Tutorial Exercise Find the center of mass of the point masses lying on the x-axis. m1 = 9, m2 = 3, m3 = 7 X1 = -5, X2 = 0, X3 = 4 Step 1 Let m; be the mass of the ith element and x; be the position of the ith element. Recall that the center of mass is given by mi xxi x i = 1 n mi i = 1 and n mi x Yi CM = 1 mi IM i = 1 Since all the point masses lie on the x-axis, we know that y = -0.89 X. Submit Skip (you cannot come back) Find Mx, My, and (x,y) for the laminas of uniform density p bounded by the graphs of the equations. y = x, y = 0, x = 4 Mx = = My (x, y) = Need Help? Read It Watch It Talk to a Tutor

Answer:

The formula for the volume of a cylinder is given by:

Volume = π * radius^2 * height

Given that the height of the cylinder is 10 centimeters and the radius is 19 centimeters, we can substitute these values into the formula and use the approximation of π as 3.14:

Volume = 3.14 * (19^2) * 10

Calculating the square of the radius:

Volume = 3.14 * 361 * 10

Multiplying the values:

Volume = 11354 * 10

Volume = 113540 cubic centimeters (rounded to the nearest hundredth)

So, the volume of the cylinder is approximately 113540 cubic centimeters.

The volume of a cylinder is calculated using the formula:

Volume = πr²h

where, π = 3.14

radius = 19 cm

Height = 10 cm

Volume = πr²h

= 3.14 × 19² × 10

= 3.14 × 361 × 10

= 11335.40 cm³

The eigenvalues are λ1 = -3 and λ2 = -4, and the corresponding eigenvectors are:

| 2 | | 1 |

| -3 | | -2 |

The coefficient matrix for the linear system is A = [−36−24], or

| -3 -2 |

| -6 -4 |

We must resolve the characteristic equation det(A - I) = 0 to get the eigenvalues, where I is the identity matrix of the same size as A:

[tex]| -3 -2 | | λ 0 | | -3-λ -2 |[/tex]

| -6 -4 | - | 0 λ | = | -6 -4-λ|

Expanding the determinant and setting it to zero, we get:

(-3-λ)(-4-λ) - (-2)(-6) = 0

λ^2 + 7λ + 12 = 0

(λ+3)(λ+4) = 0

Therefore, the eigenvalues are λ1 = -3 and λ2 = -4.

To find the eigenvectors corresponding to each eigenvalue, we solve the system (A - λI)v = 0, where v is a non-zero vector. For λ1 = -3, we have:

[tex]| -3 -2 | | v1 | | 0 |[/tex]

[tex]| -6 -4 | - | v2 | = | 0 |[/tex]

which simplifies to the equation -3v1 - 2v2 = 0, or v2 = -3/2 v1. Choosing v1 = 2, we get v2 = -3, so the eigenvector corresponding to λ1 is:

| 2 |

| -3 |

For λ2 = -4, we have:

[tex]| -3 -2 | | v1 | | 0 |[/tex]

[tex]| -6 -4 | - | v2 | = | 0 |[/tex]

which simplifies to the equation -4v1 - 2v2 = 0, or v2 = -2v1. Choosing v1 = 1, we get v2 = -2, so the eigenvector corresponding to λ2 is:

| 1 |

| -2 |

Therefore, the eigenvalues are λ1 = -3 and λ2 = -4, and the corresponding eigenvectors are:

| 2 | | 1 |

| -3 | | -2 |

respectively.

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The required answer is free motions of a mass–spring systems are modeled as nonhomogeneous linear false.

The free motions of a mass-spring system are actually modeled as homogeneous linear ordinary differential equations. Nonhomogeneous linear ordinary differential equations can arise when there are external forces or inputs acting on the system.

A differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations, this means that there are no constant terms. The solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term.

Hooke's equation holds (to some extent) in many other situations where an elastic body is deformed, such as wind blowing on a tall building, and a musician plucking a string of a guitar. An elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean.

