Consider the initial value problem mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and F(t) = 80 cos(8t) Newtons. Solve the initial value problem. x(t) = help (formulas) = 0? If it 1→[infinity]0 Determine the long-term behavior of the system (steady periodic solution). Is lim x(t): is, enter zero. If not, enter a function that approximates x(t) for very large positive values of t. For very large positive values of t, x(t) ≈ Xsp(t) = help (formulas)

Answers

Answer 1

Therefore, the solution is,x(t) = e⁻²⁺(c₁ cos(6t) + c₂ sin(6t)) + (10/13)cos(8t) - (4/13)sin(8t), where lim x(t) = 0.

Given information:

Consider the initial value problem mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N).

Assume that m = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and F(t) = 80 cos(8t) Newtons.

The given differential equation is,mx" + cx' + kx = F(t)

Substitute the given values in the equation to get,m(²)/(²) + c()/() + kx = 80cos(8t)

When the system is at rest and an external force F(t) is applied, the general solution isx(t) = xh(t) + xp(t)

Here, xh(t) represents the homogeneous solution and xp(t) represents the particular solution.

Find the homogeneous solution of the equation as,m(²)/(²) + c()/() + kx = 0

We can find the characteristic equation as, ms² + cs + k = 0

Substitute the given values, m = 2 kilograms, c = 8 kilograms per second, and k = 80 Newtons per meter.

2s² + 8s + 80 = 0s² + 4s + 40 = 0 On solving the above equation, we get the roots as,s₁, s₂ = -2 ± 6i Since the roots are complex conjugates, the homogeneous solution is given by

               xh(t) = e⁻²⁺)(c₁ cos(6t) + c² sin(6t))

Where, c₁ and c₂ are constants.Find the particular solution: xp(t)To find the particular solution, we assume that the particular solution takes the form of the forcing function

               xp(t) = Acos(8t) + Bsin(8t)xp'(t)

                           = -8Asin(8t) + 8Bcos(8t)xp''(t)

                       = -64Acos(8t) - 64Bsin(8t)

Substitute xp(t), xp'(t), and xp''(t) in the given differential equation,m(²)/(²) + c()/() + kx

        = 80cos(8t)m(-64Acos(8t) - 64Bsin(8t)) + c(-8Asin(8t) + 8Bcos(8t)) + k(Acos(8t) + Bsin(8t))

                                 = 80cos(8t)

Substitute the given values for m, c, and k and equate the coefficients of cos(8t) and sin(8t) to solve for A and B-128A + 8B + 80A = 080B + 8A + 80B = 0

On solving the above equations, we get A = 10/13 and B = -4/13 Therefore, the particular solution is,xp(t) = (10/13)cos(8t) - (4/13)sin(8t)

Therefore, the general solution is,x(t) = xh(t) + xp(t) Substituting xh(t) and xp(t),x(t) = e^(-2t)(c1 cos(6t) + c2 sin(6t)) + (10/13)cos(8t) - (4/13)sin(8t)

The given function, x(t) is 0→[∞]0.The long-term behavior of the system (steady periodic solution) is,x(t) ≈ Xsp(t) = (10/13)cos(8t) - (4/13)sin(8t)

Therefore, the limit of x(t) as t → ∞ is zero. Hence,lim x(t) = 0

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Related Questions

Write the given system of equations as a matrix equation and solve by using inverses. - 8x₁ - x₂ = kq -7x₁. x₂ = K₂ a. What are x, and x₂ when k, = 5 and k₂ = 5? b. What are x, and x₂ when k, 7 and k₂ = 3? X₁ x₂ = c. What are x, and x₂ when k, = 1 and k₂ = -37 X₁ X2

Answers

The solutions of the given equations are:

a. x1 = 10, x2 = -15

b. x1 = -11, x2 = 17

c. x1 = -45, x2 = 296

The given system of equations is as follows:

-8x1 - x2 = kq ----(1)

-7x1 + x2 = k2 ----- (2)

We can write the system of equations in matrix form:

[ -8, -1] [ -7, 1] [x1, x2] = [kq, k2]

Let [ -8, -1] [ -7, 1] be matrix A, [x1, x2] be matrix X, and [kq, k2] be matrix B.

Therefore, A X = B ⇒ X = A-1 B, where A-1 is the inverse of A.

To calculate the inverse of matrix A, we use the following formula:

A-1 = (1 / |A|) [d, -b]

[-c, a]

where |A| is the determinant of matrix A, a, b, c, d are the cofactors of the elements of matrix A.

|A| = ad - bc, and the cofactors of matrix A are:

[a11, a12]

[a21, a22]

a = ( -1 )^2 [a22]

b = (-1)^1 [a21]

c = ( -1 )^1 [a12]

d = ( -1 )^2 [a11]

Now we can find the inverse of matrix A:

A-1 = (1 / |-8 + 7|) [1, 1]

[7, -8]

 = (1 / |-1|) [1, 1]

                   [7, -8]

 = (1 / 1) [1, 1]

               [7, -8]

 = [1, 1]

     [7, -8]

By solving A-1 B, we obtain X.

