determine an appropriate interval width for a random sample of 110 observations that fall between and include each of the following: (a) 20 to 85 (b) 30 to 190 (c) 140 to 500

Answer:To show that Lorenz curves are always concave up on the interval [0, 1], we can use the definition of a Lorenz curve, which is L(x) = xp. Taking the second derivative of L(x) with respect to x gives us L''(x) = p. Since p is always greater than 0, L''(x) is always greater than 0. This means that L(x) is always concave up on the interval [0, 1].

Table of Lorenz values for p = 1.2, 1.5, 2.1, 2.5, 3, and 5:

a. The value of p that corresponds to the most equitable distribution of wealth is 1, as this would mean that the proportion of wealth held by each portion of the population is equal.

b. The value of p that corresponds to the least equitable distribution of wealth is 5, as this would mean that a small portion of the population holds a large proportion of the wealth.

The Gini Index is a measure of income inequality, where a value of 0 represents perfect equality (everyone has the same income) and a value of 1 represents perfect inequality (one person has all the income). The Gini Index is calculated as the ratio of the area between the Lorenz curve and the line of equality to the total area beneath the line of equality.

To find A and B for the Gini index we can use the following integral:

A = ∫(L(x) - x) dx from 0 to 1

B = ∫(x - L(x)) dx from 0 to 1

We can then solve the integral for each specific function of L(x) = xp to find the specific value of A and B.

Step-by-step explanation:

Answer:....

Step-by-step explanation:

The equation that would answer this question is x + (2x + 1.5) = 12. where x is the number of hours that Fin volunteers, and 2x + 1.5 represents the number of hours that each of his friends volunteers.

In this scenario, you can set up an equation that represents the total amount of time Fin and his friends volunteer. Let 'x' be the number of Finn's volunteer hours. His friends volunteer 1.5 hours more than Finn, so the number of hours each friend volunteers can be represented by x + 1.5. The total number of hours that all three friends volunteered can be expressed as:

x + (x + 1.5) + (x + 1.5) = 12

Expansion of the 2nd and 3rd terms:

x + x + 1.5 + x + 1.5 = 12

Combining the same terms:

3x + 4.5 = 12

Subtract 4.5 from both sides:

3x = 7.5

Divide both sides by 3:

x = 2.5

So Fin volunteers his 2.5 hours. 

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See the diagram below.

=================================================

Explanation:

Pick two points on this diagonal line. I'll go for (0,-3) and (1,-1)

Apply the slope formula to those coordinates.

[tex](x_1,y_1) = (0,-3) \text{ and } (x_2,y_2) = (1,-1)\\\\m = \text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}}\\\\m = \frac{\text{y}_{2} - \text{y}_{1}}{\text{x}_{2} - \text{x}_{1}}\\\\m = \frac{-1 - (-3)}{1 - 0}\\\\m = \frac{-1 + 3}{1 - 0}\\\\m = \frac{2}{1}\\\\m = 2\\\\[/tex]

A slope of 2, aka 2/1, means "move up 2, then right 1".

This "up 2, right 1" motion allows us to move from (0,-3) to (1,-1) as shown in the diagram below.

Two angle and an include side are congruent, hence by the ASA postulate triangle AEB congruent to triangle CED.

ASA similarity theorem: Two triangles are similar if two corresponding angles of one triangle are congruent to the two corresponding angles of another triangle. Also, the corresponding sides are proportional. ASA similarity is mostly known as the AA similarity theorem.

Statement: Line segments AE and EC are equal.

Reason: Given

Statement: angle A is congruent to angle B.

Reason: Given

Statement: Angles AEB & CED are equal.

Reason: vertically opposite angles are equal.

Hence, two angle an include side are congruent.

Triangles AOB & CED are congruent.

Reason: ASA Postulate.

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The probability that the next fruit is an apple is P = 0.8

We want to fnind the probability of randomly selecting an apple from the bag.

Remember that the probability is equal as the quotient between the number of apples and the total number of fruit on the bag.

Originally, there are:

4 apples.

3 peaches

2 plums.

The checkout person takes 2 plums and 1 peach, so now there are:

4 apples.

1 peach.

So there are 4 apples and 5 fruits in total

Then the probability of grabing an apple is:

P = 4/5 = 0.8

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F or G = { 5,6,7,8,9,10,11,12 } is probability  P(F or G) using the general addition rule.

What are examples and probability?

The likelihood that something will happen is called the probability. the total number of conceivable outcomes.

                                   For instance, the chance of flipping a coin and obtaining heads is 1 in 2, as there is only one way to acquire a head and there are a total of 2 possible outcomes (a head or tail). P(heads) = 12 is what we write.  the forerunners of the contemporary mathematical theory of probability (for example the "problem of points").

