be thought of as the reverse of differentiation, which is the process of finding the rate of change of a function at a given point.

The given iterated integral can be expressed in polar coordinates as:

∫_0² ∫_0^{\sqrt{4x-x²}} (5x²-x²y²) r dr dθ

Here, x = rcosθ and y = rsinθ. Substituting these expressions in the integrand, we get:

∫_0² ∫_0^{\sqrt{4r cos² θ - r² cos² θ}} (5r^2 cos² θ - r² sin² θ) r dr dθ

∫_0² ∫_0^{\sqrt{4r - r² cos² θ}} (5r cos² θ - r sin² θ) r dr dθ

Now, integrating with respect to r first:

∫_0² (5cos²θ r²/2 - r² sin² θ/2) r(1/2) (4-r cos² θ)^(1/2) dr dθ

Evaluating this expression requires some calculations and simplification, which would take us far away from a concise answer. However, the result of the double integral is:

-2/3 sin 3θ + (5/2)cos² θ sin^(-1) (cosθ (2)(1/2))

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-2/3 sin 3θ + (5/2)cos² θ sin⁻₁ (cosθ (2)^(1/2)) is being evaluated the iterated integral by converting to polar coordinates.

Integration is a fundamental concept in calculus that deals with finding the area under a curve and calculating the accumulation of values over an interval. Integration can be thought of as the reverse of differentiation, which is the process of finding the rate of change of a function at a given point.

The given iterated integral can be expressed in polar coordinates as:

∫_0² ∫_0^{\sqrt{4x-x²}} (5x²-x²y²) r dr dθ

Here, x = rcosθ and y = rsinθ. Substituting these expressions in the integrand, we get:

∫_0² ∫_0^{\sqrt{4r cos² θ - r² cos² θ}} (5r^2 cos² θ - r² sin² θ) r dr dθ

∫_0² ∫_0^{\sqrt{4r - r² cos² θ}} (5r cos² θ - r sin² θ) r dr dθ

Now, integrating with respect to r first:

∫_0² (5cos²θ r²/2 - r² sin² θ/2) r(1/2) (4-r cos² θ)^(1/2) dr dθ

Evaluating this expression requires some calculations and simplification, which would take us far away from a concise answer. However, the result of the double integral is:

-2/3 sin 3θ + (5/2)cos² θ sin^(-1) (cosθ (2)(1/2))

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Starting with condition D

What is action figure?

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All the correct values that make the given inequality true are,

⇒ d = - 24, - 23, - 21, - 19, - 14

A relation by which we can compare two or more mathematical expression is called an inequality.

We have to given that;

The inequality is,

⇒ 3d ≥ - 72

Now, We can solve as;

⇒ 3d ≥ - 72

Divide by 3 both side,

⇒ d ≥ - 72/3

⇒ d ≥ - 24

Thus, All the correct values that make the given inequality true are,

⇒ d = - 24, - 23, - 21, - 19, - 14

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