Sketch each sngle. Then find jts reference angle.
1) -210
2)-7pi/4

Please show work and steps by steps!thanks!

Answer:

39/52 / 3/4  or  75%

Step-by-step explanation:

There are 4 suits (Clubs, Hearts, Diamonds, and Spades)

There are 13 cards in each suit

52-13=39  

Hope this helps!

To reduce this fraction, divide both the numerator and denominator by their greatest common divisor, which is 13. The reduced fraction is 3/4. So, the probability of not selecting a spade is 3/4.

In a standard deck of 52 cards, there are 13 spades. To find the probability of not selecting a spade, you'll need to determine the number of non-spade cards and divide that by the total number of cards in the deck. There are 52 cards in total, and 13 of them are spades, so there are 52 - 13 = 39 non-spade cards. The probability of selecting a non-spade card is the number of non-spade cards (39) divided by the total number of cards (52). Therefore, the probability is 39/52.

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To find a solution to the differential equation y" + y = 3x with initial conditions y(0) = 2 and y'(0) = -2, we can combine the complementary solution (Ye) and the particular solution (Yp). The complementary solution is given by Ye = C1cos(x) + C2sin(x), where C1 and C2 are constants, and the particular solution is Yp = 3x. By adding the complementary and particular solutions, we obtain the complete solution to the differential equation.

The complementary solution Ye represents the general solution to the homogeneous equation y" + y = 0. It consists of two parts, C1cos(x) and C2sin(x), where C1 and C2 are determined based on the initial conditions. The particular solution Yp satisfies the non-homogeneous equation y" + y = 3x. In this case, Yp = 3x is a valid particular solution since the right-hand side of the equation is a linear function. To obtain the complete solution, we add the complementary solution and the particular solution: y(x) = Ye + Yp = C1cos(x) + C2sin(x) + 3x. To determine the values of C1 and C2, we use the initial conditions. y(0) = 2 gives C1 = 2, and y'(0) = -2 gives C2 = -2. Therefore, the solution satisfying the given initial conditions is y(x) = 2cos(x) - 2sin(x) + 3x.

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The derivative questions that meet the given criteria:

1. [tex]f(x) = (sin(x) + e^{(2x)})/(cos(x) + e^{(3x)})[/tex]

2. [tex]g(x) = sin(x) * e^{(2x)}[/tex]

3.  [tex]h(x) = sin^2{(x)}[/tex]

4. i(x) = [tex]cos(e^{(x)})[/tex]

5.  [tex]j(x) = e^{x} + e^{(2x)} + sin(x)[/tex]

Here are derivative questions that meet the given criteria:

1. Find the derivative of [tex]f(x) = (sin(x) + e^{(2x)})/(cos(x) + e^{(3x)})[/tex]

1. f'(x) = [tex][(cos(x) + e^{(3x)})(sin(x) + e^{(2x)})' - (sin(x) + e^{(2x)})(cos(x) + e^{(3x)})']/(cos(x) + e^{(3x)})^2[/tex]

2. Find the derivative of[tex]g(x) = sin(x) * e^{(2x)}[/tex]

g'(x) = [tex](sin(x) * e^{(2x)})' + (e^{(2x)} * sin(x))'[/tex]

3. Find the derivative of [tex]h(x) = sin^2{(x)}[/tex]

[tex]h'(x) = (sin^2{(x)])'[/tex]

4. Find the derivative of i(x) = [tex]cos(e^{(x)})[/tex]

[tex]i'(x) = (cos(e^{(x))})'[/tex]

5. Find the derivative of [tex]j(x) = e^{x} + e^{(2x)} + sin(x)[/tex]

j'(x) =[tex](e^x + e^{(2x)} + sin(x))'[/tex]

[tex](e^x + e^{(2x)} + sin(x))'[/tex]

The answers provided above are the derivatives of the given functions based on the specified criteria, and they are not simplified.

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The integral represents the infinitesimal lengths of small line segments along the curve, and by evaluating the integral over the appropriate interval, we can determine the total length of the curve.

The arc length formula is given by ∫√(1 + (dy/dx)^2) dx, where dy/dx is the derivative of y with respect to x. In this case, we need to find dy/dx for the given curve.