Hooke's law is only a first-order linear approximation to the real response of springs and other elastic bodies to applied forces. It must eventually fail once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. Many materials will noticeably deviate from Hooke's law well before those elastic limits are reached.
 Free motions of a mass-spring systems are modeled as homogeneous linear ordinary differential equations .  nonhomogeneous. This is because in free motion, there is no external force acting on the system, making the equation homogeneous.

A linear differential equation can be represented as a linear operator acting on y(x) where x is usually the independent variable and y is the dependent variable. Therefore, the general form of a linear homogeneous differential equation .

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The current date using the TODAY function, you can enter the formula "=TODAY()" in cell H5. This will display the current date in the cell.

Switch to the Cost Estimates worksheet. In cell A9, create a formula using the AVERAGE function that calculates the average of the values in the range A5:A7, then copy your formula to cell 09.

cell A10, create a formula using the MAX function that identifies the maximum value in the range A5:A7 and then copy your formula to cell D10. In cell A11, create a formula using the MIN function that identifies the minimum value in the range A5:A7 and then copy your formula to cell 011.

In cell B13, create a formula using the VLOOKUP function that looks up the value from cell A11 in the range A5:B7, returns the value in column 2, and specifies an exact match. Copy the formula to cell E13. Switch to the Profit Projections worksheet. In cell H5, use the TODAY function to insert the current date.

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Answer:

Total Surface Area is 20

Step-by-step explanation:

The formula for surface are with slant heigh is

SA = a^2 + 2×a×l

a = Base Edge (this case 2)

I = Slant Height (this case 4

2^2 + 2(2)(4) = 4+16=20

Answer:8

Step-by-step explanation:5 is 1st term 8 is second term 11 is third term…

The left null space is the null space of the transpose of the matrix, A^T. Since the original matrix is 3x4 and has rank 3, its nullity can be calculated as n - rank = 4 - 3 = 1. Thus, the dimension of the left null space is 1, and it is represented by R^1.

If a 3×4 matrix has rank 3, this means that there are 3 linearly independent columns. Therefore, the column space of the matrix is spanned by these 3 columns. In terms of the matrix itself, we can say that the column space is spanned by the columns corresponding to the pivot positions in the matrix after it has been reduced to row echelon form. So, in this case, we would be looking at the columns corresponding to the 3 pivot positions.

To find the left null space of the matrix, we need to find all vectors x such that A^T x = 0. Since A is a 3×4 matrix, its transpose is a 4×3 matrix. So we are looking for vector x that is in R^4 and satisfies the equation A^T x = 0. The left null space is the set of all such vectors.

To find the left null space, we can use the fact that the left null space is orthogonal to the row space of A. The row space of A is spanned by the rows corresponding to the pivot positions in the matrix after it has been reduced to row echelon form. Since the matrix has rank 3, there are only 3 pivot positions, so the row space has dimension 3.

Therefore, we can find a basis for the left null space by finding a basis for the orthogonal complement of the row space. We can use the Gram-Schmidt process to do this. Start with a basis for the row space, and then orthogonalize it by subtracting the projection onto each previous vector in the basis.

Once we have a basis for the left null space, we can determine its dimension. Since the matrix has 4 columns, the left null space has dimension 4 - rank(A) = 4 - 3 = 1. So the left null space is a one-dimensional subspace of R^4.

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Answer:

13 + 12i

Step-by-step explanation:

3 + 5i + 10 + 7i

First, we need to group it and add it.

3 + 10 because they are numbers.

7i + 5i because they have the same variable.

Next,

= 3 + 10 + 5i + 7i

= 13 + 12i

☆Hope this helps!☆

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The sum of complex numbers (3+5i) + (10+7i) is 13+12i.

To find the sum of the two complex numbers (3+5i) and (10+7i), we add their real parts and imaginary parts separately.

