The height, h, in feet of the tip of the hour hand of a wall clock varies from 9 feet to 10 feet. Which of the following equations can be used to model the height as a function of time, t, in hours? Assume that the time at t = 0 is 12:00 a.m.
h = 0.5 cosine (StartFraction pi Over 12 EndFraction t) + 9.5
h = 0.5 cosine (StartFraction pi Over 6 EndFraction t) + 9.5
h = cosine (StartFraction pi Over 12 EndFraction t) + 9
h = cosine (StartFraction pi Over 6 EndFraction t) + 9

In response to the question, we may say that  Therefore, the inverse of the given matrix is: [ 0.0 1.5 -1.0 ], [ 1.0 -2.0 2.0 ] and [-1.0 1.5 -2.0 ]

An arrangement of integers in rows and columns is known as a matrix. learns about matrices' components and dimensions and first presents them. A matrix is a rectangular grid with integers arranged in rows and columns. For example, Matrix A contains two rows and three columns. The origin of its moniker is due to its single row and 1 n row matrix arrangement. For example, a row matrix of order 1 by 4 is A = [1 2 4 5]. The row matrix P = [-4 -21 -17] of order 1-by-cubic is another example.

Let's begin by calculating the determinant of the matrix:

[tex]det(A) = 1(41 - 30) - 2(40 - 20) + 3(10 - 21)\\det(A) = 4 - 0 - 6\\det(A) = -2\\[/tex]

Since the determinant is not zero, an inverse exists. Let's find the adjugate matrix:

[tex]adj(A) = [dof(A)]^T\\[/tex]

where dof(A) is the matrix of cofactors of A.

[tex]dof(A) = [(-1)^(1+1)0, (-1)^(1+2)2, (-1)^(1+3)(-2),\\(-1)^(2+1)(-3), (-1)^(2+2)4, (-1)^(2+3)(-3),\\(-1)^(3+1)2, (-1)^(3+2)(-4), (-1)^(3+3)*4]\\dof(A) = [0, -2, 2, 3, 4, -3, 2, -4, 4]\\[/tex]

Therefore,

[tex]adj(A) = [0 -3 2; -2 4 -4; 2 -3 4]\\[/tex]

Finally, we can find the inverse matrix by multiplying the adjugate matrix by the reciprocal of the determinant:

[tex]A^(-1) = (1/det(A)) * adj(A)\\A^(-1) = (-1/2) * [0 -3 2; -2 4 -4; 2 -3 4]\\A^(-1) = [0.0 1.5 -1.0; 1.0 -2.0 2.0; -1.0 1.5 -2.0]\\[/tex]

Therefore, the inverse of the given matrix is:

[ 0.0 1.5 -1.0 ]

[ 1.0 -2.0 2.0 ]

[-1.0 1.5 -2.0 ]

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Answer

[ 1.25   -0.75

-0.5      0.5 ]

Step-by-step explanation:

Answer:

  (i)  25.83 m

  (ii)  178.9 m

  (iii) 4.592 s

Step-by-step explanation:

You want the greatest height, the range, and the time of flight of a particle launched at 45 m/s at an angle of arcsin(0.5) from the horizontal.

The vertical speed is the speed in the direction of launch multiplied by the sine of the angle from the horizontal. Here, it is ...

  (45 m/s)(sin(arcsin(0.5))) = 22.5 m/s

The maximum height is the height at which all of the initial vertical kinetic energy is converted to potential energy:

  h = v²/(2g)

  h = (22.5 m/s)²/(2·9.8 m/s²) ≈ 25.83

The maximum height is about 25.83 meters.

The time of flight is twice the time it takes to reach the maximum height. That time is the height divided by the average vertical speed, which average is half the initial vertical speed.

  T = 2 × (25.83 m)/(11.25 m/s) ≈ 4.592 s

The time of flight is about 4.592 seconds.

The range is the product of the time of flight and the horizontal speed. The horizontal speed is the speed in the launch direction multiplied by the cosine of the angle from the horizontal.

  cos(α) = √(1 -sin(α)²) = √(1 -(1/2)²) = √(3/4)

  R = (45 m/s)√(3/4)·4.592 s = 178.949 m

The horizontal range of the particle is about 178.9 meters.