Taking the derivative of y = 3x^2 + 12x with respect to x, we get dy/dx = 6x + 12.

Now, substituting this derivative into the arc length formula, we have ∫√(1 + (6x + 12)^2) dx.

To evaluate this integral, we integrate with respect to x over the interval from -3 to 1, which represents the curve between the given points.

In summary, to find the length of the curve, we set up an integral using the arc length formula and the derivative of the given curve. The integral represents the infinitesimal lengths of small line segments along the curve, and by evaluating the integral over the appropriate interval, we can determine the total length of the curve.

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They may differ depending on the behavior of the function within each subinterval.

The limits of the lower and upper sums may not be equal for a continuous and non-negative function on the interval [a, b].

The partition can have varying widths as long as the width approaches zero as the number of subintervals increases

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Answer:

  (-0.2, 1.4)

Step-by-step explanation:

You want the point on the line y = 3x +2 that is closest to the point (4, 0).

When a line is drawn from the given point perpendicular to the given line, their point of intersection will be the point we're looking for. There are several ways it can be found.

The given line has a slope of 3, so the perpendicular will have a slope of -1/3, the opposite reciprocal of 3.

One way to find that point is to write the equation for the slope from it to point (4, 0).

  (y -0)/(x -4) = -1/3

  ((3x +2) -0)/(x -4) = -1/3 . . . . . . . use the equation for y on the line

  3(3x +2) = -(x -4) . . . . . . cross multiply

  10x = -2 . . . . . . . . . . add x - 6

  x = - 0.2 . . . . . . divide by 10

  y = 3(-0.2) +2 = 2 -0.6 = 1.4 . . . . . find y from the line's equation

The closest point is (-0.2, 1.4).

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The point on the curve closest to y = 3x + 2 is (3, 11).

The given equation is y = 3x + 2 and we have to find the point on the curve which is closest to the point (4,0).

Let (a, b) be a point on the curve y = 3x + 2. Then, the distance between the point (4,0) and the point (a, b) is given by: distance = sqrt((a - 4)² + (b - 0)²)

The value of a can be obtained by substituting y = 3x + 2 in the above equation and solving for a. distance = sqrt((a - 4)² + (3a + 2)²) = f(a)Let f(a) = sqrt((a - 4)² + (3a + 2)²)

Therefore, the point on the curve y = 3x + 2 which is closest to the point (4,0) is (3, 11).

Therefore, the required point is (3, 11).

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The first approximation, denoted as oren, can be written as the product of c and d. The greatest common divisor of c and d is 1, meaning they have no common factors other than 1.

The specific values of c and d are not provided, so you would need to provide the values or determine them based on the context of the problem.

Regarding the variable u, it is not specified in your question, so it is unclear what u represents. If u is related to the approximation oren, you would need to provide additional information or context for its calculation or meaning.

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a) The series $\sum_{n=0}^\infty 4n!$ absolutely diverges.

b) The series $\sum_{n=0}^\infty (-4)^{2n+1}$ is divergent.

a) We have to check whether the following series absolutely 4n! converges or diverges. As we know that the series absolutely convergent, then we can apply the ratio test.Using ratio test, we get\[\lim_{n \to \infty}\frac{(4(n+1))!}{4n!}\]= \[\lim_{n \to \infty}\frac{(4n+4)!}{4n!}\times\frac{1}{4}\]Multiplying the numerator by 4 and then simplifying, we get \[\frac{(4n+4)(4n+3)(4n+2)(4n+1)}{4}\]\[=4(4n+3)(4n+2)(4n+1)(n!) \to \infty\]Therefore, the series absolutely diverges.b) We have to determine whether the following series absolutely (-4)2n +1 converges or diverges using the test for alternating series.The series can be written as \[\sum_{n=0}^\infty a_n\] where \[a_n=(-1)^n (-4)^{2n+1}\]i.e., \[a_n=(-1)^n (-4)^{2n}\times(-4)\] or \[a_n=(-1)^n 16^n(-4)\]We see that \[\lim_{n \to \infty}a_n\neq 0\]Hence, the series is divergent.

