Which statement about rational numbers is true?
1. All rational numbers are fractions.
2. Rational numbers never have repeating decimals.
3. Some integers are not rational numbers.
4. All fractions containing integers are rational numbers.

(a) The linear approximation is P1(x) = 4x + 1. (b)  the quadratic approximation is P2(x) = (7/2)x²  + 4x + 1.

Linear approximation, also known as tangent line approximation, is a method used to estimate the value of a function near a specific point by using the equation of the tangent line at that point. It is based on the concept that a tangent line to a curve at a certain point can closely approximate the behavior of the curve in the vicinity of that point.

The linear approximation can be derived using the first-order Taylor series expansion. Given a function f(x) and a point (a, f(a)), the linear approximation can be written as:

L(x) = f(a) + f'(a)(x - a)

In this equation, L(x) represents the linear approximation of f(x) near the point (a, f(a)). f(a) is the value of the function at the point a, and f'(a) is the derivative of the function evaluated at a.

To find the linear and quadratic approximations of f(x) = \(e^{(4x)\) * cos(3x) at x = 0, we need to evaluate the function and its derivatives at x = 0.

a) Linear Approximation:
The linear approximation P1(x) is given by:
P1(x) = f'(a)(x - a) + f(a)

First, let's find f'(x) and f(x):
f(x) = \(e^{(4x)\) * cos(3x)
f'(x) = \((4e^{(4x)\) * cos(3x)) - \((3e^{(4x)\) * sin(3x))

Now, evaluate the function and its derivative at x = 0:
f(0) = \(e^{(4 * 0)\) * cos(3 * 0) = 1 * 1 = 1
f'(0) = (\(4e^{(4 * 0)\) * cos(3 * 0)) - (\(3e^{(4 * 0)\) * sin(3 * 0)) = 4 * 1 - 3 * 0 = 4

Substituting these values into the linear approximation formula:
P1(x) = f'(0)(x - 0) + f(0)
P1(x) = 4x + 1

b) Quadratic Approximation:
The quadratic approximation P2(x) is given by:
P2(x) = (1/2)f''(a)\((x - a)^2\) + f'(a)(x - a) + f(a)

To find f''(x), we differentiate f'(x):
f''(x) = (\(16e^{(4x)\)* cos(3x)) - (\(12e^{(4x)\) * sin(3x)) - (\(9e^{(4x)\)* cos(3x)) - (\(12e^{(4x)\) * sin(3x))

Now, evaluate the function and its derivatives at x = 0:
f''(0) = (\(16e^{(4 * 0)\)* cos(3 * 0)) - (\(12e^{(4 * 0)\) * sin(3 * 0)) - (\(9e^{(4 * 0)\) * cos(3 * 0)) - (\(12e^{(4 * 0)\) * sin(3 * 0)) = 16 - 0 - 9 - 0 = 7

Substituting these values into the quadratic approximation formula:
P2(x) = (1/2)(7)(x - 0)² + 4(x - 0) + 1
P2(x) = (7/2)x²  + 4x + 1

c) Sketching the Graph:

To sketch the graph of the linear and quadratic approximations, plot the points for various values of x on the same axis. Label the axes neatly.

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Answer: A. -6.007, -2π, -√48

Step-by-step explanation:

-2π ≈ -6.28

-√48 ≈ -6.93

-6.007 = -6.007

-----------------------

-6.007>-6.28>-6.93

-6.007>-2π>-√48

The dependent variable of interest in examining whether the degree of parental involvement differs based on students' grade in school is the degree of parental involvement.

In this scenario, the independent variable is the students' grade in school (i.e., first, second, third, etc.). The dependent variable is the variable that is expected to be influenced by the independent variable. In this case, the dependent variable of interest is the degree of parental involvement.

The degree of parental involvement refers to the level or extent to which parents are engaged in their child's education and school-related activities. It can encompass various aspects, such as attending parent-teacher meetings, volunteering at school, helping with homework, and participating in school events.

By examining whether the degree of parental involvement differs based on students' grade in school, researchers or educators aim to explore any potential patterns or differences in parental involvement as children progress through different grade levels. This investigation can provide insights into how parental involvement may vary across grade levels and inform strategies to enhance parental engagement in education at specific stages of a child's schooling. Thus, the dependent variable of interest is the degree of parental involvement.

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Answer: x = 12

Step-by-step explanation: The given triangle is an equilateral triangle. Equilateral triangles have all angles and all sides equal. Thus, let's solve for x knowing all 3 equations are equal. For this, only 2 equations are necessary.

5x - 22 = 4x - 10

Adding 22 to both sides, we get:

5x = 4x + 12

Subtracting 4x to 5x, we get:

x = 12.