S={ 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14},

event F={5, 6, 7, 8, 9}​, and

event G={9, 10, 11, 12}.

F or G = { 5,6,7,8,9,10,11,12 }

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1. The unit g/cm³ is valid. So, division of measurements is possible.

2. The unit mm is valid. So, division of measurements is possible.

3. The unit L² is not valid. So, multiplication of measurements is not possible.

What are measurement units?

The group of units used to quantify different physical quantities collectively known as the units of measurement. Since ancient times, we have measured these things using many units, including length, mass, volume, current, and temperature.

1. 63g/7cm³ = 9 g/cm³

The density of a substance indicates how dense it is in a given area. Mass per unit volume is the definition of a material's density. In essence, density is a measurement of how closely stuff is packed. It is a particular physical characteristic of a specific thing.

The unit g/cm³ is a unit for measuring density.

Therefore, the division of measurements is possible.

2. The m or mm must be converted so that the units are the same.  

1 m = 1000 mm.

Convert the meters to mm:  

0.080 m = 80 mm.

480 mm²/80 mm = 6 mm

The word "area" refers to a free space. A shape's length and width are used to compute its area. Unidimensional length is expressed in terms of feet (ft), yards (yd), inches (in), etc.

The mm² is a unit for measuring area.

The m is a unit for measuring length or width.

Therefore, the division of measurements is possible.

3.  (4.5 dL) × (0.70 L)

Liquid volume is a term used to describe how much 3-D space a given amount of liquid takes up.

Volume of a liquid is measured in litres (L).

Squaring Litres doesn't make any sense.

Therefore, the multiplication of measurements is not possible.

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The solution (1, 1.5) denotes:

Each apple costs $1.

Each banana costs $1.50.

Option D is the correct answer.

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

Example:

2x + 3 = 9 is an equation.

We have,

6y = 9x _____(1)

2x + 4y = 8 ______(2)

From the equation,

x is the cost of an apple.

y is the cost of a banana.

Now,

(1, 1.5) = (x, y)

This means,

The cost of an apple = $1.

The cost of a banana = $1.5

Thus,

The solution (1, 1.5) denotes that the cost of an apple is $1 and a banana is $1.5.

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The range of values for each expression is given as follows:

1. 8 ≤ a+b ≤ 16

2. -6 ≤ a - b ≤ 2

3. 15 ≤ ab ≤ 63

4. 1/3 ≤ a/b ≤ 7/5.

For the maximum values, we have that:

For the minimum values, we have that:

The problem asks for the range of values for each expression, considering 3 < a < 7 and 5 < b < 9.

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Answer: Go up 8 and to the right 1 starting at the point (0,4)

Step-by-step explanation: y=mx+b where m=8 (your slope) and b= 4 (your y-intercept)

The matrices matrix A be a 5 by 7 , matrix B be a 7 by 6 and matrix C be a 6 by 5 matrix.

now, we need to determine the size of the matrices

AB = [tex](AB)_{5*6}[/tex]

BA is undefined

[tex]A^{T}[/tex]B  is undefined

BC = [tex](BC)_{7*5}[/tex]

ABC = [tex](ABC)_{5*5}[/tex]

CA = [tex](CA)_{6*7}[/tex]

[tex]B^{T}[/tex]A is undefined.

B[tex]C^{T}[/tex] is undefined.

given that

matrix A is [tex]A_{5*7}[/tex]

matrix B is [tex]B_{7*6}[/tex]

matrix C is [tex]C_{6*5}[/tex]

now, we need to determine the size of the matrices

AB multiplication of two matrices

multiplication of two matrices is possible only when B has the same number of columns as rows in A,

[tex]A_{m*n}[/tex]and [tex]B_{n*p}[/tex]

If it is defined as above, then the matrix AB will have m rows and p columns, i.e., A must have n columns and B must have n rows in order for AB to be defined.

[tex](AB)_{m*n}[/tex]

In addition, a matrix gets transposed when columns turn into rows and vice versa.

Thus, if

[tex]A_{m*n}[/tex]⇒[tex](AT)_{m*n}[/tex]

So, for this particular question, we have:  [tex]A_{5*7}[/tex],  [tex]B_{7*6}[/tex] , [tex]C_{6*5}[/tex]

According to the aforementioned idea, AB is therefore [tex](AB)_{5*6}[/tex] in size.

Additionally, BA and [tex]A^{T}[/tex]B are not defined.

[tex](BC)_{7*5}[/tex]

ABC=(AB)C=A(BC) because matrix multiplication is associative,

[tex](ABC)_{5*5}[/tex]

[tex](CA)_{6*7}[/tex]

[tex]B^{T}[/tex]A is undefined. because they don't have same number of columns in matrix B and rows in matrix A

B[tex]C^{T}[/tex] is undefined. because they don't have same number of columns in matrix B and rows in matrix C

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The next number in the sequence is 151.