Therefore , the solution of the given problem of angle comes out to be the particle's horizontal range is approximately 178.9 metres.

A tilt is a form in Euclidean geometry made up of two sides, but rather cylindrical faces, that divide at the barrier's centre and apex. At their junctions, two rays may combine to form an angle. Angle is another outcome of two entities interacting. Dihedral shapes best describe them. A two-dimensional curve can be created by arranging two line beams in various configurations at their ends.

Here,

Lateral momentum

The launch speed is compounded by the sine of the angle from the horizontal to obtain the vertical speed. This is it...

=> 22.5 m/s = (45 m/s)(sin(arcsin(0.5)))

I The tallest person

The height at which all of the original vertical kinetic energy is transformed into potential energy is known as the maximum height.

=> h = v²/(2g) (2g)

=> h = (22.5 m/s)²/(2·9.8 m/s²) ≈ 25.83

The highest point is roughly 25.83 metres high.

(ii) Flight time

The duration of flying is twice as long as the time required to ascend to the highest point. The height divided by the average vertical speed, which is half of the original vertical speed, yields that time.

=> T = 2 × (25.83 m)/(11.25 m/s) ≈ 4.592 s

The journey lasts for approximately 4.592 seconds.

=> (iii) Scope

The range is calculated by multiplying the flight duration by the horizontal speed. The cosine of the angle from the horizontal compounded by the speed in the direction of launch gives the horizontal speed.

=>Cos(x) = (1 - sin(x)2) = (1 -(1/2)2) = Cos(x) = (3/4)

=> R = (45 m/s)√(3/4)·4.592 s = 178.949 m

The particle's horizontal range is approximately 178.9 metres.

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Volume of Sphere B is 8 times greater than the volume of Sphere A

What does sphere refer to?

A sphere is symmetrical and round in form. All of its surface points are equally spaced from the centre, making it a three-dimensional solid. Depending on the radius, it has a surface area and a volume. No faces, corners, or edges exist on it.

                            It is sometimes referred to as the second cousin of a circle, the sphere is a three-dimensional form. Round, without boundaries, and solid in shape, a sphere is spherical. Examples of spherical shapes include a playing ball, a balloon, and even light bulbs.

Sphere A:  V = 4π (2³)/3 = 32π/3

Sphere B:  V = 4π (4³)/3 = 4π(64)/3 = 256π/3

volume of Sphere B / volume of Sphere A =  (256π/3)/(32π/3) = 8

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Answer: B - A kitten who is just born is predicted to weigh 4.05 ounces

Step-by-step explanation:

Therefore , the solution of the given problem of expressions comes out to be  x ≈ 0.6309.

Calculations that combine joining, removing, and random subdivision must be done with ever-changing factors. They could accomplish the following if they united: An programme, some data, and a mathematical problem. Formulas, components, and arithmetic operations like adds, subtractions, mistakes, and groupings can all be found in a declaration of truth.

Here,

We can rewrite this equation as follows in order to answer it using the properties of logarithms:

We can now use a substitution to simplify the problem. Let y = 8^x.

By finding x, we obtain:

=>  y = (6³ˣ))/8

This equation can be made simpler by applying the formula 8 = 23:

=> y = (6³ˣ))/(2³)

=> y = (3³ˣ)/(2³)

=> 6³ˣ - (3³ˣ)/(2³) = 0

=> (6³ˣ)*(2³) - (3³ˣ) = 0

=> (2³)/(3³ˣ) - 1 = 0

=> log((2³)/(3³ˣ)) = log(1)

=> log(2³) - log(3³ˣ) = 0

Simplifying, we get:

=> 3log(2) - 3xlog(3) = 0

By multiplying both parts by 3log(3), we obtain:

=> x = (log(2))/(log(3))

Using a calculator, we can calculate this expression to four decimal places:

=>  x ≈ 0.6309

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Answer:

4                      [Yes, really]

Step-by-step explanation:

(8/64)^-2/3      [Wow]

(1/8)^-2/3         [Simplify 8/64 to 1/8]]

(8^-1)^-2/3      [Reformat (1/8) to 8^-1]

8^(-1*(-2/3))      [Rule of exponents: When an exponent is raised to another exponent,  multiply the exponents.]