Now, let's substitute the values of kq and k2 to solve the equation:

a. When kq = k2 = 5, we have:

[1, 1] [7, -8] [5, 5] = X

= [10, -15]

Therefore, x1 = 10 and x2 = -15

b. When kq = 7 and k2 = 3, we have:

[1, 1] [7, -8] [7, 3] = X

= [-11, 17]

Therefore, x1 = -11 and x2 = 17

c. When kq = 1 and k2 = -37, we have:

[1, 1] [7, -8] [1, -37] = X

= [-45, 296]

Therefore, x1 = -45 and x2 = 296

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An 80 N crate is pushed up a ramp as shown in the diagram below. Use the information in the diagram to determine the efficiency of the system. (2 marks) 8.0 m 5.0 m Fin = 200 N

Answers

Answer:

40%

I dont want step by step



Without using a calculator, find all the roots of each equation.

x³+4x²+x-6=0

Answers

The roots of the equation x³ + 4x² + x - 6 = 0 are x = 1, x = -2, and x = -3.

To find the roots of the equation x³ + 4x² + x - 6 = 0 without using a calculator, we can use factoring or synthetic division. By trying out different values for x, we can find that x = 1 is a root of the equation. Dividing the equation by (x - 1) using synthetic division, we obtain:

1 |   1    4    1   -6

   |        1    5    6

   |........................

      1    5    6    0

The result after dividing is the quadratic expression x² + 5x + 6. To find the remaining roots, we can factor this quadratic expression:

x² + 5x + 6

= (x + 2)(x + 3)

Setting each factor equal to zero, we have:

x + 2 = 0 or x + 3 = 0

Solving these equations, we find that x = -2 and x = -3.

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there were 600 tickets for a school market . tickets for adults cost R30 and for students cost R15 .the total amount received from ticket sales was 13 200 .how many student tickets were sold​

Answers

Answer:

Step-by-step explanation:

300

A solid lies between two planes perpendicular to the x-axis at x = 0 and x = 48. The cross-sections by planes perpendicular to the X x-axis are circular disks whose diameters run from the line y = 24

Answers

The solid is a 3D object that lies between two planes perpendicular to the x-axis at x=0 and x=48. The cross-sections by planes perpendicular to the x-axis are circular disks, and the volume of the solid is 6912π cubic units.

To visualize and understand the solid, we can sketch a graph of the cross-sections. Since the cross-sections are circular disks whose diameters run from the line y = 24 to the x-axis, we can draw a circle with diameter 24 at the midpoint of each x-interval. The radius of each circle is r = 12, and the distance between the planes is 48 - 0 = 48. Therefore, the volume of each disk is given by:

V = πr^2h = π(12)^2*dx = 144π*dx

where h is the thickness of the disk, which is equal to dx since the disks are perpendicular to the x-axis. Integrating this expression over the interval [0, 48] gives:

∫[0,48] 144π*dx = 144π*[x]_0^48 = 6912π

Therefore, the volume of the solid is 6912π cubic units.

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*full question: "A solid lies between two planes perpendicular to the x-axis at x = 0 and x = 48. The cross-sections by planes perpendicular to the x-axis are circular disks whose diameters run from the line y = 24 to the top of the solid. Find the volume of the solid."



Determine the possible number of positive real zeros and negative real zeros for each polynomial function given by Descartes' Rule of Signs.

P(x)=6 x⁴-x³+5 x²-x+9

Answers

The polynomial function P(x)=6x⁴-x³+5x²-x+9 has either 2 or 0 positive real zeros and 0 negative real zeros.

Given polynomial is P(x)=6x⁴-x³+5x²-x+9.To determine the number of positive and negative real zeros of the polynomial function P(x), the Descartes' Rule of Signs is applied as follows:

Number of sign changes of the coefficients of the terms of P(x) gives the possible number of positive real zeros of the polynomial function P(x).P(x)=6x⁴-x³+5x²-x+9

The number of sign changes in the above polynomial function is 2.Therefore, P(x) has 2 or 0 positive real zeros.Number of sign changes of the coefficients of the terms of P(-x) gives the possible number of negative real zeros of the polynomial function P(x).

P(-x)=6(-x)⁴-(-x)³+5(-x)²-(-x)+9=6x⁴+x³+5x²+x+9

The number of sign changes in P(-x) is 0.Therefore, P(x) has 0 negative real zeros.So, the possible number of positive real zeros of P(x) is 2 or 0 and the possible number of negative real zeros of P(x) is 0.

Hence, The polynomial function P(x)=6x⁴-x³+5x²-x+9 has either 2 or 0 positive real zeros and 0 negative real zeros.

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Trent filled his container with 21 1/3 ounces of water. Trent then went to the gym 1/3 of the water in the container. How much water was left in the container when he left the gym?
(provide exact responses in mixed fraction form including all steps for solving).

Answers

When Trent left the gym, there were -128/9 ounces of water left in the container.