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The derivative dy/dx, obtained through implicit differentiation, is given by [tex](15y^2 - 3x cos(y)) / (5y^2 - 3).[/tex]

To find dy/dx using implicit differentiation, we differentiate both sides of the equation with respect to x. Starting with the equation [tex]3x - 5y^3 =[/tex]sin(y), we differentiate each term. The derivative of 3x with respect to x is simply 3. For the term [tex]-5y^3,[/tex] we use the chain rule, which states that [tex]d/dx(f(g(x))) = f'(g(x)) * g'(x[/tex]). Applying the chain rule, we get [tex]-15y^2 * dy/dx[/tex]. For the term sin(y), we apply the chain rule once again, which yields cos(y) * dy/dx. Setting these derivatives equal to each other, we have 3 - [tex]15y^2 * dy/dx = cos(y) * dy/dx[/tex]. Rearranging the equation, we obtain [tex](15y^2 - 3x cos(y)) / (5y^2 - 3)[/tex] as the expression for dy/dx.

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The ratio of grass to decking in terms of area, in its simplest form, is 4:5.

In the garden, 4/9 of the area is covered with grass, and the rest is decking. To find the ratio of grass to decking in terms of area, we can express it as a fraction.

Let's denote the area covered with grass as G and the area covered with decking as D.

The given information states that 4/9 of the area is grass, so we have:

G = (4/9) * Total area

Since the remaining area is covered with decking, we can express it as:

D = Total area - G

To simplify the ratio of grass to decking in terms of area, we can divide both G and D by the total area:

G/Total area = (4/9) * Total area / Total area

G/Total area = 4/9

Similarly,

D/Total area = (Total area - G)/Total area

D/Total area = (9/9) - (4/9)

D/Total area = 5/9

Therefore, the ratio is 4:5.

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a. The model equation for the average salary per year  is s(t) = 0.021 * (t^2/2) + t + 60.575

b.  The average salary in 1993 (rounded to 1 decimal place) is $63.7 thousand.

a. To find a model that gives the average salary per year, we need to integrate the given rate of change equation.

ds = 0.021t + dt

Integrating both sides with respect to t:

∫ds = ∫(0.021t + dt)

s = 0.021 * (t^2/2) + t + C

Since the average salary in 1995 was 66.1 thousand dollars, we can use this information to find the constant C. Plugging in t = 5 and s = 66.1 into the model equation:

66.1 = 0.021 * (5^2/2) + 5 + C

66.1 = 0.525 + 5 + C

C = 66.1 - 0.525 - 5

C = 60.575

Now we have the model equation for the average salary per year:

s(t) = 0.021 * (t^2/2) + t + 60.575

b. To find the average salary in 1993 (corresponding to t = 3), we can plug t = 3 into the model:

s(3) = 0.021 * (3^2/2) + 3 + 60.575

s(3) = 0.021 * 4.5 + 3 + 60.575

s(3) = 0.0945 + 3 + 60.575

s(3) = 63.6695

Therefore, the average salary in 1993 (rounded to 1 decimal place) is $63.7 thousand.

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To find the area above the curve y = -[tex]e^{x}[/tex] + [tex]e^{2x-3}[/tex] and below the x-axis for x > 0, we can use integration. The graph will help visualize the area and provide a numerical result.

To begin, let's first rewrite the equation of the curve as y = [tex]e^{2x-3}[/tex] - [tex]e^{x}[/tex]The area we need to find is the region above this curve and below the x-axis, limited to x > 0.

To determine the area using integration, we need to find the x-values where the curve intersects the x-axis. We set y equal to zero and solve for x:

0 = [tex]e^{2x-3}[/tex]-[tex]e^{x}[/tex]

Unfortunately, this equation does not have an algebraic solution that can be easily obtained. However, we can still find the area by approximating it numerically using integration.

By graphing the function, we can visually estimate the x-values where the curve intersects the x-axis. These values can be used as the limits of integration. Integrating the function over this interval will give us the desired area.

Once the graph is plotted, we can use numerical methods or graphing software to evaluate the integral and find the area. The result will provide the value of the area above the curve and below the x-axis for x > 0.

Remember, it is crucial to accurately determine the limits of integration from the graph to obtain an accurate result.

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The circumference of the circle is  56.52 units.