Thus 5*12 - 22 = 38, and 4*12 - 10 = 38, and so is 3*12 + 2 = 38. Thus, we know this is correct.

Answer:

x = 12

Step-by-step explanation:

Hello!

All three angles are congruent, indicating that this is an equilateral triangle.

The rules of this triangle say that all sides are congruent.

We can solve for x by setting any 2 sides equal to each other. I'm picking AB and BC.

The value of x is 12.

Scatterplot D is likely the best representation of a positive non-linear correlation.

Non-linear correlation is the relationship between two or more variables in which the change in one variable does not correspond linearly with the change in the other variable. It is used to describe a wide range of complex relationships between variables that cannot be accurately described by linear relationships. Non-linear correlation can be used to identify outliers, to understand the relative importance of different factors, and to detect non-linear relationships between variables.

This can be seen from the pattern of the points, which curve upwards from the lower left corner to the upper right corner. This curved pattern indicates that as the independent variable increases, the corresponding dependent variable also increases in a non-linear fashion. Therefore, as the independent variable continues to increase, the dependent variable will increase more quickly. In contrast, the other scatterplots shown are either linear or lack any significant pattern.

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Answer:

$4.50 each

Step-by-step explanation:

sorry if I'm wrong

Answer:

the asnwer is 68

Step-by-step explanation:

8 times 8 is 68

The required fraction should be greater than 1. Option B is correct.

Fraction is defined as the number of compositions that constitutes the Whole.

Here,
Alaina multiplies a fraction by 8, which results in a product greater than 8.
Let a/b is the required fraction,
If a / b is greater than zero and less than one, then the product of fraction and 8 will be less than 8. but when a/b is greater than 1 the product of the fraction and 8 will be greater than 8.

Thus, the required fraction should be greater than 1. Option B is correct.

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Each of the expressions has been simplified by using properties of exponents as shown below.

In Mathematics, an exponent refers to a mathematical operation that is typically used in conjunction with an algebraic expression in order to raise a quantity to the power of another.

This ultimately implies that, an exponent is represented by the following mathematical expression;

bⁿ

Where:

the variables b and n are numerical values (numbers) or an algebraic expression.

n is referred to as a superscript or power.

By applying the multiplication and division law of exponents for powers to each of the expressions, we have the following:

(1.25 × 10⁷) + 63,000,000 = (1.25 × 10⁷) + (6.3 × 10⁷) = (1.25 + 6.3) × 10⁷ = 7.55 × 10⁷

12,000 + 7 × 10⁴ = 1.2 × 10⁴ + 7 × 10⁴ = (1.2 + 7) × 10⁴ = 8.2 × 10⁴

5.88 × 10⁵ - 3.44 × 10⁵ = (5.88 - 3.44) × 10⁵ = 2.44 × 10⁵

6 × 10⁸ ÷ 120 = 6 × 10⁸ ÷ 1.2 × 10² = (6 ÷ 1.2) × 10⁸⁻² = 5 × 10⁶

9 × 10¹² ÷ 4.5 × 10⁴ = (9 ÷ 4.5) × 10¹²⁻⁴ = 2 × 10⁸

1.3 × 10³ · 2,200 = 1.3 × 10³ × 2.2 × 10³ = (1.3 × 2.2) × 10³⁺³ = 2.86 × 10⁶

(6.3 × 10⁴) + 1.3 × 10⁴ = (6.3 + 1.3) × 10⁴ = 7.6 × 10⁴

3,400 · 2 × 10⁴ = 3.4 × 10³ × 2 × 10⁴ = (3.4 × 2) × 10³⁺⁴ = 6.8 × 10⁷

9.78 × 10⁵ - 732,000 = (9.78 - 7.32) × 10⁵ = 2.46 × 10⁵

3.5 × 10² · 2 × 10⁴ = (3.5 × 2) × 10²⁺⁴ = 7 × 10⁶

1.1 × 10⁸ ÷ 22,000 = 1.1 × 10⁸ ÷ 2.2 × 10⁴ = (1.1 ÷ 2.2) × 10⁸⁻⁴ = 0.5 × 10⁴ = 5 × 10³.

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\(\text {Hey! To solve this problem you must...}\)

\(\text {Simplify the Equation:}\)

\(\text {Multiply (x+8) by 2.}\)

\(\text {x becomes 2x. 8 becomes 16}\)

\(2x+16>x+14 \text { will be your answer once you simplify it.}\)

\(\text {The Next Step is to Subtract x:}\)

\(\text {Q: Why Subtract x instead of adding x?}\)

\(\text {A: We need x to cancel out. So x-x will cancel out each other.}\)

\(\text {Note: In equations like these x-x is another way of saying 1-1.}\)

\(2x+16-x>x+14-x\\x+16>14 \text { would be your answer once you subtract x.}\)

\(\text {The Final Step is to Subtract 16.}\)

\(\text {Note: Subtracting 16 will cancel out 16-16 and just leave x by itself.}\) \(\text {That's what we are working towards.}\)

\(x+16-16>14-16\\x>-2 \text { would be your final answer.}\)

\(\text {Best of Luck!}\)

\(\text {-LimitedX}\)

Hey there!