To solve the problem, let's first find 1/3 of 21 1/3 ounces of water.

1/3 of 21 1/3 can be calculated by multiplying 21 1/3 by 1/3:

(21 1/3) * (1/3) = (64/3) * (1/3) = 64/9

So, 1/3 of the water in the container is 64/9 ounces.

To find the amount of water left in the container, we need to subtract 1/3 of the water from the total amount.

Total amount of water = 21 1/3 ounces

Amount of water taken at the gym = 1/3 of 21 1/3 = 64/9 ounces

Water left in the container = Total amount of water - Amount of water taken at the gym

                                 = 21 1/3 - 64/9

To subtract these fractions, we need to have a common denominator.

The common denominator of 3 and 9 is 9.

Rewriting 21 1/3 with a denominator of 9:

21 1/3 = (63/3) + 1/3 = 63/3 + 1/3 = 64/3

Now, subtracting the fractions:

64/3 - 64/9

To subtract these fractions, they need to have the same denominator. The least common multiple (LCM) of 3 and 9 is 9.

Converting both fractions to have a denominator of 9:

(64/3) * (3/3) = 192/9

64/9 - 192/9 = -128/9

Therefore, when Trent left the gym, there were -128/9 ounces of water left in the container.

Since having a negative amount of water doesn't make sense in this context, we can say that the container was empty when Trent left the gym.

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Find the inverse function of f(x)= 1/x+6. F^−1(x)=

Answers

Given the function f(x)= 1/(x+6) We are to find the inverse function of the given function,

i.e., f^-1(x).To find the inverse of a function, we need to interchange the x and y and solve for y. So, we have:=> x = 1/(y+6) => y+6 = 1/x => y = 1/x - 6

Therefore, the inverse function of f(x) = 1/(x+6) is f^-1(x) = 1/x - 6.

Since the answer requires a 250-word count, we can explain the concept of inverse function.

What is the inverse function? A function which performs the opposite operation of another function is known as the inverse function.

The inverse function of a given function may be obtained by replacing x with y in the given function and solving for y. If the inverse function exists, the domain of the original function is equal to the range of the inverse function and the range of the original function is equal to the domain of the inverse function.

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Give your answer as a fraction in its simplest form. 7/7+ 71/14 = 14 + 14

Answers

Answer:

169 / 14

Step-by-step explanation:

7/1 + 71/14 = 7/1 * 14/14 + 71/14

= 98/14 + 71/14

= (98 + 71) / 14

= 169 / 14

So, the answer is 169 / 14

Special Right Triangles Practice U3L2
1. What is the value of h?
8_/2
2. What are the angle measures of the triangle?
30°, 60°, 90°
3. What is the value of x?
5_/2
4. A courtyard is shaped like a square with 250-ft-long sides.
354.6 ft
5. What is the value of x?
5_/3
6. What is the height of an equilateral triangle with sides that are 12 cm long?
10.4 cm

Answers

The height of an equilateral triangle with sides that are 12 cm long is approximately 10.4 cm.

An equilateral triangle is a triangle whose sides are equal in length. All the angles in an equilateral triangle measure 60 degrees. The height of an equilateral triangle is the line segment that goes from the center of the triangle to the opposite side, perpendicular to that side. In order to find the height of an equilateral triangle, we can use a special right triangle formula: 30-60-90 triangle ratio.

Let's look at the 30-60-90 triangle ratio:
In a 30-60-90 triangle, the length of the side opposite the 30-degree angle is half the length of the hypotenuse, and the length of the side opposite the 60-degree angle is √3 times the length of the side opposite the 30-degree angle. The hypotenuse is twice the length of the side opposite the 30-degree angle.

Using the 30-60-90 triangle ratio, we can find the height of an equilateral triangle as follows:

Since all the sides of an equilateral triangle are equal, the height of the triangle is the length of the side multiplied by √3/2. Therefore, the height of an equilateral triangle with sides that are 12 cm long is:

height = side x √3/2
height = 12 x √3/2
height = 6√3
height ≈ 10.4 cm
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A dib with 24 members is to seledt a committee of six persons. In how many wars can this be done?

Answers

There are 134,596 ways to select a committee of six persons from a dib with 24 members.

To solve this problem, we can use the concept of combinations. A combination is a selection of items without regard to the order. In this case, we want to select six persons from a group of 24.

The formula to calculate the number of combinations is given by:

C(n, r) = n! / (r! * (n-r)!)

Where n is the total number of items and r is the number of items we want to select.

Applying this formula to our problem, we have:

C(24, 6) = 24! / (6! * (24-6)!)

Simplifying this expression, we get:

C(24, 6) = 24! / (6! * 18!)

Now let's calculate the factorial terms:

24! = 24 * 23 * 22 * 21 * 20 * 19 * 18!

6! = 6 * 5 * 4 * 3 * 2 * 1

Substituting these values into the formula, we have:

C(24, 6) = (24 * 23 * 22 * 21 * 20 * 19 * 18!) / (6 * 5 * 4 * 3 * 2 * 1 * 18!)