What is the successive differences?

The method of successive differences involves finding the differences between consecutive terms in a sequence and using that information to predict the next term in the sequence.

To use this method for the sequence 3, 7, 17, 33, 55, 83, 117:

Find the differences between consecutive terms:

7 - 3 = 4

17 - 7 = 10

33 - 17 = 16

55 - 33 = 22

83 - 55 = 28

117 - 83 = 34

Check if the differences are constant. In this case, they are increasing by 6 each time.

Use this information to predict the next term in the sequence:

117 + 34 = 151

So, the next number in the sequence is 151.

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The intercepts and the asymptotes of h(x) using the graph is  x-intercept -4 and y-intercept =-1

The y-intercept of a function y = f(x) is a point where its graph would meet the y-axis and is obtained by substituting x = 0. Understand the y-intercept and its formula with derivation

From the given graph

Y-intercept is the point where the curves crosses the y-axis

From the graph, the coordinate of the y-intercept is (0, -1)

X-intercept is the point where the curves crosses the x-axis

From the graph, the coordinate of the x-intercept is (-4, 0)

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Answer:

The vertical asymptote is -5

Step-by-step explanation:

You basically have to set it to 0 and then solve the equation. You should get x= -5

Answer:

a) e^x (3 - e^x)^4 dx

Using the formula [ f'(x)[f(x)"]dx =[ [f(x)]^n+1 / n+1] + c, where n = 1,

we have:

e^x (3 - e^x)^4 dx = (e^x)^5 (3 - e^x)^4 / 5 + c

= (e^5x - 4e^4x + 6e^3x - 4e^2x + e^x) / 5 + c

b) 3e^2x √(1 + e²x) dx

Using the formula [ f'(x)[f(x)"]dx =[ [f(x)]^n+1 / n+1] + c, where n = 1,

we have:

3e^2x √(1 + e²x) dx = (3e^2x)^2 * (1 + e²x)^(3/2) / 2 + c

= (9e^4x + 3e^2x) / 2 + c

c) 3e^-2x / (1 + e^-2x)^3 dx

Using the formula [ f'(x)[f(x)"]dx =[ [f(x)]^n+1 / n+1] + c, where n = 1,

we have:

3e^-2x / (1 + e^-2x)^3 dx = -(3e^-2x)^2 / (1 + e^-2x)^2 + c

= -(9e^-4x) / (e^-4x + 2e^-2x + 1) + c

d) 4 cos 2x sin³ 2x dx

Using the formula [ f'(x)[f(x)"]dx =[ [f(x)]^n+1 / n+1] + c, where n = 1,

we have:

4 cos 2x sin³ 2x dx = -4 cos 2x (sin 2x)^4 / 4 + c

= -(cos 2x) (1 - cos 4x)^2 / 2 + c

e) sec² 3x tan³ 3x dx

Using the formula [ f'(x)[f(x)"]dx =[ [f(x)]^n+1 / n+1] + c, where n = 1,

we have:

sec² 3x tan³ 3x dx = -sec² 3x (tan 3x)^4 / 4 + c

= -sec² 3x (sec² 3x - 1)^2 / 4 + c

f) 2+tan ² x / cos² x dx

Using the formula [ f'(x)[f(x)"]dx =[ [f(x)]^n+1 / n+1] + c, where n = 1,

we have:

2+tan ² x / cos² x dx = ln|sec x| + c

It's worth noting that all these integrals are indefinite, which means that the constant c is arbitrary, and the actual antiderivative depends on the problem context.

Step-by-step explanation:

Answer:

8th term = 98

Step-by-step explanation:

Given equation,

→ Tn = 2n² - 3n - 6

Now the 8th term will be,

→ Tn = 2n² - 3n - 6

→ T8 = 2(8)² - 3(8) - 6

→ T8 = 2(64) - 24 - 6

→ T8 = 128 - 30

→ [ T8 = 98 ]

Hence, the 8th term is 98.

Answer:

g(f(- 4)) = 40

Step-by-step explanation:

to evaluate g(f(- 4)) , evaluate f(- 4) and substitute the value obtained into g(x)

f(- 4) = (- 4)² + 3(- 4) + 6 = 16 - 12 + 6 = 4 + 6 = 10 , then

g(10) = 5(10) - 10 = 50 - 10 = 40

The amount  of marginal product of the 13th worker is 10 units of output, which is the difference between the quantity of output when L = 12 (122) and when L = 13 (132).