\(2(x+8)>18+x-4\)

In our first step, we will need to simplify the inequality.

\(2x+16>x+14\)

Now we have to subtract x from both sides of the inequality.

\(2x+16-x>x+14-x=x+16>14\\x+16>14\)

In our last step, we will need to subtract 16 from both sides of the inequality.

\(x+16-16>14-16=x>-2\\x>-2\text{(ANSWER)}\)

Hope this helps!

\(\text{Please rate this answer and press the thank button!}\\\text{-TestedHyperr}\)

Holy will take 13/5 or 2.6 hours to finish the bookcase.

The relation between time and work is

Work Done = Time Taken × Rate of Work ·

or, Rate of Work = 1 / Time Taken ·

or, Time Taken = 1 / Rate of Work ·

Now it is given that,

Time taken by Fritz to build a bookcase =  8 hours

So, Bookcase build by Fritz in 1 hour = 1/8

Similarly,

Time taken by Holly to build a bookcase =  10 hours

So, Bookcase build by Holly in 1 hour = 1/10

So, bookcase build in 3 hours

Fritz = 3/8

Holly = 3/10

Total bookcase build in 3 hours =  3/8 + 3/10 =  27/40

So. remaining work =  1 - 27/40 =  13/40

Thus, Time taken by Fritz to complete the bookcase =  8 * (13/40) hours

                                                                                         =  13/5 or 2.6 hours

Thus, Holy will take 13/5 or 2.6 hours to finish the bookcase.

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Answer:

D

Step-by-step explanation:

D is the only one that makes sense

Answer:

0.08, or 8%

Step-by-step explanation:

From 0.54 subtract 0.46.  This yields 0.08.  This tells us that 8% of customers would prefer to stay in a hotel.  We don't know how many customers there are altogether, so cannot answer with a specific number of customers, just this percentage.

To solve this problem, we need to use the formula for the z-score:z = (x - μ) / (σ / sqrt(n)), where x is the sample mean (which is the average score in 16 selected games), μ is the population mean (which is 43.06 points), σ is the population standard deviation (which is 15 points), and n is the sample size (which is 16).

Given the average score in the NFL is 43.06 points with a standard deviation of 15 points. Since we are looking at the average of 16 selected games, we need to calculate the standard error, which is the standard deviation divided by the square root of the sample size (in this case, 16):

Standard error = 15 / √16 = 15 / 4 = 3.75

Now, we will calculate the z-score for 38 points, which represents how many standard errors 38 points is away from the average score:

Z-score = (38 - 43.06) / 3.75 = -1.35

Using a z-score table or calculator, we find the probability of a z-score being less than -1.35 is approximately 0.0885 or 8.85%.

Since we want the probability that the average of 16 selected games scored larger than 38 points, we need to find the complement of this probability:

Probability (average score > 38) = 1 - 0.0885 = 0.9115 or 91.15%

So, the probability that the average of 16 selected games scored larger than 38 points is approximately 91.15%.

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Answer:

427

Step-by-step explanation:

The area of the field is 2.384 hectares and price of the field is $28,608

The area of a quadrilateral given its sides and the diagonal:

Area = 1/4 × √[4s^2d^2 - (b^2 + d^2 - a^2 - c^2)^2]

where s = (a + b + c + d)/2 is the semi perimeter,

a, b, c, d are the lengths of the sides

d is the length of the diagonal.

Substituting the given values,

s = (80 + 150 + 120 + 250)/2

= 300

a = 80, b = 150, c = 120,

d = √(250^2 + 80^2)

= 259.8076

Area = 1/4 × √[4(80+150+120+250)^2*(259.8076)^2 - (80^2 + 259.8076^2 - 150^2 - 120^2)^2]

≈ 23844.17 square meters

To convert square meters to hectares, we divide by 10,000:

Area = 23844.17 / 10,000 hectares

≈ 2.384 hectares

Therefore, the area of the field is approximately 2.384 hectares.

The price of the field = 12,000/hectare,

Total price = 2.384 hectares × $12,000/hectare

= $28,608

Therefore, the price of the field is $28,608.

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Answer:

5 square2 −3 − 7

Step-by-step explanation:

i calculated it all

Answer:

d

Step-by-step explanation:

...........