Simplifying further, we can cancel out the common terms in the numerator and denominator:

C(24, 6) = (24 * 23 * 22 * 21 * 20 * 19) / (6 * 5 * 4 * 3 * 2 * 1)

Calculating the values, we get:

C(24, 6) = 134,596

Therefore, there are 134,596 ways to select a committee of six persons from a dib with 24 members.

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if y = w*y*z and w is growing at 2%, y is growing 4%, and z is
growing at -1%, what is the approximate growth rate of y?

Answers

The approximate growth rate of y is 4% per year

To determine the approximate growth rate of y, we need to consider the growth rates of the variables involved: w, y, and z.

Let's denote the growth rates as follows:

G_w: Growth rate of w

G_y: Growth rate of y

G_z: Growth rate of z

We are given that:

G_w = 2% = 0.02 (per year)

G_y = 4% = 0.04 (per year)

G_z = -1% = -0.01 (per year)

Now, we can use the concept of logarithmic differentiation to approximate the growth rate of y. Taking the natural logarithm of both sides of the equation y = w * y * z, we have:

ln(y) = ln(w) + ln(y) + ln(z)

Differentiating both sides with respect to time (t), we get:

(1/y) * dy/dt = (1/w) * dw/dt + (1/y) * dy/dt + (1/z) * dz/dt

Simplifying the equation, we have:

dy/dt = (1/w) * dw/dt + dy/dt + (1/z) * dz/dt

Substituting the growth rates, we have:

dy/dt = (1/w) * (0.02) + (0.04) + (1/z) * (-0.01)

Since we are interested in the approximate growth rate of y, we can ignore the terms involving dw/dt and dz/dt, as they are small compared to dy/dt. Thus, we can approximate the growth rate of y as:

Approximate growth rate of y = dy/dt = 0.04

Therefore, the approximate growth rate of y is 4% per year.

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Group 5. Show justifying that if A and B are square matrixes that are invertible of order n, A-¹BA ABA-1 then the eigenvalues of I and are the same.

Answers

In conclusion, the eigenvalues of A^(-1)BA and ABA^(-1) are the same as the eigenvalues of B.

To show that the eigenvalues of A^(-1)BA and ABA^(-1) are the same as the eigenvalues of B, we can use the fact that similar matrices have the same eigenvalues.

First, let's consider A^(-1)BA. We know that A and A^(-1) are invertible, which means they are similar matrices. Therefore, A^(-1)BA and B are similar matrices. Since similar matrices have the same eigenvalues, the eigenvalues of A^(-1)BA are the same as the eigenvalues of B.

Next, let's consider ABA^(-1). Again, A and A^(-1) are invertible, so they are similar matrices. This means ABA^(-1) and B are also similar matrices. Therefore, the eigenvalues of ABA^(-1) are the same as the eigenvalues of B.

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Which of the following sets of vectors are bases for R^3?
(a) (3, 1, −4), (2, 5, 6), (1, 4, 8)
(b) (1, 6, 4), (2, 4, −1), (−1, 2, 5)

Answers

The set of vectors (3, 1, −4), (2, 5, 6), (1, 4, 8) forms a basis for R^3.

The set of vectors (1, 6, 4), (2, 4, −1), (−1, 2, 5) forms a basis for R^3.

To determine if a set of vectors forms a basis for R^3, we need to check if the vectors are linearly independent and if they span R^3.

(a) For the set of vectors (3, 1, −4), (2, 5, 6), (1, 4, 8):

To check for linear independence, we can set up the equation:

c1(3, 1, −4) + c2(2, 5, 6) + c3(1, 4, 8) = (0, 0, 0)

Solving this system of equations, we find that c1 = 0, c2 = 0, and c3 = 0, which means the vectors are linearly independent.

To check if they span R^3, we can see if any vector in R^3 can be written as a linear combination of the given vectors. Since the vectors are linearly independent and there are three vectors in total, they span R^3.

(b) For the set of vectors (1, 6, 4), (2, 4, −1), (−1, 2, 5):

To check for linear independence, we set up the equation:

c1(1, 6, 4) + c2(2, 4, −1) + c3(−1, 2, 5) = (0, 0, 0)

Solving this system of equations, we find that c1 = 0, c2 = 0, and c3 = 0, which means the vectors are linearly independent.

To check if they span R^3, we can see if any vector in R^3 can be written as a linear combination of the given vectors. Since the vectors are linearly independent and there are three vectors in total, they span R^3.

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Nicholas is inviting people to his parents' anniversary party and wants
to stay at or below his budget of $3,300 for the food. The cost will be
$52 for each adult's meal and $24 for each child's meal.

Answers

To stay within his budget of $3,300 for the food, Nicholas needs to carefully consider the number of adults and children he invites to the party based on the cost per meal.

To determine the number of adult and child meals Nicholas can afford within his budget of $3,300, we need to set up equations based on the cost of the meals.

