solve this problem and I will give u brainlst.

Answer:

C

Step-by-step explanation:

note that

[tex]6^{0}[/tex] = 1

[tex]6^{1}[/tex] = 6

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

6³ = 216

that is 6 raised to the power of d gives the corresponding values of c

then explicit formula is

[tex]a_{d}[/tex] = [tex]6^{d}[/tex]

The second smallest integer value of x which would make the given expression as a terminating decimal is 4.

Here we have been given that the variable

[tex]\frac{1}{x^2 + x}[/tex] is a terminating decimal

We need to find the smallest such number. A number can be a terminating decimal if the denominator of the number in fractional form can be expressed as

2ᵃ X 5ᵇ where a and b are whole numbers. Hence we can say that

x² + x = 2ᵃ X 5ᵇ

or, x(x+1) = 2ᵃ X 5ᵇ

This implies that we need to find a pair of consecutive numbers that are factors of 2 or 5,

The first pair is 1,2 as 1X2 = 2

The second pair would be 4,5. 4(4 + 1) = 20

Hence we get the value of x to be 4

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Complete Question

There exist several positive integers x such that [tex]\frac{1}{x^2 + x}[/tex] is a terminating decimal. What is the second smallest such integer?

The equation of the line is expressed in slope-intercept form as:

y = -5/6x - 7.

The equation of a line can be written in slope-intercept form as y = mx + b, where we have:

m = the slope

b = the y-intercept.

Find the slope (m):

Slope (m) = rise/run = -5/6

The y-intercept (b) is -7.

Substitute m = -5/6 and b = -7 into y = mx + b:

y = -5/6x - 7

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Answer:

88

Step-by-step explanation:

180 - 92 (circle theorem) quadrilateral add up to 180

a]Cannot occur due to violation of baryon number conservation, b] Can occur,  c] Can occur, d] Can occur but violates lepton number conservation, e] Can occur and f] Cannot o cur due to violation of charge and baryon number conservations.

The answer are as follows-  a) This reaction can occur as it conserves charge, baryon number, and lepton number.

b) This reaction cannot occur as it violates conservation of charge. The right side has one more positive charge than the left side.

c) This reaction cannot occur as it violates conservation of charge. The left side has zero charge while the right side has one positive charge.

d) This reaction can occur as it conserves charge and lepton number. However, it violates conservation of baryon number as the left side has a baryon number of one while the right side has a baryon number of zero.

e) This reaction can occur as it conserves charge, lepton number, and baryon number.

f) This reaction cannot occur as it violates conservation of charge. The left side has a positive charge while the right side has a neutral charge.

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The calculated value of the size of the angle m is 71 degrees

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

The quadrilateral

The sum of angles in a quadrilateral is 360

So, we have

86 + 180 - 67 + 90 + m = 360

Evaluate the like terms

So, we have

289 + m = 360

Subtract 289 from both sides

So, we have

m = 71

Hence, the size of the angle m is 71 degrees

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The functions x^2 and x^4 form a fundamental set of solutions for the differential equation x^2y" - 5xy' + 8Y = 0 on the interval (0, Infinity), as they are linearly independent and satisfy the differential equation.

To further confirm this, let's examine the properties of the given functions. A fundamental set of solutions consists of linearly independent functions that satisfy the given differential equation. In this case, x^2 and x^4 are linearly independent because they cannot be written as a scalar multiple of one another. Moreover, these functions satisfy the differential equation, which can be demonstrated by substituting them into the equation and verifying that it holds true.

Additionally, the Wronskian being equal to 0 in the specified interval is a key factor in determining the linear independence of the solutions. Since W(x^2, x^4) = 0 for 0 < X < Infinity, this indicates that the given functions form a fundamental set of solutions for the given differential equation on the specified interval.