Real part of the sum = Real part of (3+5i) + Real part of (10+7i) = 3 + 10 = 13

Imaginary part of the sum = Imaginary part of (3+5i) + Imaginary part of (10+7i) = 5i + 7i = 12i

Therefore, the sum of the two complex numbers is:

(3+5i) + (10+7i) = 13 + 12i

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The polar coordinates are (4√3, 5π/6). The correct answer is D. (4√3, 5π/6). The other given options are incorrect.

To convert Cartesian coordinates (-6, 2√3) to polar coordinates, we use the formulas:

r = √(x^2 + y^2)
θ = tan^-1 (y/x)

Plugging in the values, we get:

r = √((-6)^2 + (2√3)^2) = √(36 + 12) = 2√13
θ = tan^-1 (2√3/-6) = -π/3

However, since the point is in the second quadrant, we need to add π to the angle, giving us:

θ = -π/3 + π = 2π/3

Therefore, the polar coordinates of (-6, 2√3) can be expressed in two ways:

A. (4 √3, 3 π/4)
B. (3 √3, 3 π/4)

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To evaluate the integral using a linear change of variables, we need to:

(A) Find a transformation: Let u = x + y and v = x - y. The inverse transformation is x = (u + v)/2 and y = (u - v)/2.

(B) Transform the polygon region: The vertices (2, 0), (0, 2), (-2, 0), and (0, -2) transform to (2, 2), (2, -2), (-2, -2), and (-2, 2), forming a rectangular region with u = [-2, 2] and v = [-2, 2].

(C) Compute the Jacobian: J(u, v) = |∂(x, y)/∂(u, v)| = |(1/2, 1/2; 1/2, -1/2)| = 1/2.

(D) Convert the iterated double integral: ∬R e^(x² - y²) dA = ∬_{-2}^2 ∬_{-2}^2 e^((u+v)²/4 - (u-v)²/4) * (1/2) du dv.

(E) The final answer: The integral evaluates to 1/2 ∬_{-2}² ∬_{-2}² e^(uv) du dv = 2π(e⁴ - 1).

The integral evaluates to 2π(e⁴ - 1) using a linear change of variables with the given transformation, transformed region, Jacobian, and converted integral.

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The exact values of P(X = k) for k = 0, 1, 2, ..., 16 are:

P(X = 0) = 1/65536

P(X = 1) = 1/4096

P(X = 2) = 15/8192

P(X = 3) = 455/65536

P(X = 4) = 3003/262144

P(X = 5) = 1001/65536

P(X = 6) = 2002/65536

P(X = 7) = 1716/65536

P(X = 8) = 6435/262144

P(X = 9) = 5005/262144

P(X = 10) = 3003/262144

P(X = 11) = 455/65536

P(X = 12) = 1001/65536

P(X = 13) = 15/8192

P(X = 14) = 1/4096

P(X = 15) = 1/65536

P(X = 16) = 1/65536

The number of possible outcomes when rolling a fair die once is 6, with 3 odd numbers (1, 3, and 5) and 3 even numbers (2, 4, and 6). Therefore, the probability of rolling an odd number is 3/6 = 1/2 and the probability of rolling an even number is also 1/2.

The number of odd numbers that appear in 16 tosses of a fair die is a binomial random variable with parameters n = 16 and p = 1/2. The probability mass function of X, the number of odd numbers, is given by:

P(X = k) = (16 choose k) [tex]*[/tex] (1/2)¹⁶, for k = 0, 1, 2, ..., 16.