8^(2/3)             [Simplify (-1(-2/3)) to (2/3)]

64^(1/3)            [8^(2/3) means square the 8 and then take the cube root]

4                      [The cube root of 64 is 4 (4*4*4 = 64]

Hence, in response to the provided question, we can say that As a result,  arithmetic progression the usual difference is d = -30.

The difference between subsequent terms in a string will never vary, and this is referred like an arithmetic progression. The sequences 5, 7, 9, 11, 13, through 15 are examples of arithmetic progressions with a tolerance of 2. Arithmetic progression (A.P.) is a progression that has a set tolerance between any 2 consecutive values. Numerical progression may fall into one of two main types: finite-length arithmetic series A finite geometry progression is a series with just a fixed number of terms. The series' terms can be used to calculate the early, late, leniency, and number of term.

A) The sequence is, indeed, arithmetic.

We can find the common difference by subtracting any two consecutive terms:

109 - 9 = 100

209 - 109 = 100

309 - 209 = 100

As a result, the typical difference is d = 100.

A) The sequence is, indeed, arithmetic.

We can find the common difference by subtracting any two consecutive terms:

-32 - (-2) = -30 \s-62 - (-32) = -30 \s-92 - (-62) = -30

As a result, the usual difference is d = -30.

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Answer:

[tex]32x-16[/tex]

Step-by-step explanation:

[tex](2x+3)^{2} - (2x-5)^{2}[/tex]

expand the equation.

[tex](2x+3)(2x+3) - (2x-5)(2x-5)[/tex]
then distribute.
[tex](4x^{2} +6x + 6x+9) - (4x^2 - 10x - 10x + 25)[/tex]

finally simplify the equation.

= [tex]32x-16[/tex]

Answer:

Starting with the expression n² - 2 - (n - 2)², we can simplify it by expanding the second term using the formula for the square of a binomial:

n² - 2 - (n - 2)² = n² - 2 - (n² - 4n + 4)

Next, we can combine like terms:

n² - 2 - (n² - 4n + 4) = n² - n² + 4n - 6

Simplifying further, we get:

n² - 2 - (n - 2)² = 4n - 6

To prove that this expression is always even, we can show that it can be written in the form 2k, where k is an integer.

So, let's rewrite the expression as:

4n - 6 = 2(2n - 3)

Since 2n - 3 is an integer (because n is an integer greater than 1), we have shown that 4n - 6 can be written in the form 2k, where k is an integer.

Therefore, we have proved algebraically that n² - 2 - (n - 2)² is always an even number.

The sum of the perimeters of all squares is 80 ft.

What is perimeter?

Perimeter is the distance around the outer boundary of a two-dimensional shape, such as a rectangle, square, triangle, or circle. It is the total length of all the sides of the shape.

Starting with a square with a perimeter of 40 ft, we can find the length of one side by dividing the perimeter by 4:

[tex]\frac{40ft}{4} = 10ft[/tex]

So, the original square has a side length of 10 ft.

Connecting the midpoints of the sides of this square creates a new square with half the side length of the original square. Each side of this new square is the hypotenuse of a right triangle with legs equal to half the side length of the original square (i.e., 5 ft).

Using the Pythagorean theorem, we can find the length of each side of the new square:

[tex]s^2 = (5 ft)^2 + (5 ft)^2[/tex]

[tex]s^2 = 50 ft^2[/tex]

[tex]s = \sqrt{50}ft[/tex]

So, the second square has a side length of [tex]\sqrt{50}ft[/tex], which is approximately 7.07 ft.

Continuing this process, each subsequent square will have half the side length of the previous square. Therefore, the side length of the third square will be half the side length of the second square, which is:

[tex]\frac{1}{2}[/tex] * [tex]\sqrt{50}ft[/tex] = [tex]\sqrt{25}ft[/tex] = 5 ft.

The side length of the fourth square will be half the side length of the third square, which is: [tex]\frac{1}{2}[/tex] * 5 ft = 2.5 ft. And so on.