Remember that for a circle whose center is at (a, b) and that has a radius R is written as:

(x - a)² + (y - b)² = R²

Here we have the circle equation:

x² - 2x + 10y + y² = 7 - 16x

We can rewrite this as:

x² - 2x + 16x + y² + 10y = 7

x² + 14x + y² + 10y = 7

Now we can add 7² and 5² in both sides to get:

x² + 14x + 7² +  y² + 10y + 5² = 7+ 5² + 7²

(x + 7)² + (y + 5)² = 81 = 9²

So the radius of the circle is 9 units, then the circumference is:

C = 2*3.14*9 = 56.52 units.

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The expected value for the given probability model is 16.400. To calculate the expected value, we multiply each value by its corresponding probability and sum up the results

In this case, we have three values: 3, 10, and 50, with probabilities 0.25, 0.35, and 0.07, respectively.

The expected value is obtained by the following calculation:

Expected value = [tex]\((3 \cdot 0.25) + (10 \cdot 0.35) + (50 \cdot 0.07) = 0.75 + 3.5 + 3.5 = 7.75 + 3.5 = 11.25 + 3.5 = 14.75 + 1 = 15.75\)[/tex]

Rounding to three decimal places, we get the expected value as 16.400.

In summary, the expected value for the given probability model is 16.400. This is calculated by multiplying each value by its probability and summing up the results. The expected value represents the average value we would expect to obtain over a large number of repetitions or trials.

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The first question asks about the number of different onto (surjective) functions possible from a set of 6 elements to a set of 8 elements.

To find the number of onto functions from a set of 6 elements to a set of 8 elements, we can use the concept of counting. An onto function is one where every element in the codomain (the set of 8 elements) is mapped to by at least one element in the domain (the set of 6 elements). Since there are 8 elements in the codomain, and each element can be mapped to by any of the 6 elements in the domain, we have 6 choices for each element. Therefore, the total number of onto functions is calculated as 6^8.

To determine the number of functions that are not one-to-one from a set of 2 elements to a set of 8 elements, we need to consider the definition of a one-to-one function. A function is one-to-one (injective) if each element in the domain is mapped to a unique element in the codomain.

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(2) To find the critical points of the function f(x, y) = x + 2y² - 4xy, we need to determine the values of (x, y) where the partial derivatives with respect to x and y are both equal to zero.

Taking the partial derivative of f(x, y) with respect to x, we get ∂f/∂x = 1 - 4y. Setting this equal to zero gives 1 - 4y = 0, which implies y = 1/4. Taking the partial derivative of f(x, y) with respect to y, we get ∂f/∂y = 4y - 4x. Setting this equal to zero gives 4y - 4x = 0, which implies y = x. Therefore, the critical point occurs at (x, y) = (1/4, 1/4).  (3) The given question seems to be incomplete as it mentions "the following double integrals are casier to eval- uate in one o."

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a. The rectangular prism with point P = (2, 3, 4) in ℝ³ is drawn on the provided grid.

b. The coordinates for all the points and their labels are as follows:

- Point A: (2, 0, 0)

- Point B: (2, 3, 0)

- Point C: (2, 0, 4)

- Point D: (2, 3, 4)

- Point E: (0, 3, 0)

- Point F: (0, 3, 4)

- Point G: (0, 0, 4)

- Point H: (0, 0, 0)

In the rectangular prism, the x-coordinate represents the distance along the x-axis, the y-coordinate represents the distance along the y-axis, and the z-coordinate represents the distance along the z-axis.

Point P, given as (2, 3, 4), has x = 2, y = 3, and z = 4. By using these values, we can determine the coordinates of the other points in the rectangular prism.

The points labeled A, B, C, D, E, F, G, and H represent the vertices of the prism. Point A has the same x-coordinate as P but is located at y = 0 and z = 0.

Similarly, points B, C, and D have the same x-coordinate as P but different y and z values. Points E, F, G, and H have different x-coordinates but the same y and z values.

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Using the Fundamental Theorem of Calculus, the integral of (4x - 1) dx can be evaluated as (2x^2 - x) + C, where C is the constant of integration.

To compute the integral (4x - 1) dx in terms of areas, we can relate it to the graph of y = 4x - 1. The integral represents the area under the curve of the function over a given interval. In this case, we want to find the area between the curve and the x-axis.