To find the exact value of P(X = k), we need to substitute k into this formula and evaluate it. For example:

P(X = 0) = (16 choose 0) [tex]*[/tex] (1/2)¹⁶ = 1/65536

P(X = 1) = (16 choose 1) [tex]*[/tex] (1/2)¹⁶ = 16/65536 = 1/4096

P(X = 2) = (16 choose 2) [tex]*[/tex] (1/2)¹⁶ = 120/65536 = 15/8192

and so on, until

P(X = 16) = (16 choose 16) [tex]*[/tex] (1/2)¹⁶ = 1/65536

Therefore, the exact values of P(X = k) for k = 0, 1, 2, ..., 16 are:

P(X = 0) = 1/65536

P(X = 1) = 1/4096

P(X = 2) = 15/8192

P(X = 3) = 455/65536

P(X = 4) = 3003/262144

P(X = 5) = 1001/65536

P(X = 6) = 2002/65536

P(X = 7) = 1716/65536

P(X = 8) = 6435/262144

P(X = 9) = 5005/262144

P(X = 10) = 3003/262144

P(X = 11) = 455/65536

P(X = 12) = 1001/65536

P(X = 13) = 15/8192

P(X = 14) = 1/4096

P(X = 15) = 1/65536

P(X = 16) = 1/65536

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The probability of spinning a 5 first and then a 1 is:

(1/6) * (1/6) = 1/36

Expressed as a percent, this is:

(1/36) * 100% = 2.78%

So the probability of landing on a 5 and then a 1 is 2.78%

Answer: The given statement "if every column of an augmented matrix contains a pivot then the corresponding system is consistent" is true. This is because when every column of an augmented matrix contains a pivot, it implies that there are no free variables in the system of equations represented by the matrix.

Step-by-step explanation: Since every variable has a pivot in the augmented matrix, there is a unique solution to the system of equations. This is the definition of a consistent system - one that has at least one solution.                                                                                                                  In summary, the statement is true because the presence of a pivot in every column of an augmented matrix guarantees a unique solution to the system of equations, which is the definition of a consistent system.

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The probability that Niamh chooses B after she had the hint is given as follows:

0.30 = 30%.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The sum of all the probabilities is given as follows:

1 = 100%.

Hence, initially, considering that there are four choices, the probability of each is given as follows:

1/4 = 0.25.

A decays by 0.15, while the increase in each of the other probabilities is given as follows:

0.15/3 = 0.05.

Hence the probability of B is given as follows:

p = 0.25 + 0.05

p = 0.3.

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The price at which unitary elasticity occurs is $157.14 per unit.

To find the price at which unitary elasticity occurs, we need to first determine the elasticity of demand with respect to price. The elasticity of demand is defined as the percentage change in quantity demanded divided by the percentage change in price.

Let's rearrange the demand equation to solve for q

q = 86,000 - 550p

We can then take the derivative of q with respect to p:

dq/dp = -550

This tells us that the rate of change of quantity demanded with respect to price is a constant -550. To find the price at which unitary elasticity occurs, we need to find the price where the absolute value of the elasticity is equal to 1.

Using the formula for elasticity of demand

e = (dq/dp) × (p/q)

At unitary elasticity, e = -1, so:

-1 = (dq/dp) × (p/q)

Substituting in the expression for dq/dp and q, we get

-1 = (-550)  (p / (86,000 - 550p))

Simplifying this equation gives

p = $157.14

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The price at which unitary elasticity occurs is $157.14 per unit.

To find the price at which unitary elasticity occurs, we need to first determine the elasticity of demand with respect to price. The elasticity of demand is defined as the percentage change in quantity demanded divided by the percentage change in price.

Let's rearrange the demand equation to solve for q

q = 86,000 - 550p

We can then take the derivative of q with respect to p:

dq/dp = -550

This tells us that the rate of change of quantity demanded with respect to price is a constant -550. To find the price at which unitary elasticity occurs, we need to find the price where the absolute value of the elasticity is equal to 1.