In conclusion, the functions x^2 and x^4 form a fundamental set of solutions for the differential equation x^2y" - 5xy' + 8Y = 0 on the interval (0, Infinity), as they are linearly independent and satisfy the differential equation. The Wronskian, W(x^2, x^4), confirms their linear independence in the given interval.

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To determine the coordinates of vector AT in this base, we first need to find the coordinates of points B' and C.

Let's start by finding the coordinates of B'. Since B' divides side CA in the ratio 4:4 from C, we can use the following formula to find its coordinates:

B' = (4C + 4A)/8

Simplifying this expression, we get:

B' = (C + A)/2

Similarly, we can find the coordinates of C using the fact that it divides AB in the ratio 3:5 from A:

C = (3A + 5B)/8

Now, we can use these coordinates to find the equation of lines BB' and CC.

The equation of line BB' can be found using the point-slope form:

BB': y - yB = (yB' - yB)/(xB' - xB) * (x - xB)

Substituting the coordinates of B and B', we get:

BB': y - 0 = (yB'/2)/(xB'/2) * (x - 1)

Simplifying this expression, we get:

BB': y = (yB'/xB') * x - yB'/2

Similarly, we can find the equation of line CC:

CC: y = (yC'/xC') * x - yC'/2

Now, we can find the coordinates of point T by solving the system of equations formed by the equations of lines BB' and CC:

(yB'/xB') * x - yB'/2 = (yC'/xC') * x - yC'/2

Simplifying this expression, we get:

x = (yC' - yB') / ((yC'/xC') - (yB'/xB'))

Substituting this value of x into the equation of line BB', we get:

y = (yB'/xB') * ((yC' - yB') / ((yC'/xC') - (yB'/xB'))) - yB'/2

Simplifying this expression, we get:

y = ((yB' * xC') - (yC' * xB')) / (2 * (xC' - xB'))

Now, we can find the coordinates of point T:

T = (x, y)

Substituting the coordinates of B', C, and T into the expression for vector AT, we get:

AT = T - A

Simplifying this expression, we get:

AT = ((x - xA), (y - yA))
In triangle ABC, let B' and C' be points on sides CA and AB, respectively. B' divides CA in a 4:4 ratio from C, meaning CB':B'A = 4:4, and C' divides AB in a 3:5 ratio from A, meaning AC':C'B = 3:5.

Let A be the origin, and let AB = a and AC = b be the vectors forming a base in the plane. Since B' divides CA in half, the position vector of B' is the midpoint of CA, so B' = (1/2)b. Similarly, C' divides AB in a 3:5 ratio, so C' = (5/8)a.

Now, let's consider triangle B'C'T. Since T is the intersection of BB' and CC', we can write the vectors BT and CT in terms of B'T and C'T, respectively:

BT = B'T + TB
CT = C'T + TC

Since B'T and C'T are parallel to b and a, respectively, we can write:

BT = k1 * b
CT = k2 * a

Here, k1 and k2 are constants. Now, using the ratios CB':B'A and AC':C'B, we can write:

k1 * b = 4 * (TB - (1/2)b)
k2 * a = 5 * (TC - (5/8)a)

Finally, we want to find the vector AT. Since AT = TB + TC, we can substitute the expressions for TB and TC from the above equations and solve for AT:

AT = (k1 * b + (1/2)b) + (k2 * a + (5/8)a)

The coordinates of the vector AT are given in terms of the base vectors a and b, with the constants k1 and k2 accounting for the position of the point T in the triangle.

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The solution of the equations is (2, -2)

Given is an equation we need to solve it,

[tex]3 ^ x * 3 ^ y = 1 \\\\2^{(2x - y)} - 64 = 0[/tex]

[tex]\begin{bmatrix}3^x\cdot \:3^y=1\\ 2^{2x-y}-64=0\end{bmatrix}[/tex]

[tex]\mathrm{Substitute\:}x=-y[/tex]

[tex]\begin{bmatrix}2^{2\left(-y\right)-y}-64=0\end{bmatrix}[/tex]

[tex]\begin{bmatrix}8^{-y}-64=0\end{bmatrix}[/tex]

[tex]x=-\left(-2\right)[/tex]

x = 2 and y = 2

Hence, the solution of the equations is (2, -2)

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The dealers included in the sample would be the 5th dealer, the 9th dealer, the 13th dealer, the 17th dealer, and so on, depending on the total number of dealers on the list. This method of sampling is a systematic approach that helps ensure a representative and unbiased sample while still being efficient and random.