The perimeters of the squares are:

The perimeter of the first square is 40 ft.

The perimeter of the second square is 4 * [tex]\sqrt{50}ft[/tex], which is approximately 28.28 ft.

The perimeter of the third square is 4 * 5 ft = 20 ft.

The perimeter of the fourth square is 4 * 2.5 ft = 10 ft.

We can see that the perimeter of each subsequent square is half the perimeter of the previous square. So, the series for the perimeters of the squares is a geometric series with a first term of 40 ft and a common ratio of 1/2.

Using the formula for the sum of an infinite geometric series, we can find the sum of the perimeters of all squares:

S = a / (1 - r)

where a is the first term (40 ft) and r is the common ratio (1/2). Plugging in these values, we get:

S = 40 ft / (1 - 1/2)

S = 80 ft

Therefore, the sum of the perimeters of all squares is 80 ft.

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Answer:

33

Step-by-step explanation:

First, you multiply 3 into the other equation (remember, anytime there is a parentheses by each like there is shown, it will be multiplication.) So -3 * -7 is 21 because a negative and a negative is a positive. Then -3 * -4 which is 12. Add 21 and 12 together and you get 33!

I hoped this helped <3

Answer:

Given:

Initial velocity, u = 0 (since the car is stopped)

Final velocity, v = 80 m/s

Time taken, t = 5 s

Required:

Acceleration, a

Equation:

We can use the following equation to calculate the acceleration:

a = (v - u) / t

where u is the initial velocity, v is the final velocity, and t is the time taken.

Solution:

Substituting the given values, we get:

a = (80 - 0) / 5

a = 16 m/s²

Answer:

The acceleration of the car is 16 m/s².

(^2 is just another version of ²)

Hope this helped! Sorry if it didn't. If you need more help, ask me! :]

A system of inequalities to represent the constraints of this problem are x ≥ 0 and y ≥ 0.

A graph of the system of inequalities is shown on the coordinate plane below.

In order to write a system of linear inequalities to describe this situation, we would assign variables to the number of HD Big View television produced in one day and number of Mega Tele box television produced in one day respectively, and then translate the word problem into algebraic equation as follows:

Since the HD Big View television takes 2 person-hours to make and the Mega TeleBox television takes 3 person-hours to make, a linear equation to describe this situation is given by:

2x + 3y = 192.

Additionally, TVs4U’s total manufacturing capacity is 72 televisions per day;

x + y = 72

For the constraints, we have the following system of linear inequalities:

x ≥ 0.

y ≥ 0.

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The expressions for the dimensions of the rectangular composite figure, indicates;

a) The simplified expression for the shaded region is; (3·x + 36) cm²

b) x = 17

c) Please find attached the drawing of the figure, showing the dimensions, created with MS Word

d) The area of the shaded region cannot be 39 cm², because that will make the length of the small rectangle -1 cm.

A composite figure is a figure that comprises of two or more simpler figures.  

The figure can be considered as a composite figure, comprising of a small rectangular unshaded region within a shaded larger rectangle.

The area of the shaded region is the difference between the area of the large rectangle and the area of the unshaded rectangle, which can be found as follows;

Area of a rectangle = Length × Breadth

Length of the large rectangle = (x + 5) cm

Breadth of the large rectangle = 6 cm

Area of large rectangle = 6 cm × (x + 5) cm = (6·x + 30) cm²

Area of the unshaded area = 3 cm × (x - 2) cm = (3·x - 6) cm²

Area of the shaded region, A, is therefore;

A = (6·x + 30) cm² - (3·x - 6) cm² = (3·x + 36) cm²

b) The shaded area = 87 cm², therefore;

(3·x + 36) = 87

3·x = 87 - 36 = 51

x = 51/3 = 17

c) The dimensions of the figures are therefore;

x = 17

Please find attached the drawing of the shape created with MS Word

d) The area of the shaded region is; (3·x + 36)

Therefore, if the area of the shaded region is 39, we get;

(3·x + 36) = 39

x = (39 - 36)/3 = 1

x = 1

The length of the unshaded region is; (x - 2)

The length of the unshaded region becomes; (1 - 2) = -1, which is not a natural number and therefore, not possible

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The change in the amount of shore after 7 years and 4 months is -5.4975 feet.