Let's assume Nicholas invites x adults and y children to the party.

The cost of adult meals will be $52 multiplied by the number of adults: 52x.

The cost of child meals will be $24 multiplied by the number of children: 24y.

Since Nicholas wants to stay at or below his budget of $3,300, we can set up the following inequality:

52x + 24y ≤ 3300

Now, let's analyze the situation further. Since Nicholas cannot invite a fraction of a person, the number of adults and children must be whole numbers (integers). Additionally, the number of adults and children cannot be negative.

Considering these conditions, we can determine the possible combinations of adults and children that satisfy the inequality. We can start by assuming different values for x (the number of adults) and then calculate the corresponding number of children (y) that would keep the total cost within the budget.

For example, if Nicholas invites 50 adults (x = 50), the maximum number of child meals he can afford can be found by rearranging the inequality:

24y ≤ 3300 - 52x

24y ≤ 3300 - 52(50)

24y ≤ 3300 - 2600

24y ≤ 700

y ≤ 700/24

y ≤ 29.17

Since the number of children must be a whole number, Nicholas can invite a maximum of 29 children.

By exploring different values of x and calculating the corresponding y values, Nicholas can determine the combinations of adults and children that will keep the total cost of meals at or below his budget.

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Note: This is the only question on the search engine

Find/Describe at least three traces and then sketch the 3D

surface.

A) x^2/9 − y^2 + z^2/25 = 1

B) 4x^2 + 2y^2 + z^2 = 4

Answers

A) The equation x^2/9 - y^2 + z^2/25 = 1 represents an elliptical cone. Let's examine some traces:

x = 0:

Substituting x = 0 into the equation, we have -y^2 + z^2/25 = 1. This represents a hyperbola in the yz-plane.

y = 0:

Substituting y = 0 into the equation, we have x^2/9 + z^2/25 = 1. This represents an ellipse in the xz-plane.

z = 0:

Substituting z = 0 into the equation, we have x^2/9 - y^2 = 1. This represents a hyperbola in the xy-plane.

B) The equation 4x^2 + 2y^2 + z^2 = 4 represents an elliptical paraboloid. Let's examine some traces:

x = 0:

Substituting x = 0 into the equation, we have 2y^2 + z^2 = 4. This represents an ellipse in the yz-plane.

y = 0:

Substituting y = 0 into the equation, we have 4x^2 + z^2 = 4. This represents an ellipse in the xz-plane.

z = 0:

Substituting z = 0 into the equation, we have 4x^2 + 2y^2 = 4. This represents an ellipse in the xy-plane.

Unfortunately, as a text-based interface, I am unable to provide a sketch of the 3D surface. I recommend using graphing software or tools to visualize the surfaces.

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Select the correct answer from each drop-down menu.
Consider quadrilateral EFGH on the coordinate grid.


Graph shows a quadrilateral plotted on a coordinate plane. The quadrilateral is at E(minus 4, 1), F(minus 1, 4), G(4, minus 1), and H(1, minus 4).
In quadrilateral EFGH, sides
FG

and
EH

are because they . Sides
EF

and
GH

are . The area of quadrilateral EFGH is closest to square units.
Reset Next

Answers

Answer: 30 square units

Step-by-step explanation: In quadrilateral EFGH, sides FG ― and EH ― are parallel because they have the same slope. Sides EF ― and GH ― are parallel because they have the same slope. The area of quadrilateral EFGH is closest to 30 square units.

Answer the following question about quadrilateral DEFG. Which sides (if any) are congruent? You must show all your work.

Answers

To determine which sides of quadrilateral DEFG are congruent, we need more information about the shape and measurements of the quadrilateral.

Without any additional information, it is not possible to determine the congruency of the sides. A quadrilateral is a polygon with four sides. In general, a quadrilateral can have different side lengths, and without specific measurements or properties provided for DEFG, we cannot determine if any sides are congruent. Congruent sides are sides that have the same length. In a quadrilateral, there are several possibilities for congruent sides, such as:

A parallelogram, where opposite sides are congruent.

A rectangle, where all four sides are congruent.

A rhombus, where all four sides are congruent.

A square, where all four sides are congruent and all angles are right angles. Without information about the shape or properties of DEFG, we cannot make any conclusions about the congruency of its sides. To determine the congruency of sides, we would typically need information such as side lengths, angle measurements, or specific properties of the quadrilateral.

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What are the increasing intervals of the graph -2x^3-3x^2+432x+1

Answers

Answer:

  decreasing: (-∞, -9) ∪ (8, ∞)

  increasing: (-9, 8)

Step-by-step explanation:

You want the intervals where the function f(x) = -2x³ -3x² +432x +1 is increasing and decreasing.

Derivative

The slope of the graph is given by its derivative:

  f'(x) = -6x² -6x +432 = -6(x +1/2)² +433.5

Critical points

The slope is zero where ...