Using systematic random sampling, every fourth dealer is selected starting with the 5th dealer in the list. This means that the first dealer in the sample would be the 5th dealer on the list. Then, every fourth dealer after that would also be included in the sample. In this case, you will start with the 5th dealer and select every fourth dealer afterward. Here's the step-by-step explanation:

1. Start with the 5th dealer on the list (since that's your starting point).
2. Move 4 dealers down the list (because you're selecting every 4th dealer) and select the next dealer.
3. Repeat step 2 until you reach the end of the list.

By following these steps, you'll get the dealers included in the sample.

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As per the given graph, there are 8 fewer students walk to school in Class A than in Class B

In this case, we are looking at two classes, Class A and Class B, and the number of students who walk to school in each class. The graph should show a picture or symbol for each student who walks to school.

Now, to answer the question of how many fewer students walk to school in Class A than in Class B, we need to compare the number of symbols or pictures for each class on the graph.

One way to do this is to count the number of symbols or pictures for each class and then subtract the number of students who walk to school in Class A from the number of students who walk to school in Class B.

This will give us the number of students that walk to school in Class B but not in Class A, which is the same as the number of fewer students who walk to school in Class A.

If there are 10 symbols for Class A and 18 symbols for Class B, then we can say that there are 8 fewer students who walk to school in Class A than in Class B.

We get this by subtracting 10 from 18, which gives us 8.

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The value of y at x = π is  [tex]y(\pi) = -e^{(-\pi/2)}sin(\pi/2 + 1)[/tex].

The given differential equation is a second-order linear homogeneous equation with constant coefficients. The characteristic equation is r² + 2r + 2 = 0, which has complex conjugate roots -1 + i and -1 - i. Therefore, the general solution is:

[tex]y(x) = e^{(-x/2)}(c_1cos(x/2) + c_2sin(x/2))[/tex]

Using the initial conditions y(0) = 0 and y'(0) = 1, we can solve for c₁ and c₂ as follows:

y(0) = 0 => c₁ = 0

[tex]y'(x) = -1/2 * e^{(-x/2)*sin(x/2)} + 1/2 * e^{(-x/2)*cos(x/2)[/tex]

y'(0) = 1 => 1/2 * c₂ = 1 => c₂ = 2

Therefore, the particular solution is:

[tex]y(x) = e^{(-x/2)} * 2 * sin(x/2) = 2e^{(-x/2)} * sin(x/2)[/tex]

Plugging in x = π, we get:

[tex]y(\pi) = 2e^{(-\pi/2)} * sin(\pi/2) = -e^{(-\pi/2) }|* sin(\pi/2 + 1)[/tex]

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If the drawing of a circular water-cover, has the diameter of 1.5 feet, then it's circumference is 4.71 feet.

The "Circumference" of circle is the distance around the edge of a circle. It is the perimeter or the length of the boundary of the circle. The formula for the circumference(C) of a circle is given by : C = πd,

where "C" = circumference, "d" = diameter of circle, and π (pi) is a mathematical constant approximately equal to 3.14,

To find the circumference of a circular water-cover with a diameter of 1.5 feet, we substitute the value of diameter in formula of circumference:

We get,

⇒ C = π × d,

⇒ C = π × (1.5 feet),

Using  π as 3.14, We get,

⇒ C = 3.14 × (1.5 feet),

⇒ C = 4.71 feet,

Therefore, the circumference of the circular water-cover is 4.71 feet.