First, let us convert the values into the same quantities

7 1/3 years = 7 + 1/3 years = 7.33 years

The change in shore after 7 years and 4 months can be calculated as

Change in amount of shore = (average change per year) x (number of years)

Change in amount of shore = (-3/4 foot/year) x (7.33 years)

Change in amount of shore = -5.4975 feet.

Therefore, the change in the amount of shore after 7 years and 4 months is approximately -5.4975 feet.

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Therefore , the solution of the given problem of unitary comes out to be

option A is correct option.

The task can be finished by combining the data expression obtained to use this nanosection methodology with in all supplemental data from two people who employed a particular variable tactic. In plain English, this indicates that, if the desired result occurs, either the stated individual will be known or, in fact, the colour by both enormous processes would be skipped. A refundable fee of Rupees ($1.01) may be necessary for forty pens.

Here,

Given :

there is a relationship between a person’s age and their preference for a particular style of dress.

So,

The question can be included in the interview

and so,

option A. The questions are acceptable because the researcher is studying the relationship between age and preferred dress style.

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A. The questions are acceptable because the researcher is studying the relationship between age and preferred dress style.

What is probability?

Probability is a branch of mathematics that deals with the study of random events and the likelihood of their occurrence.

conducting a research study, it is important to carefully consider the questions being asked in the survey to ensure that they are relevant to the research question and that they do not cause any harm or offense to the participants. In the case of the clothing designer's survey, asking about age and dress size is relevant to the study's goal of investigating the relationship between age and preferred dress style. Therefore, including these questions in the survey is acceptable.

However, it is also important to consider the potential concerns or discomfort that some participants may have about answering these questions. To address these concerns, the survey questions should be worded in a respectful and non-intrusive manner.

Therefore, A. The questions are acceptable because the researcher is studying the relationship between age and preferred dress style.

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To determine which company would give Amy a stable long-term investment, we need to compare the standard deviations of the stock prices for both companies. A lower standard deviation indicates that the stock price is more stable and has less variability.

From the given information, we can see that the standard deviation of stock prices for Garden Statues Express is $1.05, while the standard deviation of stock prices for Masterful Pocketwatches is $9.83. This means that Garden Statues Express has a much lower standard deviation, and thus a more stable stock price, compared to Masterful Pocketwatches.

Therefore, if Amy is looking for a stable long-term investment, she should consider investing in Garden Statues Express.

By answering the above question, we may state that As a result, there equation were 108 scouts who attended the jamboree.

A math equation is a mechanism for connecting two claims by using the equals sign (=) to indicate equivalence. A mathematical statement that establishes the equivalence of two mathematical expressions is known as an equation in algebra. The equal symbol, for example, splits the numbers 3x + 5 and 14. A mathematical formula can be used to describe the relationship between two phrases written on opposite sides of a letter. The logo and programme are frequently the same. 2x - 4 Equals 2, for example.  

Because each tin is divided by four scouts, the number of fruit tins necessary will be one-fourth of the number of scouts. As a result, the quantity of tins of fruit required is x/4.

The total number of tins opened, according to the query, is 117. Based on the preceding data, we can construct the following equation:

x/2 + x/3 + x/4 = 117

To remove the denominators, multiply both sides by 12 (the LCM of 2, 3, and 4).

6x + 4x + 3x = 1404

When we simplify, we get:

13x = 1404

By multiplying both sides by 13, we get:

x = 108

As a result, there were 108 scouts who attended the jamboree.

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A2AGE, A2HT, A2WGT, A2FTL, A2HC, A2CC, A2NC, A2WC, A2SW, A2SH, A2SKH, A2SEH, A2NHT, A2PPD, A2ARM, A2WH, A2GND

The ANSUR II study was conducted in 2012 and consisted of body measurements from 6068 U.S. Army personnel. In the data set, AGE is in years, WEIGHT is in kilograms (kg), GENDER is identified as 1=male and 0=female, WRITING HAND is 1=right and 2=left and 3=both, and the other body measurements are in millimeters (mm). The following data list names are provided for TI-83/84: A2AGE, A2HT, A2WGT, A2FTL, A2HC, A2CC, A2NC, A2WC, A2SW, A2SH, A2SKH, A2SEH, A2NHT, A2PPD, A2ARM, A2WH, A2GND.