Using the formula for elasticity of demand

e = (dq/dp) × (p/q)

At unitary elasticity, e = -1, so:

-1 = (dq/dp) × (p/q)

Substituting in the expression for dq/dp and q, we get

-1 = (-550)  (p / (86,000 - 550p))

Simplifying this equation gives

p = $157.14

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This text presents information about two exponential functions f and g. Function f passes through the points (-1, 5) and (2, -1.5), and intercepts the x-axis at (1, 0) and the y-axis at (0, 2). Function g is a decreasing exponential function with a y-intercept of 5 and no x-intercept. The text asks to compare the end behavior of these two functions as x approaches negative and positive infinity. End behavior refers to the behavior of the function as x approaches either positive or negative infinity.

The decision variables for this problem are:

This problem has the following constraints:

Production time cannot exceed the available time on each machine:

5x1,1 + 8x2,1 ≤ 600

4x1,2 + 5x2,2 ≤ 600

Production cannot be negative:

x1,1 ≥ 0

x1,2 ≥ 0

x2,1 ≥ 0

x2,2 ≥ 0

Demand must be met for each product:

x1,1 + x1,2 ≥ 120

x2,1 + x2,2 ≥ 150

Demand cannot exceed the maximum demand for each product:

x1,1 + x1,2 ≤ 200

x2,1 + x2,2 ≤ 130

Therefore, this problem has 4 constraints.

The decision variables for this problem are x1,1 (the number of units of product 1 produced on machine 1), x1,2 (the number of units of product 1 produced on machine 2), x2,1 (the number of units of product 2 produced on machine 1), and x2,2 (the number of units of product 2 produced on machine 2).

Therefore, this problem has 4 decision variables.

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False. If wealth has increasing marginal utility for an individual, it implies that the person derives greater satisfaction from each additional unit of wealth.

False. If wealth has increasing marginal utility for an individual, it implies that the person derives greater satisfaction from each additional unit of wealth. However, risk-averse individual typically experiences diminishing marginal utility of wealth, which means they derive less satisfaction from each additional unit of wealth. Risk-averse individuals are more cautious with their decisions, preferring lower-risk options to avoid potential losses .unwilling to take risks or wanting to avoid risks as much as possible

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Note that the histogram showing the age of campers and the frequency of attendance is attached accordingly.

A histogram is a graph that uses rectangles to represent the frequency of numerical data. The vertical axis of a rectangle reflects the distribution frequency of a variable (the quantity or frequency with which that variable appears).

It is used to summarise discrete or continuous data on an interval scale. It is frequently used to depict the key aspects of data distribution in a handy format.

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The value of the car 20 years after it was purchased is approximately $4,100.

Depreciation refers to the decrease in the value of an asset over time due to wear and tear, obsolescence, or other factors. In the context of a car, depreciation means that its value decreases as it is used and ages.

To calculate the value of the car 20 years after it was purchased, we need to find out how many times the value is halved in 20 years. Since 3.5 years is the time it takes for the value to be halved, we can divide 20 by 3.5 to get the number of times the value is halved.

20 / 3.5 = 5.71 (rounded to two decimal places)

So, the value of the car after 20 years would be:

$29,000 / (2^5.71) = $4,090 (rounded to the nearest hundred dollars)

Therefore, the value of the car 20 years after it was purchased is approximately $4,100.

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The correct answer is d. none of these choices.
In a Poisson distribution, the mean is equal to the variance. The median may or may not be equal to the standard deviation, as it depends on the specific values and shape of the distribution.

The Poisson distribution is a random distribution. It gives the probability of an event occurring at any time (k) at a given time or place. The Poisson distribution has only one parameter, the number of events, λ (lambda).

For example, a call center receives an average of 180 calls per hour, 24 hours a day. The call is free; accepting one does not change the outcome of the next coming. The number of calls received per minute follows a Poisson probability distribution with an average of 3: the most common numbers are 2 and 3, but 1 and 4 are also possible with a probability of as little as zero, and the result is very small maybe 10. Another example is the number of radio disturbance events during the observation period.

