Point R is located at (1, 2) on a coordinate grid. Point S is located at (4, 5) on the same

coordinate grid. What is the distance from point R to point S, rounded to the nearest tenth?

A. 3. 2 units

B. 4. 6 units

C. 7. 6 units

D.

10. 0 units

Answer:

[tex]7 {t}^{2} + 2t - 9 = [/tex]

[tex](7t + 9)(t - 1)[/tex]

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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(0,8) is the y-intercept of the given function g(x).

The y-intercept of g from the equation for f(x).

However, to find the value of g(0) using the given equation:

g(0) = -4f(0) + 12

Since f(0) = 1, we can substitute:

g(0) = -4(1) + 12 = 8

So the value of g at x=0 is 8, and the y-intercept of the function g(x) will be 8.

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Complete question:

What is the y-intercept of function g if g ⁡ ( x ) = - 4 ⁢ f ⁡ ( x ) + 12 and f(0) = 1 ?

To ensure that the first child in line has the greatest number of candy after all 100 pieces are distributed, they must receive as many pieces as possible. We know that each child is guaranteed at least one piece of candy, which means that the 16 children will receive a minimum of 16 pieces altogether.

To find the maximum number of pieces the first child can receive, we can start by assuming that each of the remaining 15 children receive only one piece of candy. This would leave a total of 85 pieces for the first child to receive.

However, we want to maximize the number of pieces the first child can receive while still ensuring that each of the other children receives at least one piece. We can achieve this by giving the second child 2 pieces of candy, the third child 3 pieces, and so on, until we get to the 15th child who receives 15 pieces. This leaves a total of 40 pieces for the first child to receive, which is the maximum amount they can receive while still guaranteeing that each of the other children receives at least one piece.

Therefore, the first child must receive at least 40 pieces of candy to ensure that they have the greatest number after all 100 pieces are distributed.

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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 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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The given identity sin(0) + sin(o) = 2 sin(2*0 - o) * cos(o) is false when checked using complex exponentials and Euler's formula.

To check the given identity using complex exponentials, we'll make use of Euler's formula, which states:

e^(ix) = cos(x) + i*sin(x)

Let's rewrite the given identity in terms of complex exponentials:

sin(0) + sin(o) = 2 sin(2*0 - o) * cos(o)

Now, we'll apply Euler's formula:

(1/2i)(e^(i0) - e^(-i0)) + (1/2i)(e^(io) - e^(-io)) = 2(1/2i)(e^(i(2*0 - o)) - e^(-i(2*0 - o))) * (1/2)(e^(io) + e^(-io))

Simplify the expression:

(1/2i)(e^(i0) - e^(-i0) + e^(io) - e^(-io)) = (1/2i)(e^(i(2*0 - o)) - e^(-i(2*0 - o))) * (1/2)(e^(io) + e^(-io))

We notice that the left side of the equation does not match the right side, which means that the given identity is not true. Therefore, the answer is:

2. FALSE

The given identity sin(0) + sin(o) = 2 sin(2*0 - o) * cos(o) is false when checked using complex exponentials and Euler's formula.

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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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The amount will become zero after 400 months.

Given that, Mr. Tallchief have $400,000 in his account after his retirement,

He intends to withdraw 4,000 from the account every month after the retirement,

We need to find the equation that represents the amount in the account n months after his retirement.

So,

f(n) = 400,000 - 4000n

This equation will give the withdrawal of money each month,

Now, when the account will have no money in it,

A = 0,

0 = 400,000 - 4000n

400,000 = 4000n

n = 400

Hence, the amount will become zero after 400 months.

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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 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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If Meghan has y dollars and Joseph has half as much money as Meghan then the expression y ÷ 2 represents the amount of money Joseph has. Thus, the most appropriate option which is the answer to the given question is B.

An expression is defined as a mathematical phrase with two or more variables with any of the mathematical operations between them. The following are a few examples of expressions: 3x + 45y, 9u, 55 - a, and so on.

In the given question, We are given,

Amount of money Meghan has = $y

Amount of money Joseph has = half as much as Meghan

Thus to calculate the money owned by Joseph, we have to divide the money of Meghan by 2.