The first five rows of the data set are provided here for reference:

AGE HEIGHT WEIGHT FOOT LENGTH HEAD CIRC CHEST CIRC SHOULDER WIDTH SITTING HT ARM SPAN WRITING HAND GENDER (1=M)
41 1776 81.5 273 583 1074 493 928 1782 1 1
35 1702 72.6 263 568 1021 479 884 1745 2 1
42 1735 92.9 270 573 1120 544 917 1867 2 1
31 1655 79.4 267 576 1114 518 903 1708 1 1
21 1914 94.6 305 566 1048 524 919 2035 1 1

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Answer:

range = 60

Step-by-step explanation:

the range of the data set is the difference between the largest and smallest values in the set.

the largest value = 140 ( denoted by the line on furthest right )

the smallest value = 80 ( denoted by line on left )

range = 140 - 80 = 60

The identity of cos (2π-x) = -cosx

Trigonometric Identities are the equalities that involve trigonometry functions and holds true for all the values of variables given in the equation.

for example;

cos 2 θ + sin 2 θ = 1

1 + cot 2 θ = csc 2 θ

1 + tan 2 θ = sec 2 θ

For cos(2π -x)

π = 180°

2π = 2× 180

= 360°

Therefore cos ( 360-x) will have the identity -cos x.

This is because 360° has an equivalent as 0 in trigonometry.

Therefore cos( 0-x)

= -cosx

therefore the identity of cos (2π-x) is -cos(x).

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Answer:

the mean is 13

Step-by-step explanation:

the mean is all the numbers added (130) divided the number of numbers in the set(10)

In the case of the example given, the finance charge is $5.32, which is the amount of money that will be charged for borrowing the money for the given period.

Finance charges refer to the fees and interest associated with borrowing money or making a purchase on credit. Finance charges are usually expressed as a percentage of the total amount borrowed or purchased, and they are calculated based on the interest rate and the length of the loan term. In the case of the example given, the finance charge is $5.32, which is the amount of money that will be charged for borrowing the money for the given period. The finance charge is calculated by multiplying the interest rate by the amount borrowed and then dividing it by the total number of days in the loan term.

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Answer:

angle=80 degrees

Step-by-step explanation:

if we know the arc is 160 degrees than we know we want to find the interior angle, then we must divide 160/2 to get the measure of the indicated angle.

The equation that represents the situation is y = x + 6.

An equation is a mathematical statement that shows the equality between two expressions. It consists of two sides separated by an equal sign (=) and can involve numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.

The equation that represents the situation is B. y = x + 6.

The problem states that the total cost, y, is the book price, x, plus a $6 shipping fee. We can express this as:

y = x + 6

Option A, y = 6x, suggests that the total cost is only determined by the book price multiplied by 6, which is not what the problem states.

Option C, y = , is not a valid equation as it does not include any terms to determine the value of y.

Option D, y = x - 6, suggests that the total cost is the book price minus $6, which is not what the problem states either.

Therefore, the equation that represents the situation is y = x + 6.

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When these two numbers are subtracted, the difference is 2 3/4, which is the total change in the volume of water in the birdbath after two days. This can be written as a mixed number, which would be 2 3/4.

The total change in the volume of water in the birdbath after two days can be determined by subtracting the starting volume of water (1 1/2) from the ending volume of water (4 1/4).

The total change in the volume of water in the birdbath after two days is an important figure to calculate in order to understand the amount of water that has been added or removed from the birdbath. To calculate this figure, one must subtract the starting volume of water from the ending volume of water. In this scenario, the starting volume of water was 1 1/2 and the ending volume of water was 4 1/4. When these two numbers are subtracted, the difference is 2 3/4, which is the total change in the volume of water in the birdbath after two days. This can be written as a mixed number, which would be 2 3/4. This figure is important to know as it can help one determine how much water has been added or removed from the birdbath over a certain period of time.

the complete question is :

A birdbath is filled to the top. On Day 1, the volume of water in the bowl decreases by 7/8 cup. On Day 2, the volume in the bowl decreases by 3/4 cup. What number represents the total change in the volume of water in the bowl after two days? Write your answer as a mixed number.