  -6(x +1/2)² = -433.5

  (x +1/2)² = 72.25

  x +1/2 = ±8 1/2

  x = -9, +8

Intervals

The graph will be decreasing for x < -9 and x > 8, since the leading coefficient is negative. It will be increasing between those values:

  decreasing: (-∞, -9) ∪ (8, ∞)

  increasing: (-9, 8)

__

Additional comment

A cubic (or any odd-degree) function with a positive leading coefficient generally increases over its domain, with a possible flat spot or interval of decrease. When the leading coefficient is negative, the function is mostly decreasing, with a possible interval of increase, as here.

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Which of these shapes will tessellate without leaving gaps?

triangle
circle
squares
pentagon

Answers

Answer:

squares

Step-by-step explanation:

A tessellation is a tiling of a plane with shapes in such a way that there are no gaps or overlaps. Squares have the unique property that they can fit together perfectly, edge-to-edge, without any spaces in between. This allows for a seamless tiling pattern that can cover a plane without leaving any gaps or overlaps.

On the other hand, triangles and pentagons cannot tessellate the plane without leaving gaps. Although there are tessellations possible with triangles and pentagons, they require a combination of different shapes to fill the plane without leaving gaps.

A circle, being a curved shape, cannot tessellate a plane without leaving gaps or overlaps. Circles cannot fit together perfectly in a regular pattern that covers the plane without any gaps.

Therefore, squares are the only shape from the ones you mentioned that can tessellate without leaving gaps.

Answer:Triangles, squares and hexagons

Step-by-step explanation:

What is -3/8 + 6/10 =
You need common denominators before you can add or subtract a fraction

Answers

The sum of -3/8 and 6/10 is 9/40.

When adding or subtracting fractions, it is necessary to have a common denominator. The common denominator allows us to combine the fractions by adding or subtracting their numerators while keeping the same denominator.

In this case, we have the fractions -3/8 and 6/10. To find a common denominator, we need to determine the least common multiple (LCM) of the denominators, which are 8 and 10.

The LCM of 8 and 10 is 40. So, we rewrite the fractions with a common denominator of 40:

-3/8 = -15/40 (multiplying the numerator and denominator of -3/8 by 5)

6/10 = 24/40 (multiplying the numerator and denominator of 6/10 by 4)

Now that both fractions have a common denominator of 40, we can add or subtract their numerators:

-15/40 + 24/40 = 9/40

Therefore, the sum of -3/8 and 6/10 is 9/40.

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Simplify each expression. Use positive exponents.

(mg⁵)⁻¹

Answers

The simplified expression for (mg⁵)⁻¹ is 1/(mg⁵), obtained by applying the rule of raising a power to a negative exponent.

To simplify the expression (mg⁵)⁻¹, we can apply the rule of raising a power to a negative exponent.

The rule states that for any non-zero number a, (aⁿ)⁻¹ is equal to 1 divided by aⁿ.

Applying this rule to our expression, we have:

(mg⁵)⁻¹ = 1/(mg⁵)

Therefore, the simplified expression is 1/(mg⁵).

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III. Simplify the following compound proposition using the rules of replacement. (15pts) 2. A = {[(PQ) AR] V¬Q} → (QAR)

Answers

The simplified form of the compound proposition is {(P ∨ ¬Q) ∧ (¬R ∨ ¬Q)} → (Q ∨ R).

To simplify the given compound proposition using the rules of replacement, let's start with the given proposition:

A = {[(P ∧ Q) ∨ R] → ¬Q} → (Q ∧ R)

We can simplify the expression P ∨ Q as equivalent to ¬(¬P ∧ ¬Q) using the rule of replacement. Applying this rule, we can simplify the given proposition as:

A = {[(P ∨ ¬R) ∨ ¬Q] → (Q ∨ R)}

Next, we simplify the expression [(P ∨ ¬R) ∨ ¬Q]. We know that [(P ∨ Q) ∨ R] is equivalent to (P ∨ R) ∧ (Q ∨ R). Therefore, we can simplify [(P ∨ ¬R) ∨ ¬Q] as:

(P ∨ ¬Q) ∧ (¬R ∨ ¬Q)

Putting everything together, we have:

A = {(P ∨ ¬Q) ∧ (¬R ∨ ¬Q)} → (Q ∨ R)

Thus, The compound proposition is written in its simplest form as (P Q) (R Q) (Q R).

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Show that if (an) is a convergent sequence then for, any fixed index p, the sequence (an+p) is also convergent.

Answers

If (an) is a convergent sequence, then for any fixed index p, the sequence (an+p) is also convergent.

To show that if (an) is a convergent sequence, then for any fixed index p, the sequence (an+p) is also convergent, we need to prove that (an+p) has the same limit as (an).

Let's assume that (an) converges to a limit L as n approaches infinity. This can be represented as:

lim (n→∞) an = L

Now, let's consider the sequence (an+p) and examine its behavior as n approaches infinity:

lim (n→∞) (an+p)

Since p is a fixed index, we can substitute k = n + p, which implies n = k - p. As n approaches infinity, k also approaches infinity. Therefore, we can rewrite the above expression as:

lim (k→∞) ak

This represents the limit of the original sequence (an) as k approaches infinity. Since (an) converges to L, we can write:

lim (k→∞) ak = L

Hence, we have shown that if (an) is a convergent sequence, then for any fixed index p, the sequence (an+p) also converges to the same limit L.