The graph of y = 4x - 1 is a straight line with a slope of 4 and a y-intercept of -1. The integral of (4x - 1) dx corresponds to the sum of the areas of infinitesimally thin rectangles bounded by the x-axis and the curve.

Each rectangle has a width of dx (an infinitesimally small change in x) and a height of (4x - 1). Summing up the areas of all these rectangles from the lower limit to the upper limit of integration gives us the total area under the curve. Evaluating this integral using the antiderivative of (4x - 1), we obtain the expression (2x^2 - x) + C, where C is the constant of integration.

In conclusion, the integral (4x - 1) dx represents the area between the curve y = 4x - 1 and the x-axis, and using the Fundamental Theorem of Calculus, we can evaluate it as (2x^2 - x) + C, where C is the constant of integration.

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Hence, there are 2,408 ways to place the blue king and the yellow king on an empty chessboard so that they do not attack each other.

To determine the number of ways to place a blue king and a yellow king on an empty chessboard such that they do not attack each other, we can consider the possible positions for the blue king.

Since there are 64 squares on a chessboard, we have 64 choices for the blue king's position. Once the blue king is placed, there are 49 remaining squares where the yellow king can be placed. However, we need to ensure that the yellow king is not in a position to attack the blue king.

If the blue king is placed on a corner square (4 corner squares available), then there are 8 squares adjacent to the blue king where the yellow king cannot be placed. Therefore, for each corner square placement of the blue king, we have 41 choices for the yellow king's position.

If the blue king is placed on a square along the edge of the board (24 edge squares available), then there are 11 squares adjacent to the blue king where the yellow king cannot be placed. So, for each edge square placement of the blue king, we have 38 choices for the yellow king's position.

If the blue king is placed on an inner square (36 inner squares available), then there are 12 squares adjacent to the blue king where the yellow king cannot be placed. Hence, for each inner square placement of the blue king, we have 37 choices for the yellow king's position.

Therefore, the total number of ways to place the blue king and the yellow king on the chessboard such that they do not attack each other is:

(4 * 41) + (24 * 38) + (36 * 37) = 164 + 912 + 1,332 = 2,408 ways.

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Therefore, the displacement of the wheel during this interval is approximately 141.372 cm.

To find the displacement of the wheel during this interval, we need to determine the distance traveled by a point on the rim of the wheel.

Given:

Radius of the wheel: 45.0 cm

The wheel rolls without slipping

The wheel rolls through one-half of a revolution

Since the wheel rolls without slipping, the distance traveled by a point on the rim of the wheel is equal to the circumference of the wheel for each complete revolution. Therefore, the distance traveled for one-half of a revolution is equal to half the circumference of the wheel.

The circumference of a circle can be calculated using the formula: C = 2πr, where r is the radius of the circle.

Using the given radius of the wheel, we can calculate the circumference:

C = 2π(45.0 cm) ≈ 2π(45.0) cm ≈ 282.743 cm (rounded to three decimal places)

Since the wheel rolls through one-half of a revolution, the displacement is equal to half the circumference of the wheel:

Displacement = 0.5 × 282.743 cm ≈ 141.372 cm (rounded to three decimal places)

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The wheel's displacement is equal to the 282.6 cm that it has covered in its voyage.

We must ascertain the wheel's distance traveled and the displacement's direction.

Since the wheel has completed one-half of a revolution, the distance it has gone is equal to half its circumference. The formula: can be used to determine a circle's circumference:

Circumference = 2 * π * radius

In this case, the radius of the wheel is 45.0 cm. Let's calculate the circumference:

Circumference = 2 * π * 45.0 cm

Circumference ≈ 2 * 3.14 * 45.0 cm

Circumference ≈ 282.6 cm

So, the distance traveled by the wheel is approximately 282.6 cm.

The wheel's displacement is the angular separation between its starting point, where it first makes contact with the ground, and its finishing point, where it stops after rolling through one-half of a rotation. The point of contact with the floor does not move since the wheel is moving without slipping.

Therefore, the wheel's displacement is equal to the 282.6 cm that it has covered in its voyage.

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The points on the curve where the tangent line is horizontal or vertical are (0, π/2) and (2, 3π/2).