..........

correct ans is D

Answer:

what

Step-by-step explanation:

Question: Find the length and width of a rectangle that has the given perimeter and a maximum area. perimeter: p unitsMain Answer:The maximum area of a rectangle with a given perimeter occurs when the rectangle is a square.

Thus, to find the length and width of a rectangle that has the given perimeter and a maximum area, we must solve for the dimensions of a square using the given perimeter.The formula for the perimeter (P) of a rectangle is given as: P = 2l + 2w, where l represents the length of the rectangle and w represents the width.To solve for the length and width of a rectangle that has a given perimeter (p) and a maximum area,

we must first solve for the length and width of the corresponding square using the given perimeter (p). We can do this by dividing the perimeter by 4 (since a square has four equal sides).Length of one side of the square is `p/4`.As the square is the rectangle with maximum area, we can say that l = w= `p/4` units.Explanation:Thus, the length and width of the rectangle are both equal to `p/4` units, and the rectangle with these dimensions has the given perimeter and maximum area.

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Answer: Yes, 59 is a prime number

Step-by-step explanation: 59 is a prime number, it has only two factors, such as one and the number itself. Hence, the factors of 59 are 1 and 59.

Yes, 59 is a prime number.

What is a prime number?

Any natural number greater than 1 that is not the sum of two smaller natural numbers is referred to as a prime number. A composite number is a natural number greater than one that is not prime.

In other terms, prime numbers are positive integers greater than one that only has the number itself and 1 as factors. 2, 3, 5, 7, 11, 13, and other prime numbers are just a few examples. Never forget that 1 is neither a prime number nor a composite. Apart from 1, the other numbers can all be categorised as prime and composite numbers. Except for 2, which is the smallest prime number and the only even prime number, all prime numbers are odd.

The number in question, which is 59,  has only two factors: 1 and 59.

Therefore 59 is a prime number.

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Given table shows the counts of respondents of a survey that asked 1000 adults about their employment status and whether they owned stocks:

Employment Status Owning Stocks Not Owning Stocks Employed 400 300

Unemployed 50 250 Retired 150 100We have to complete the following parts:

a) How many of the respondents are unemployed or do not own any stocks?

b) If a respondent is chosen at random, what is the probability that the respondent owns stocks?

c) If a respondent is chosen at random, what is the probability that the respondent is retired and does not own stocks?d) If a respondent is chosen at random, what is the probability that the respondent is employed or retired?

e) If a respondent is chosen at random, what is the probability that the respondent is employed, given that the respondent does not own stocks?

Solution:

a) The total number of respondents who are unemployed or do not own any stocks is 50 + 250 + 100 = 400.

b) Probability that a respondent owns stocks = Number of respondents owning stocks / Total number of respondents= (400 + 50 + 150) / 1000= 0.6

c) Probability that a respondent is retired and does not own stocks= Number of retired respondents not owning stocks / Total number of respondents= 100 / 1000= 0.1

d) Probability that a respondent is employed or retired= (Number of employed respondents / Total number of respondents) + (Number of retired respondents / Total number of respondents)= (400 / 1000) + (150 / 1000)= 0.55

e) Probability that a respondent is employed, given that the respondent does not own stocks= (Number of employed respondents not owning stocks / Total number of respondents not owning stocks)= 300 / 350= 0.86

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Answer:

160

Step-by-step explanation:

4 × 10 × 4 = 160

(Random words to hit the character limit please ignore)

Answer:

I believe it is 2s

Step-by-step explanation:

a variable is a symbol which functions as a placeholder, so in this case, the placeholder is "s"

According to the solution we have come to find that, The expression that represents the perimeter of the triangle is 16.6t + 9.7u - 14.4 centimeters.

What is triangle?

A triangle is a three-sided polygon formed by three line segments that connect three non-collinear points in a plane.

The perimeter of the triangle is the sum of the lengths of all three sides.

Using the given expressions for the side lengths, we can simplify them as follows:

Side 1: (8.2t - 8.7) cm

Side 2: (8.4t + 3.1) cm

Side 3: (9.7u - 8.8) cm

To find the perimeter, we add the lengths of all three sides:

Perimeter = Side 1 + Side 2 + Side 3

= (8.2t - 8.7) cm + (8.4t + 3.1) cm + (9.7u - 8.8) cm

= 8.2t + 8.4t + 9.7u - 8.7 + 3.1 - 8.8 cm

= 16.6t + 9.7u - 14.4 cm

Therefore, the expression that represents the perimeter of the triangle is 16.6t + 9.7u - 14.4 centimeters.

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Answer:

x² + 27x + 180

Step-by-step explanation:

(12 + x)(15 +x) = 12*15 + 12x + 15x + x² = x² + 27x + 180