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The given question is incomplete, the complete question is

Find the circumference of the drawing of a circular water-cover, which has the diameter of 1.5 feet.

The derivative is a. y' = [7 * cos(7x) * tan(3x) - 3 * sin(7x) * sec²(3x)] / [tan²(3x)] and the derivative of second funtction is b. (ln(π) * [tex]\pi^(3x^2-7)[/tex]) * (6x) + (9 - x²) / (x²+9)² - 32 / sqrt(1 - (32x+1)²).

a. y = sin(7x)/tan(3x)

To differentiate this function, we can use the quotient rule, which states that if we have a function in the form f(x) = g(x)/h(x), where g(x) and h(x) are differentiable functions, the derivative of f(x) is given by:

f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))²

In this case, g(x) = sin(7x) and h(x) = tan(3x). Let's differentiate both g(x) and h(x) first:

g'(x) = d/dx [sin(7x)] = 7 * cos(7x)

h'(x) = d/dx [tan(3x)] = 3 * sec²(3x)

Now we can substitute these derivatives into the quotient rule formula:

y' = [(7 * cos(7x) * tan(3x)) - (sin(7x) * 3 * sec²(3x))] / (tan(3x))²

Simplifying further, we get:

y' = [7 * cos(7x) * tan(3x) - 3 * sin(7x) * sec²3x)] / [tan²(3x)]

b. y = [tex]\pi^{(3x^2-7)[/tex] + x/(x²+9) + cos⁻¹(32x+1)

To differentiate this function, we can use the sum and chain rules. Let's differentiate each term separately:

For the first term, y₁ = [tex]\pi^{(3x^2-7)[/tex]:

y₁' = d/dx [[tex]\pi^{(3x^2-7)[/tex]]

Using the chain rule, the derivative is:

y₁' = (ln(π) * [tex]\pi^{(3x^2-7)[/tex]) * (6x)

For the second term, y₂ = x/(x²+9):

y₂' = d/dx [x/(x²+9)]

Using the quotient rule, the derivative is:

y₂' = [(1 * (x²+9)) - (x * 2x)] / (x²+9)²

Simplifying further, we get:

y₂' = (9 - x²) / (x²+9)²

For the third term, y₃ = cos⁻¹(32x+1):

y₃' = d/dx [cos⁻¹(32x+1)]

Using the chain rule, the derivative is:

y₃' = -32 / sqrt(1 - (32x+1)²)

Now, we can add all the derivatives together to find the derivative of the function:

y' = y₁' + y₂' + y₃'

y' = (ln(π) * [tex]\pi^{(3x^2-7)[/tex])) * (6x) + (9 - x²) / (x²+9)² - 32 / sqrt(1 - (32x+1)²)

The complete question is:
a. Differentiate the function: [tex]y=\frac{sin(7x)}{tan(3x)}[/tex].

b. Differentiate the function: [tex]\pi^{(3x^2-7)[/tex] + x/x²+9+cos⁻¹(32x+1)

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The simplified value represented in the complex number form |(8+9i)(8+9i)| is equal to 145.

The product of the complex numbers are,

|(8+9i)(8+9i)|

We can expand the expression (8 + 9i)(8 + 9i) using the FOIL method,

(8 + 9i)(8 + 9i)

= 8(8) + 8(9i) + 9i(8) + 9i(9i)

= 64 + 72i + 72i + 81(i²)

Value of i² = -1.

= 64 + 144i - 81

= -17 + 144i

Then, to find the absolute value of this complex number, we take the square root of the sum of the squares of its real and imaginary parts .

Modulus of complex number is,

|(-17 + 144i)|

= √((-17)² + (144)²)

= √(289 + 20736)

= √(21025)

= 145

Therefore, the value of the complex number |(8 + 9i)(8 + 9i)| is equal to 145.

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This shows that the difference between the eigenvalues of x for vector A and B is related to the commutator [A, B] and the eigenvector of x for matrix B.