And we get the amount of money Joseph has = y ÷ 2

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The complete question might have been:

Meghan has y dollars. Joseph has half as much money as Meghan. Which expression represents the amount of money Joseph has?

A. y + 2

B. y * 2

C. y ÷ 2

D. y - 2

The probability that the paint thickness at a particular point is less than 0.2mm is approximately 0.14.

a) To find the value of A, we need to use the fact that the total area under the probability density function must be equal to 1:

∫0.5 0.125 f(x) dx = 1

Using the given formula for f(x), we get:

∫0.5 0.125 A(0.5 - (x - 0.25)2) dx = 1

Expanding the square inside the integral, we get:

∫0.5 0.125 A(0.5 - x2 + 0.5x - 0.0625) dx = 1

Simplifying and integrating, we get:

A(0.5x - 1/3 x3 + 0.5(1/2)x2 - 0.0625x)∣∣0.125^0.5 = 1

Substituting the limits of integration and simplifying, we get:

A(1/48) = 1

Therefore, A = 48.

The probability density function can be sketched by plotting the function f(x) against x for values of x between 0.125 and 0.5. It will look like a bell-shaped curve with its maximum value at x = 0.25 and decreasing to 0 at x = 0.125 and x = 0.5.

b) The cumulative distribution function (CDF) is defined as:

F(x) = P(X ≤ x) = ∫(-∞)x f(t) dt

To construct the CDF, we need to integrate the probability density function from 0.125 to x:

F(x) = ∫x 0.125 f(t) dt

Using the formula for f(x), we get:

F(x) = 48∫x 0.125 (0.5 - (t - 0.25)2) dt

Simplifying the integral and integrating, we get:

F(x) = 16(x - 0.25) + 3/2(x - 0.25)3 - 1/16(x - 0.25)4

The cumulative distribution function can be sketched by plotting F(x) against x for values of x between 0.125 and 0.5. It will start at 0 when x = 0.125 and approach 1 when x = 0.5, and will be an increasing curve with a maximum slope at x = 0.25.

c) To find the probability that the paint thickness at a particular point is less than 0.2mm, we need to evaluate the cumulative distribution function at x = 0.2:

P(X ≤ 0.2) = F(0.2) = 16(0.2 - 0.25) + 3/2(0.2 - 0.25)3 - 1/16(0.2 - 0.25)4

P(X ≤ 0.2) ≈ 0.14

Therefore, the probability that the paint thickness at a particular point is less than 0.2mm is approximately 0.14.

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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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Answer:

7)a. Brad--2/3, Lola--1/3

b. 2x + x = $630

3x = $630, so x = $210

Brad--$420, Lola--$210

8)a. William--3/7, Emma--4/7

b. 3x + 4x = $12,600

7x = $12,600, so x = $1,800

William--$5,400, Emma--$7,200

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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Answer:

In fraction form:

4.06/100

In decimal form:

0.0406

-12x will be the value of the linear portion in this quadratic equation. Thus, option A is correct.

A linear portion will establish a condition that the value should have the power of the variable as 1. j

In the given equation 7[tex]x^{2}[/tex] - 12x + 16 = 0 which is a trinomial equation:

7[tex]x^{2}[/tex] will have a power of 2

- 12x have a power of 1

16 has a power of o.

The condition of the linear equation states that the value of power should be equal to one. Therefore, option A is correct.

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The question is incomplete, Complete question probably will be  is:

Select the term that describes the linear portion in this quadratic equation.

7[tex]x^{2}[/tex] - 12x + 16 = 0

a. -12x

b. 7x2

c. 16

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 triangle is obtuse, since the square of the longest side is greater than the sum of the squares of the other two sides.

The sum of the areas of the squares formed on the legs of the triangle equals the area of the square formed on the hypotenuse:

a²+b²=c²

a, b and c are side lengths

a=12,b=37 and c=40

12²+37²=40²

144+1369=1600

1513 is not equal to 1600

Since 1513 < 1600, we know that:

This means that the triangle is obtuse, since the square of the longest side is greater than the sum of the squares of the other two sides.

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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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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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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 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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Answer:

88

Step-by-step explanation:

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

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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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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