To find the points where the tangent line is horizontal or vertical, we need to determine the values of r and θ that satisfy these conditions. First, let's consider the horizontal tangent lines.

A tangent line is horizontal when the derivative of r with respect to θ is equal to zero. Taking the derivative of r = 1 + cos(θ) with respect to θ, we have

dr/dθ = -sin(θ). Setting this equal to zero, we get -sin(θ) = 0, which implies that sin(θ) = 0. The values of θ that satisfy this condition are θ = 0, π, 2π, etc. However, we are given that 0 ≤ θ < 2, so the only valid solution is θ = π. Substituting this back into the equation r = 1 + cos(θ), we find r = 2.

Next, let's consider the vertical tangent lines. A tangent line is vertical when the derivative of θ with respect to r is equal to zero. Taking the derivative of r = 1 + cos(θ) with respect to r, we have

dθ/dr = -sin(θ)/(1 + cos(θ)). Setting this equal to zero, we have -sin(θ) = 0. The values of θ that satisfy this condition are θ = π/2, 3π/2, 5π/2, etc. Again, considering the given range for θ, the valid solution is θ = π/2. Substituting this back into the equation r = 1 + cos(θ), we find r = 0.

Therefore, the points on the curve where the tangent line is horizontal or vertical are (0, π/2) and (2, 3π/2).

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The expected number of students who take advanced placement courses in a random sample of 150 students, in a highly academic suburban school system where 45% of girls and 40% of boys take advanced placement classes, is approximately 127 students.

In a highly academic suburban school system, where 45% of girls and 40% of boys take advanced placement classes, the expected number of students who take advanced placement courses in a random sample of 150 students can be calculated by multiplying the probability of a student being a girl or a boy by the total number of girls and boys in the sample, respectively.

To find the expected number of students who take advanced placement courses in a random sample of 150 students, we first calculate the expected number of girls and boys in the sample.

For girls, the probability of a student being a girl is 45%, so the expected number of girls in the sample is 0.45 multiplied by 150, which gives us 67.5 girls.

For boys, the probability of a student being a boy is 40%, so the expected number of boys in the sample is 0.40 multiplied by 150, which gives us 60 boys.

Next, we add the expected number of girls and boys in the sample to get the total expected number of students who take advanced placement courses. Adding 67.5 girls and 60 boys, we get 127.5 students.

Since we can't have a fraction of a student, we round down the decimal to the nearest whole number. Therefore, the expected number of students who take advanced placement courses in a random sample of 150 students is 127 students.

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The particle's motion is therefore periodic, with a period of[tex]2\pi[/tex], and its path is an ellipse centered at the origin with major axis of length 4 and minor axis of length 3 in case of parametric equations.

The given parametric equations define the motion of a particle in the xy-plane, which are;4 cos(t)3 sin(t), where t represents the time in seconds. Parametric equations. In mathematics, a set of parametric equations is used to describe the coordinates of points that are determined by one or more independent variables that are related to a number of dependent variables by way of a set of equations.

When an independent variable is altered, the values of the dependent variables change accordingly.ParticleIn classical mechanics, a particle refers to a small object that has mass but occupies no space. It is used in kinematics to describe the motion of objects with negligible size by assuming that their mass is concentrated at a point in space. Therefore, a particle in motion refers to a moving point mass.The motion of a particle can be represented using parametric equations. In the given equation [tex]4 cos(t) 3 sin(t)[/tex], the particle is moving in the xy-plane and its path is given by the equation x = [tex]4 cos(t)[/tex] and y = [tex]3 sin(t)[/tex].

The particle's motion is therefore periodic, with a period of [tex]2\pi[/tex], and its path is an ellipse centered at the origin with major axis of length 4 and minor axis of length 3.

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We are given two curves y = x and y = 4x. In order to find the area of the region enclosed by the curves, we need to find the points of intersection between the curves and then integrate the difference of the two curves with respect to x from the leftmost point of intersection to the rightmost point of intersection.

Let us find the point(s) of intersection between the curves. y = x and y = 4x. We equate the two expressions for y to get x. x = 4x ⇒ 3x = 0 ⇒ x = 0.

Thus, the point of intersection is (0,0).