To show that if x is an eigenvector of matrix A belonging to an eigenvalue λ, then x is also an eigenvector of matrix B belonging to an eigenvalue μ, we can start with the eigenvector equation for matrix A:

A x = λ x

Multiplying both sides by matrix B, we get:

B (A x) = B (λ x)

Using the associative property of matrix multiplication, we can rewrite the left side as:

(B A) x = (A B) x

Substituting the eigenvector equation for matrix A, we get:

(λ B) x = (A B) x

Since x is nonzero, we can divide both sides by x:

λ B = A B

This shows that if x is an eigenvector of matrix A belonging to eigenvalue λ, then it is also an eigenvector of matrix B belonging to eigenvalue μ = λ.

The matrices A and B are related through the commutator [A, B] = AB - BA. We can rewrite the equation λ B = A B as:

λ B - A B = [A, B] B

Since x is nonzero, we can multiply both sides by x:

λ B x - A B x = [A, B] B x

Using the eigenvector equation for matrix A and the fact that x is an eigenvector of matrix A, we get:

λ x - μ x = [A, B] B x

Simplifying, we get:

(λ - μ) x = [A, B] B x

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The height of the jar is 20 centimeters when the radius is 4 centimeters.

The area that any three-dimensional solid occupies is known as its volume. These solids can take the form of a cube, cuboid, cone, cylinder, or sphere.

Let V be the total volume of the jar. Since the jar is one-fourth full, we know that the remaining three-fourths are empty.

Thus, we can write:

V = (4/3)πr²h

We can also write the volume of the baby food as:

20π = (1/4)πr²h

Simplifying this equation, we get:

80 = r²h

Now, we can substitute this value of r²h in the equation for the total volume of the jar:

V = (4/3)πr²h

V = (4/3)πr²(80/r²)

V = (4/3)π(80)

V = 320π

Therefore, the total volume of the jar is 320π cubic centimeters.

Now, we can use the formula for the volume of a cylinder to find the height of the jar:

320π = πr²h

320 = 16h

h = 20

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Answer:

5 centimeters

Step-by-step explanation:

The formula for the volume of a cylinder is:

[tex]\boxed{V=\pi r^2 h}[/tex]

where:

If a cylindrical jar is one-fourth full of baby food, and the volume of the baby food is 20π cm³, then the volume of the jar is 80π cm³.

[tex]\begin{aligned}\textsf{Volume of the jar}& = 4 \cdot 20\pi \\&=80 \pi \; \sf cm^3 \end{aligned}[/tex]

To calculate the height of the jar when its radius is 4 cm, substitute r = 4 and V = 80π into the formula for the volume of a cylinder, and solve for h:

[tex]\begin{aligned}V&=\pi r^2 h\\\implies 80\pi & = \pi (4)^2h\\80\pi & = 16\pi h\\80& = 16 h\\h&=80 \div 16\\h&=5\; \sf cm\end{aligned}[/tex]

Therefore, the height of the jar when the radius of the jar is 4 cm is:

In this case, the theoretical and experimental probabilities are the same for the numbers 1 and 6, so the solution to the problem is 1 and 6.

The theoretical probability of an event is the expected probability based on mathematical analysis or theory.

The experimental probability of an event is the probability that is observed through actual experiments or trials.

To determine which number had the same theoretical and experimental probability in the experiment conducted by Ms. McClain with an n-sided number cube, you need to follow these steps:

Determine the total number of trials conducted in the experiment, denoted as T.

Determine the number of times each number landed during the experiment, denoted as n1, n2, n3,..., nN, where N is the number of sides on the cube.

Calculate the theoretical probability of each number, denoted as P1, P2, P3,..., PN, using the formula:

Pi = 1/N, where i is the number on the cube.

Calculate the experimental probability of each number, denoted as E1, E2, E3,..., EN, using the formula:

Ei = ni / T, where i is the number on the cube.