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Answer:

X° = 72.6459

Step-by-step explanation:

To solve x we must use tan b/c it contain both side,

which is opposite and adjecent

tan ( x°) =16/5

tan ( x°) =16/5tan ( x°) = 3.2

tan ( x°) =16/5tan ( x°) = 3.2X °= tan^-1(3.2)

tan ( x°) =16/5tan ( x°) = 3.2X °= tan^-1(3.2)X° = 72.6459 round to 72.65°

CD is the altitude to side AB of right [tex]\triangle[/tex]ABC, where m[tex]\angle[/tex]ACB = [tex]90^o[/tex] The value of BC is 7.28 units.

Value in math is a concept that describes the magnitude, or size, of a number. It can refer to absolute value, which is the actual number, or it can refer to relative value, which is the number compared to other numbers. Value is important in math because it is used to compare and measure different quantities. For example, in addition and subtraction, the value of the numbers being added or subtracted determines the answer. In multiplication, the value of the factors determines the product. Value is also important for performing calculations, such as finding averages, which requires knowledge of numbers and their relative values.
The given triangle is a right triangle, with ∠acb as the right angle. Using the Pythagorean Theorem, we can find the length of the side BC. The Pythagorean Theorem states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Therefore, BC² = AC² + BD²
Substituting the given values in the equation,
BC² = 52 + (5 1/3)²
Simplifying the equation,
BC² = 25 + 27.69
Therefore, BC² = 52.69
Taking the square root of both sides,
BC = √52.69
Therefore, BC = 7.28 units.
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When writing a rational number as a decimal, the decimal may either terminate or repeat indefinitely.

If the decimal repeats, there is a pattern of digits that repeat after a certain point. To indicate the repeating pattern, a bar is placed over the digits that repeat. This bar is typically placed over the smallest repeating pattern, which may be one or more digits.

For example, in the decimal representation of 1/3, the digit 3 repeats indefinitely, so the number is written as 0.333... with a bar over the 3. In the decimal representation of 2/7, the pattern 142857 repeats indefinitely, so the number is written as 0.285714285714... with a bar over the repeating pattern.

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Complete Question:

Writing Rational Numbers as Repeating Decimals. Highlight the number that repeats.

Let x be a random variable with pdf f(x) = 1/13, 21 P(X > 30) = 0.31.

We are given that X is a random variable with a probability density function (pdf) of f(x) = 1/13 for the interval 21  x  34.

We are asked to find P(X > 30), which means we need to find the probability of the random variable X being greater than 30. To do this, we will calculate the area under the PDF in the interval [30, 34].

Step 1: Determine the width of the interval [30, 34].
Width = 34 - 30 = 4

Step 2: Calculate the area under the PDF in the interval [30, 34].
Since the pdf is a constant value (1/13) within the given interval, we can calculate the area as follows:
Area = f(x) * width
Area = (1/13) * 4

Step 3: Round off the result to the second decimal place.
Area ≈ 0.31 (rounded to two decimal places)

So, P(X > 30) ≈ 0.31.

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The probability that xy = 36 is 3/100.

Since x and y are discrete uniform random variables over {1,2,3,4,5,6,7,8,9,10}, we have:

P(x = i) = 1/10 for i = 1,2,...,10

P(y = j) = 1/10 for j = 1,2,...,10

We need to find P(xy = 36), which means that xy = 36. Since x and y can only take on integer values between 1 and 10, the only possible pairs of (x,y) that satisfy xy = 36 are (6,6) and (9,4) (or (4,9)).

Therefore:

P(xy = 36) = P((x=6) and (y=6)) + P((x=9) and (y=4)) + P((x=4) and (y=9))

= P(x=6) * P(y=6) + P(x=9) * P(y=4) + P(x=4) * P(y=9)

= (1/10)(1/10) + (1/10)(1/10) + (1/10)*(1/10)

= 3/100

So the probability that xy = 36 is 3/100.