This result holds true because shifting the index of a convergent sequence does not affect its convergence behavior. The terms in the sequence (an+p) are simply the terms of (an) shifted by a fixed number of positions.

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inference for a single proportion comparing to a known proportion choose which calculation you desire

Answers

Inference for a single proportion comparing to a known proportion involves calculating a statistical measure to determine if the observed proportion is significantly different from a known proportion.

When conducting inference for a single proportion, we are interested in comparing the proportion of a specific characteristic in a sample to a known proportion in the population. This known proportion can come from previous studies, historical data, or established benchmarks.

To perform this comparison, we use statistical calculations to assess whether the observed proportion in the sample is significantly different from the known proportion. This helps us make inferences about the population based on the sample data.

The calculation used in this type of inference depends on the specific question being addressed and the characteristics of the data. Common statistical tests include the z-test and the chi-squared test, depending on the nature of the data and the sample size.

These tests involve comparing the observed proportion to the known proportion, taking into account factors such as sample size and variability.

By performing the appropriate statistical calculations, we can determine the statistical significance of the difference between the observed and known proportions. This allows us to make conclusions about whether the observed proportion is significantly different from the known proportion, providing valuable insights for decision-making and drawing conclusions about the population of interest.

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5. A shopper in a store is 2.00m from a security mirror and sees his image 12.0m behind the mirror. [ 14 points ] a. What is the focal length of the mirror? [4 points ] b. Is the security mirror concave or convex? Explain how you know. [3 points ] c. What is the magnification of the mirror? [ 4 points ] d. Describe the image of the shopper as real or imaginary, upright or inverted, and enlarged or reduced. [ 3 points] New equations in this chapter : n₁ sin 0₁ = n₂ sin 0₂ sinớc= n2/n1 m || I s' h' S h || = S + = f

Answers

The required answers are:

a) The focal length of the mirror is -2.4 m.

b) The mirror is concave.

c) The magnification of the mirror is 6.00.

d) The image is real, upright, and magnified.

a. To find the focal length of the mirror, we can use the mirror equation:

1/f = 1/s + 1/s'

Where:

f is the focal length of the mirror,

s is the object distance (distance of the shopper from the mirror), and

s' is the image distance (distance of the image from the mirror).

Given:

s = 2.00 m

s' = -12.0 m (negative sign indicates the image is behind the mirror)

Plugging in the values:

1/f = 1/2.00 + 1/(-12.0)

Simplifying the equation:

1/f = -5/12

Taking the reciprocal of both sides:

f = -12/5 = -2.4 m

Therefore, the focal length of the mirror is -2.4 m.

b. The mirror is concave. We know this because the image distance (s') is negative, which indicates that the image is formed on the same side as the object (in this case, behind the mirror). In concave mirrors, the focal length is negative.

c. The magnification of the mirror can be determined using the magnification formula:

m = -s'/s

Given:

s = 2.00 m

s' = -12.0 m

Plugging in the values:

m = -(-12.0) / 2.00 = 6.00

Therefore, the magnification of the mirror is 6.00.

d. Based on the information given, we can describe the image of the shopper as follows:

- The image is real because it is formed by the actual convergence of light rays.

- The image is upright because the magnification is positive.

- The image is enlarged because the magnification is greater than 1 (magnification = 6.00).

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Find the future value of an annuity due of $100 each quarter for 8 1 years at 11%, compounded quarterly. (Round your answer to the nearest cent.) $ 5510.02 X

Answers

The future value of an annuity due of $100 each quarter for 8 years at 11%, compounded quarterly, is $5,510.02.

To calculate the future value of an annuity due, we need to use the formula:

FV = P * [(1 + r)^n - 1] / r

Where:

FV = Future value of the annuity

P = Payment amount

r = Interest rate per period

n = Number of periods

In this case, the payment amount is $100, the interest rate is 11% per year (or 2.75% per quarter, since it is compounded quarterly), and the number of periods is 8 years (or 32 quarters).

Plugging in these values into the formula, we get:

FV = 100 * [(1 + 0.0275)^32 - 1] / 0.0275 ≈ $5,510.02

Therefore, the future value of the annuity due is approximately $5,510.02.