Compare each theoretical probability Pi to its corresponding experimental probability Ei.

The number that has the same theoretical and experimental probability is the solution to the problem.

For example, if the experiment was conducted with a 6-sided cube, and the table shows the following results:

Number 1 2 3 4 5 6

Trials 10 12 8 11 9 10

Then, you can calculate the theoretical probability of each number:

P1 = 1/6 = 0.1667

P2 = 1/6 = 0.1667

P3 = 1/6 = 0.1667

P4 = 1/6 = 0.1667

P5 = 1/6 = 0.1667

P6 = 1/6 = 0.1667

And the experimental probability of each number:

E1 = 10/60 = 0.1667

E2 = 12/60 = 0.2

E3 = 8/60 = 0.1333

E4 = 11/60 = 0.1833

E5 = 9/60 = 0.15

E6 = 10/60 = 0.1667.

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The correct answer is A. f(2,1) = -8, relative minimum and B. f(1,2) = 9, relative minimum.

To find relative extrema of the function f(x,y) = x³ - 12xy + 8y³, we first find the partial derivatives f_x and f_y:

f_x = 3x² - 12y
f_y = -12x + 24y²

Set both partial derivatives equal to 0 to find critical points:

3x² - 12y = 0 => x² = 4y
-12x + 24y² = 0 => x = 2y²

Solving these equations simultaneously, we get the critical points (2,1) and (1,2). To determine if these points are relative minima or maxima, we use the second derivative test. Compute the second partial derivatives:

f_xx = 6x
f_yy = 48y
f_xy = f_yx = -12

Evaluate the discriminant D = (f_xx * f_yy) - (f_xy * f_yx) at each critical point:

D(2,1) = (12 * 48) - (-12 * -12) = 576 - 144 = 432 > 0, and f_xx(2,1) = 12 > 0, so it's a relative minimum with value f(2,1) = -8.

D(1,2) = (6 * 96) - (-12 * -12) = 576 - 144 = 432 > 0, and f_xx(1,2) = 6 > 0, so it's a relative minimum with value f(1,2) = 9.

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"Why did you choose the outfit you wore today?" is an example of a open-ended question question.

An illustration of an open-ended question is "why did you choose the outfit you wore today?" An open-ended question is one that invites a range of responses and motivates the reply to give more specific information.

In this situation, a question is open-ended and is meant to inspire a unique response based on the person's preferences, views, or situation. In research, interviews, counseling, and other situations where gathering in-depth and varied information is desired, open-ended questions are frequently employed.

They can help researchers and practitioners better understand people's experiences and perspectives by offering insightful information about their thoughts, feelings, and behaviors.

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The wind has a magnitude of approximately 20.25 km/h and is blowing in a direction approximately 56.31° east of south. Magnitude is the hypotenuse of a right triangle with westerly and southerly components.

To find the magnitude and direction of the wind with a westerly component of 11 km/h and a southerly component of 17 km/h, we can use the Pythagorean theorem and trigonometry.

The magnitude of the wind is given by the hypotenuse of a right triangle with legs 11 km/h and 17 km/h. Using the Pythagorean theorem, we get:

magnitude = [tex]\sqrt{(11^2 + 17^2)} \approx 20.25 \;km/h[/tex]

To find the direction of the wind, we can use trigonometry. The angle θ between the wind direction and the east direction can be found using the inverse tangent function:

[tex]tan(\theta)[/tex] = opposite/adjacent = 17/11

[tex]\theta = atan(17/11) \approx 56.31^{\circ}[/tex]

Therefore, the wind has a magnitude of approximately 20.25 km/h and is blowing in a direction approximately 56.31° east of south.

In summary, to find the magnitude and direction of wind with given westerly and southerly components, we can use the Pythagorean theorem and trigonometry.

The magnitude is given by the hypotenuse of a right triangle with legs equal to the westerly and southerly components, while the direction is given by the angle between the wind direction and the east direction, which can be found using the inverse tangent function.