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This formula will give you the correct t statistic for comparing the means of two independent samples without assuming equal variances

To calculate the t statistic for the difference between means of independent samples without pooling, you can use the following formula:

t = (xbar1 - xbar2) / √[(s₁² / n₁) + (s₂² / n₂)]

Here,
- xbar1 and xbar2 are the sample means of the two groups,
- s₁² and s₂² are the sample variances for the two groups,
- n₁ and n₂ are the sample sizes for the two groups.

This formula will give you the correct t statistic for comparing the means of two independent samples without assuming equal variances (i.e., without pooling).

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the rock hits the ground after 25/8 seconds or approximately 3.125 seconds.

How to solve the question?

To find when the rock hits the ground, we need to determine the value of x when h equals zero, since at that time the height of the rock will be at ground level.

Setting h=0, we get:

0 = -16x² + 25

Solving for x, we can use the quadratic formula:

x = (-b ± √(b²-4ac))/2a

where a = -16, b = 0, and c = 25.

Plugging in these values, we get:

x = (-0 ± √(0²-4(-16)(25)))/2(-16)

Simplifying:

x = ±√(625)/8

x = ±25/8

Since time cannot be negative, we take the positive value:

x = 25/8

Therefore, the rock hits the ground after 25/8 seconds or approximately 3.125 seconds.

We can also verify our result by graphing the function h= -16x² + 25 and observing where the graph crosses the x-axis, which represents the ground level.

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The limit of the sequence as n approaches infinity is -1. Since the limit exists and is finite, the sequence a_n = cos(n*pi/(n+1)) converges. The limit is -1.

To determine whether the sequence, [tex]an= cos(n*pi/n+1)[/tex] converges or diverges, we first note that[tex]n*pi/n+1 = n*[/tex](pi/(n+1)). As n approaches infinity, pi/(n+1) approaches zero. Thus, we can rewrite the sequence as[tex]an = cos(n*(pi/(n+1)))[/tex]

We know that the cosine function oscillates between -1 and 1, and as n gets larger, the argument [tex]n*(pi/(n+1))[/tex] becomes more and more dense in the interval[tex][0, pi][/tex]. Thus, we can say that the sequence oscillates between -1 and 1 infinitely many times as n approaches infinity.

Therefore, the sequence diverges as it does not approach a single limit value.
To determine if the sequence [tex]a_n = cos(n*pi/(n+1))[/tex]converges or diverges, we need to find the limit as n approaches infinity.

Step 1: Write down the sequence formula:
[tex]a_n = cos(n*pi/(n+1))[/tex]

Step 2: Calculate the limit of the sequence as n approaches infinity:
[tex]lim (n → ∞) cos(n*pi/(n+1))[/tex]

Step 3: Analyze the argument inside the cosine function:
As n approaches infinity, the fraction n/(n+1) approaches 1. Therefore, the argument inside the cosine function [tex](n*pi/(n+1))[/tex] approaches pi.

Step 4: Calculate the limit of the cosine function:
cos(pi) = -1

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The new position of the rectangle after the 90-degree clockwise rotation will be A'B'C'D' shown in figure.

A translation is a geometric transformation in which each point in a figure or space moves in the same direction.

If a point A (h, k) is rotated about the origin through 90° in clockwise direction. So, the new position will become A' (k, -h).

After rotating the rectangle ABCD 90° clockwise about the origin, the new position of the rectangle will be A'B'C'D'. The coordinates of the new vertices can be found by applying a 90° clockwise rotation transformation to each of the original coordinates of the vertices.

Coordinates of original vertices are A (1, 5), B (4, 5), C (1, 2), and D (4, 2) the coordinates of the new vertices A'B'C'D' can be calculated as follows:

A (1, 5)  ⇒  A' (5, -1)

B (4, 5)  ⇒ B' (5, -4)

C (1, 2)   ⇒ C' (2, -1)

D (4, 2)   ⇒ D' (2, -4)

So, the new position of the rectangle after the 90-degree clockwise rotation will be A'B'C'D'.

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