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Solve. Please show your work
3m/(2m-5)-7/(3m+1)=3/2
explain it like you are teaching me please

Answers

Answer:

[tex] \frac{3m}{2m - 5} - \frac{7}{3m + 1} = \frac{3}{2} [/tex]

Multiply both sides by 2(2m - 5)(3m + 1) to clear the fractions:

6m(3m + 1) - 14(2m - 5) = 3(2m - 5)(3m + 1)

Distribute and combine like terms:

18m² + 6m - 28m + 70 = 3(6m² - 13m - 5)

18m² + 6m - 28m + 70 = 18m² - 39m - 15

-22m + 70 = -39m - 15

Add 39m to both sides, and subtract 70 from both sides:

17m = -85

Divide both sides by -17:

m = -5

An investment of $3495.69 earns interest at 7.1% per annum compounded annually for 4 years. At that time the interest rate is changed to 9.3% compounded monthly. How much will the accumulated value be 3 years after the change? The accumulated value is $ (Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed) Find the present value for the amount given in the table. The present value is \$ Gabe opened an RRSP deposit account on December 1,2008 , with a deposit of $2100. He added $2100 on July 1 . 2010 , and $2100 on May 1, 2012. How much is in his account on August 1,2016 , if the deposit earns 5.6% p.a. compounded monthly? The amount in the account is $ (Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.) What sum of money will grow to $5295.05 in three years at 9.1% compounded annually? The sum of money is $ (Round to the nearest cent as needed. Round all intermediate values to six decimal places as needed.) The Continental Bank advertises capital savings at 7.2% compounded annually while TD Canada Trust offers premium savings at 7.05% compounded monthly. Suppose you have $1500 to invest for two years. (a) Which deposit will earn more interest? (b) What is the difference in the amount of interest? (a) The savings account will earn more interest. (b) The difference is $ (Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.)

Answers

1. The accumulated value 3 years after the change will be $6126.23.

2. The amount in the account on August 1, 2016, will be $7892.22.

3. The sum of money needed to grow to $5295.05 in 3 years at 9.1% compounded annually is $4055.84.

4. The difference in the amount of interest earned is $4.27.

How to calculate accumulated value

1The accumulated value after 4 years at 7.1% per annum compounded annually is:

[tex]A = 3495.69 * (1 + 0.071)^4 = 4604.0790[/tex]

After 4 years, the interest rate is changed to 9.3% compounded monthly.

The effective monthly rate is:

[tex]i = (1 + 0.093/12)^12 - 1 = 0.007616[/tex]

After 3 years at this rate, the accumulated value is:

[tex]A = 4604.0790 * (1 + 0.007616)^36 = 6126.2337[/tex]

Therefore, the accumulated value 3 years after the interest rate change is $6126.23.

To calculate present value of the deposits

[tex]FV = 2100 * (1 + 0.056/12)^n[/tex]

The first deposit of $2100 was made in December 2008, which is 11*12 = 132 months before August 2016.

The second deposit of $2100 was made in July 2010, which is 6*12 = 72 months before August 2016.

The third deposit of $2100 was made in May 2012, which is 51*12 = 612 months before August 2016.

Therefore, the present value of the deposits is:

[tex]PV = 2100 * (1 + 0.056/12)^132 + 2100 * (1 + 0.056/12)^72 + 2100 * (1 + 0.056/12)^612 ≈ 7892.22[/tex]

Therefore, the amount in the account on August 1, 2016, is $7892.22.

Let the initial sum be x

[tex]x * (1 + 0.091)^3 = 5295.05[/tex]

Solving for x, we get:

[tex]x = 5295.05 / 1.091^3 ≈ 4055.84[/tex]

Therefore, the sum of money needed to grow to $5295.05 in 3 years at 9.1% compounded annually is $4055.84.

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A shipping company charges a flat rate of $7 for packages weighing five pounds or less, $15 for packages weighing more than five pounds but less than ten pounds, and $22 for packages weighing more than ten pounds. During one hour, the company had 13 packages that totaled $168. The number of packages weighing five pounds or less was three more than those weighing more than ten pounds. The system of equations below represents the situation.

Answers

Answer:

Step-by-step explanation:Let's define the variables:

Let "x" be the number of packages weighing five pounds or less.

Let "y" be the number of packages weighing more than ten pounds.

Based on the given information, we can set up the following equations:

Equation 1: x + y = 13

The total number of packages is 13.

Equation 2: 7x + 15y + 22z = 168

The total cost of the packages is $168.

Equation 3: x = y + 3

The number of packages weighing five pounds or less is three more than those weighing more than ten pounds.

To solve this system of equations, we can use the substitution method or elimination method. Let's use the substitution method here:

From Equation 3, we can rewrite it as:

y = x - 3

Now we substitute this value of y in Equation 1:

x + (x - 3) = 13

2x - 3 = 13

2x = 13 + 3

2x = 16

x = 16/2

x = 8

Substituting the value of x back into Equation 3:

y = x - 3

y = 8 - 3

y = 5

So, we have x = 8 and y = 5.

To find the value of z, we substitute the values of x and y into Equation 2:

7x + 15y + 22z = 168

7(8) + 15(5) + 22z = 168

56 + 75 + 22z = 168

131 + 22z = 168

22z = 168 - 131

22z = 37

z = 37/22

z ≈ 1.68

Therefore, the number of packages weighing five pounds or less is 8, the number of packages weighing more than ten pounds is 5, and the number of packages weighing between five and ten pounds is approximately 1.68.

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