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Answer:

In fraction form:

4.06/100

In decimal form:

0.0406

The length of secant segment ED = 39 units

We know that the Intersecting Secants Theorem states that 'when two secants of a circleintersect at an exterior point, then the product of the one secant segment and its external secant segment is equal to the product of  the other secant segment and its external secant segment.'

Here, ABC and EDC are secants of a circle.

Using  Intersecting Secants Theorem,

AB × BC = ED × DC

Here, BC = 13, DC = 12 and AB = ED - 3

Substituting values in above equation we get,

(ED - 3) × 13 = ED × 12

13ED - 39 = 12ED

ED = 39 units

Therefore, ED = 39 units

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The volume of the solid of revolution is (31/30)π.

To find the volume of the solid of revolution, we need to use the method of cylindrical shells. We will integrate over the height of the region, which is from y=0 to y=1.

First, let's find the points of intersection between the curves:

[tex]x - x^3\\ = x - x^2x^3 - x^2\\ = 0x^2(x-1) \\= 0x=0, x=1[/tex]

So the region we need to rotate is bound by the curves x=0, x=1, y=x-x^3 and y=x-x^2.

Next, we need to express the curves in terms of x and y as follows:

[tex]x = y + y^3\\x = y + y^2[/tex]

To use the method of cylindrical shells, we need to express the radius of each shell as a function of y. The radius of each shell is the distance from the y-axis to the curve at a given height y.

The distance from the y-axis to the curve [tex]x = y + y^3[/tex] is simply [tex]x = y + y^3.[/tex]Therefore, the radius of each shell is r = y + y^3.

The distance from the y-axis to the curve [tex]x = y + y^2 is x = y + y^2.[/tex]Therefore, the radius of each shell is[tex]r = y + y^2.[/tex]

The volume of each shell is given by the formula V = 2πrhΔy, where h is the height of the shell (which is simply Δy) and Δy is the thickness of each shell.

Thus, the total volume of the solid of revolution is given by the integral:

[tex]V = ∫[0,1] 2π(y+y^3)(y+y^2) dy\\V = 2π ∫[0,1] (y^4 + 2y^3 + y^2) dy\\V = 2π [(1/5)y^5 + (1/2)y^4 + (1/3)y^3] [0,1]\\V = 2π [(1/5) + (1/2) + (1/3)]V = (31/30)π[/tex]

Therefore, the volume of the solid of revolution is (31/30)π.

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The initial value problem y''+ty'-2y=6-t, y(0) =0, y'(0) =1 whose Laplace transform exists by taking the Laplace transform of the given differential equation, simplifying it, and then using partial fractions to separate the terms.  The solution to the initial value problem is: y(t) = 2 - 4e^{-2t} + 2e^{t} where y(0) = 0 and y'(0) = 1.

To solve the initial value problem, we first need to take the Laplace transform of the given differential equation:

L{y''} + L{ty'} - L{2y} = L{6-t}

Using the properties of Laplace transforms, we can simplify this equation to: s^2 Y(s) - s y(0) - y'(0) + s Y(s) - y(0) - 2 Y(s) = 6/s - L{t}

Substituting in the initial values y(0) = 0 and y'(0) = 1, we get: s^2 Y(s) + s Y(s) - 2 Y(s) = 6/s - L{t} Simplifying further, we can write this equation as: Y(s) = (6/s - L{t}) / (s^2 + s - 2)

To find the inverse Laplace transform of this equation, we need to factor the denominator as (s+2)(s-1) and then use partial fractions to separate the terms: Y(s) = (2/s) - (4/(s+2)) + (2/(s-1))

Taking the inverse Laplace transform of each term, we get: y(t) = 2 - 4e^{-2t} + 2e^{t} Therefore, the solution to the initial value problem is: y(t) = 2 - 4e^{-2t} + 2e^{t} where y(0) = 0 and y'(0) = 1.