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According to the given information, it will take Hector 15 weeks to reach 10 km if he starts at 2.5 km and adds 0.5 km every week.

What is an equation?

An equation is a mathematical statement that shows that two expressions are equal. It typically consists of two sides separated by an equals sign (=), where each side contains an expression that may contain variables, constants, or mathematical operations. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true.

The equation that correctly determines how many weeks it will take Hector to reach 10 km is 0.5x + 2.5 = 10, where x is the number of weeks.

To verify, we can solve for x:

0.5x + 2.5 = 10

0.5x = 10 - 2.5

0.5x = 7.5

x = 7.5/0.5

x = 15

Therefore, it will take Hector 15 weeks to reach 10 km if he starts at 2.5 km and adds 0.5 km every week.

The other equations are not equivalent to Hector's weekly addition of 0.5 km:

0.5 – 2.5 = 10x is not equivalent because it does not involve adding 0.5 every week.

2.5x + 0.5 = 10 is not equivalent because it does not take into account Hector's starting distance of 2.5 km.

2.5x – 0.5 = 10 is not equivalent because it does not involve adding 0.5 every week and it also does not take into account Hector's starting distance of 2.5 km.

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The statement "The basic calculation for obtaining a response bias is not to divide the number of legitimately completed questionnaires returned by the total number of sample members provided a chance to participate." is False

Instead, the basic calculation for obtaining a response bias is to subtract the response rate from 100%. The response rate is calculated by dividing the number of legitimately completed questionnaires returned by the total number of sample members provided a chance to participate.

So, the formula for calculating response bias is:
Response bias = 100% - (number of legitimately completed questionnaires returned / total number of sample members provided a chance to participate)

In other words, response bias is the percentage of sample members who did not participate or did not complete the questionnaire legitimately. It is important to calculate response bias in order to understand how representative the sample is of the population.

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If the company has limited resources or is not particularly concerned with precision, it may decide to forego the additional information and rely on its prior probabilities to make decisions

Bayes TheoremBayes Theorem is a statistical technique that enables one to find the probability of an event given another event that is known to have happened. The theorem states that the probability of an event A happening given that another event B has already occurred is equivalent to the probability of event B happening given that event A has already occurred multiplied by the probability of A happening divided by the probability of B happening. The revised decision tree can be produced to make a decision. However, before that, we need to find the sum of getting a Raise report for each outcome of the prior probabilities. This is shown in the table below:Consultant`s Prediction (in %)Actual Market OutcomePrior ProbabilitiesRiseStableFallRaise0.60.70.20.10.70.140.1260.0280.0140.20.30.20.60.240.0600.0180.1260.03 0.03150.0060.00210.07Total 0.34 0.36 0.30 1.00To revise the consultant's prediction using Bayes theorem, we can use the following formula:P(Stable | Raise) = (P(Raise | Stable) * P(Stable)) / (P(Raise | Stable) * P(Stable) + P(Raise | Rise) * P(Rise) + P(Raise | Fall) * P(Fall)) = (0.140 * 0.30) / (0.140 * 0.30 + 0.126 * 0.34 + 0.028 * 0.36) = 0.2906 or 29.06%The probability of getting a stable report given a raise report is 29.06%. This value can also be calculated for getting a fall report given a raise report using the same formula, which gives a probability of 24.51%.To make a decision on whether to use the consultant firm or not, we can use the revised decision tree. The tree shows that the probability of getting a raise report is only 34%, and given that we get a raise report, the probability of getting a stable report is 29.06% while the probability of getting a fall report is 24.51%.If the company decides to use the consultant firm, they will have to pay for the additional information. This cost needs to be weighed against the benefits of having this information, which will depend on the goals and priorities of the company. If the company values precision and accuracy in its decision-making process and is willing to pay for it, then it may choose to use the consultant firm. However, if the company has limited resources or is not particularly concerned with precision, it may decide to forego the additional information and rely on its prior probabilities to make decisions.

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