Now we can integrate the difference of the two curves with respect to x from x = 0 to x = 1. A(x) = ∫[0,1](4x - x)dxA(x) = ∫[0,1]3xdxA(x) = (3/2)x² |[0,1]A(x) = (3/2)(1² - 0²)A(x) = (3/2) units².

Therefore, the area of the region enclosed by the curves is 3/2 square units.

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A harsh total represents the summary total of codes from all records in a batch that do not represent a meaningful total.

A hash total is defined as the numerical sum of one or more fields in a file, including data not normally used in calculations, such as account number.

A control total is defined as the an accounting term used for confirming key data such as the number of records and total value of records in an operation.

The harsh total is different from the control total because it has no intrinsic meaning.

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The football factory should produce 10,000 footballs to maximize total profits.

To maximize total profits, the football factory should produce 10,000 footballs.
Here's how we got this answer:
First, let's calculate the total cost of producing x footballs:
Total cost = Fixed cost + (Variable cost per unit x number of units)
Total cost = $20,000 + ($1 x x)
Total cost = $20,000 + $x
Next, let's calculate the revenue earned from selling x footballs:
Revenue = Sale price per unit x number of units
Revenue = ($21 - $0.001x) x x
Revenue = $21x - $0.001x^2
Finally, let's calculate the total profit:
Profit = Revenue - Total cost
Profit = ($21x - $0.001x^2) - ($20,000 + $x)
Profit = $20x - $0.001x^2 - $20,000
To find the number of footballs that maximizes total profit, we need to take the derivative of the profit function and set it equal to 0:
d(Profit)/dx = 20 - 0.002x = 0
x = 10,000
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The only point of intersection where the line has a slope of 2 is (2,3). Therefore, the point of tangency is (2,3).

The line 2y + x = 10 can be rewritten as y = -x/2 + 5. The circle x² + y² - 2x - 4y = 0 can be rewritten as (x-1)² + (y-2)² = 5. The radius of the circle is ✓(5).

To find the point of tangency, we need to find the point where the line and the circle intersect. We can do this by substituting the equation of the line into the equation of the circle. This gives us:

(x-1)² + ((-x/2 + 5)-2)² = 5

(x-1)² + (-x/2 + 3)² = 5

This is a quadratic equation in x. We can solve it by factoring or by using the quadratic formula. The solutions are:

x = 2 or x = -4

When x = 2, y = -x/2 + 5 = 3. When x = -4, y = -x/2 + 5 = 7.

Therefore, the points of intersection are (2,3) and (-4,7).

The only point of intersection where the line has a slope of 2 is (2,3). Therefore, the point of tangency is (2,3).

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Answer:

Not true

Step-by-step explanation:

Absolute value describes the positive distance from 0. Since |3| = 3, then |3| is not greater than 4.

The integral ∫(1/√(2r))dr can be evaluated using basic integral rules. The result is √(2r) + C, where C represents the constant of integration.

To evaluate the integral ∫(1 / √(2r)) dr, we can use the power rule for integration. The power rule states that ∫x^n dx = (x^(n+1)) / (n+1) + C, where C is the constant of integration. In this case, we have x = 2r and n = -1/2.

Applying the power rule, we have:

∫(1 / √(2r)) dr = ∫((2r)^(-1/2)) dr

To integrate, we add 1 to the exponent and divide by the new exponent:

= (2r)^(1/2) / (1/2) + C

Simplifying further, we can rewrite (2r)^(1/2) as √(2r) and (1/2) as 2:

= 2√(2r) + C

So, the final result of the integral is √(2r) + C, where C is the constant of integration.

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The local maxima and minima of a function correspond to points where its derivative changes sign or is equal to zero.

The relationship between the local maxima and minima of a function and its derivative is defined by critical points. A critical point occurs when the derivative of the function is either zero or undefined.

At a critical point, the function may have a local maximum, local minimum, or an inflection point. If the derivative changes sign from positive to negative at a critical point, the function has a local maximum.

Conversely, if the derivative changes sign from negative to positive, the function has a local minimum. When the derivative is zero at a critical point, the function may have a local maximum, local minimum, or a point of inflection.

However, it's important to note that not all critical points correspond to local extrema, as there could be points of inflection or undefined behavior.

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