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To find the relative and absolute extrema of the function f(x) = x^3/(x^2 - 48), we first find the derivative:

f'(x) = (3x^2(x^2 - 48) - 2x(x^3))/(x^2 - 48)^2

= (x^4 - 144x)/(x^2 - 48)^2

We can see that f'(x) is defined for all x except x = 0 and x = ± 6√2. To find the critical points, we set f'(x) = 0:

(x^4 - 144x)/(x^2 - 48)^2 = 0

x(x^3 - 144)/(x^2 - 48)^2 = 0

The numerator is zero when x = 0 or x = ±6, but x = 0 and x = ±6 are not in the domain of f(x). Therefore, there are no critical points in the domain of f(x).

Next, we check the endpoints of the domain of f(x), which are x = ±∞. We take the limit as x approaches infinity:

lim x→∞ f(x) = lim x→∞ (x^3/(x^2 - 48))

= lim x→∞ (x/(1 - 48/x^2)) (by dividing numerator and denominator by x^2)

= ∞

Similarly, we take the limit as x approaches negative infinity:

lim x→-∞ f(x) = lim x→-∞ (x^3/(x^2 - 48))

= lim x→-∞ (x/(1 - 48/x^2))

= -∞

Therefore, there is no absolute maximum but there is an absolute minimum at x = -∞.

Since there are no critical points in the domain of f(x), there are no relative extrema. Therefore, the function has an absolute minimum at x = -∞ and does not have any maximums or minimums in the domain.

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This explanation is supported by the provided information and does not involve any cartographic errors or legislative changes affecting the electoral system.

In the 2008 election, Florida had 27 electoral votes. The data on this map can be explained by changes in the state's population and federal legislation affecting the electoral system. Population shifts, as revealed by census data, can lead to states gaining or losing representatives and electoral votes. In this case, if Florida experienced a significant population increase, it could result in additional representatives being allocated, thus increasing the number of electoral votes. On the other hand, changes in federal legislation can also impact the organization of the electoral system. However, there is no specific information provided about such legislative changes affecting Florida's electoral votes in this question. Therefore, the most plausible explanation for the data shown on the map is the population increase in Florida, leading to the state gaining representatives and electoral votes. In conclusion, the most likely reason behind the change in Florida's electoral votes is the increase in population, which resulted in additional representatives and electoral votes being allocated to the state.

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The final cost of the chair is $76.

Given that a chair has a sticker price of $100, with 20% discount, $20 off and $2 mail in rebate.

So, the price will be =

$100 - $20 - $100 × 0.20 - $2

= $80 - $2 - $2

= $80 - $4

= $76

Hence, the final cost of the chair is $76.

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To find the value of k for g(x) = f(x) + k, we compared the graphs of f(x) and g(x). We estimated the distance between the graphs at a common point, x=2, and found k to be approximately 3.25. So, the correct option is A).

We can determine the value of k by comparing the graphs of f(x) and g(x).

The graph of f(x) is a vertical asymptote at x=2, and it approaches zero as x moves away from 2 in either direction.

The graph of g(x) is also a vertical asymptote, but it occurs at x=-3. Moreover, the graph of g(x) is identical to the graph of f(x) shifted upwards by k units.

To find the value of k, we need to find the difference in y-values between the two graphs at any point. Let's take the point x=2, which is on the graph of f(x).

f(2) = 1 / (3(2) - 2) = 1/4

g(2) = f(2) + k = 1/4 + k

Since the graphs of f(x) and g(x) have the same shape and differ only by a vertical shift, we can see that the distance between the graphs at x=2 is equal to k.

Looking at the graph, we can estimate that the distance between the graphs at x=2 is approximately 3 units. Therefore, we have

k = g(2) - f(2) = (1/4 + 3) - 1/4 = 3 1/4

So the value of k is approximately 3.25. So, the correct answer is A).

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