The marginal product of the 13th worker can be calculated as follows:

Marginal product = Quantity of output for L = 13 - Quantity of output for L = 12

Marginal product = 132 - 122

Marginal product = 10 units of output

The marginal product of the 13th worker is the additional output created by hiring one additional worker. This is calculated by taking the difference between the quantity of output when the number of workers is 12 and when it is 13. In this case, the quantity of output when L = 12 is 122, and when L = 13 it is 132, so the marginal product is 10 units of output. This calculation shows that the 13th worker is adding 10 units of output to the firm's total production. This is an important concept in production economics, as it helps firms determine the optimal number of workers to hire in order to maximize their output and profits.

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The area of the shaded region is obtained as (π/√2) units².

What is area?

An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure.

The diagram is labelled.

The values for area of z and y needs to be obtained.

The result will be in the form |y| + z as when the integration will be done, y will come out to be negative.

Now, x + z is the area of rectangle ABCD.

Verify whether x and y are equal in magnitude -

[tex]\begin{aligned}& x=\int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \frac{\cos \theta}{\sin ^2 \theta} d \theta=\left[\frac{-1}{\sin \theta}\right]_{\frac{\pi}{4}}^{\frac{\pi}{2}} \\& =-1-(\sqrt{2}) \\& =\sqrt{2}-1\end{aligned}[/tex]

[tex]\begin{aligned}& y=\int_{\frac{\pi}{2}}^{\frac{3\pi}{4}} \frac{\cos \theta}{\sin ^2 \theta} d \theta=\left[\frac{-1}{\sin \theta}\right]_{\frac{\pi}{2}}^{\frac{3\pi}{4}} \\& =-(\sqrt{2})-(-1) \\& =-\sqrt{2}+1\end{aligned}[/tex]

This is equal to |y| = x.

So, |y| + z is equivalent to writing x + z.

Now the formula for area of rectangle ABCD is -

Area = length × breadth

Area = √2 - [(3π/4) - (π/4)]

Area = √2 - (π/2)

Area = (π/√2)

Therefore, the area is found to be (π/√2) units².

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The probability of randomly selecting a marble that is not purple is 70%, so the correct option is A.

We know that there are:

For a total number of 2 + 4 + 15 + 9 = 30

There are a total of 30 marbles on the bag, the probability of randomly selecting a marble that is not purple, is equal to the quotient between the number of marbles that are not purple and the total number of marbles.

21 marbles are not purple, then the probability is:

P = (21/30)*100%

P = 0.7*100%

P = 70%

The correct option is a.

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The fastest speed of the bird of species A is 185 mph and the fastest speed of the bird of species B is 37 mph

An equation is an expression showing the relationship between numbers and variables.

Let a represent the top speed of bird A and b represent the top speed of bird B.

A bird of species ​ A, when diving, can travel 5 times as fast as a bird of species B top speed, hence:

a = 5b

a - 5b = 0     (1)

If the total speeds for these two birds is 222 miles per hour, hence:

a + b = 222    (2)

To solve by elimination method, subtract equation 2 from 1, hence:

-6b = -222

Dividing by -6:

b = 37

Put b = 37 in equation 2:

a + 37 = 222

a = 185

The speed of the bird are 185 miles per hour and 37 miles per hour

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The inequality is solved to and represented in interval notation as follows

The inequality is made of two sets and can be separated as

−4 ≤ −2(y − 1)

−2(y − 1) < 2

solving the first

−4 ≤ −2(y − 1)

−4 ≤ −2y + 2

−4 - 2 ≤ −2y

-6 ≤ −2y

divide through by -2

y ≤ 3

solving the second

−2(y − 1) < 2

−2y + 2 < 2

−2y < 2 - 2

−2y < 0

divide through by -2

y > 0

the inequality in interval notation is

(-∞, 3] [0, ∞)

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The value of function showing the total cost of hiring your neighbor (y) for a certain amount of hours (x) is,

⇒ y = 10 + 20x

Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.

We have to given that;

A neighbor charges $10 to mow your lawn plus $20 per hour.

Let the total cost of hiring your neighbor = y

And, Number of amount of hours = x

Hence, We can formulate;

The function which showing the total cost of hiring your neighbor (y) for a certain amount of hours (x) is,

⇒ y = 10 + 20x

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Answer:

9 is correct

Step-by-step explanation:

2+3=5+4=9 , believe

Assuming David’s height off the floor is continuous and differentiable, the statement that is true is: B. The Mean Value Theorem does not apply.

Based on the details we can vividly states that the Mean Value Theorem does not apply based on the fact that the function that help to described David's height off the floor is not continuous and differentiable.

This  function would only be differentiable if David's movement off the couch and landing on the floor could be described by a mathematical function, which is highly unlikely.

Therefore the correct option is B.

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A function with an infinite number of critical points can be expressed as a continuous function with a derivative that is equal to zero at every point in its domain.

For example, the function f(x) = 0 has an infinite number of critical points since its derivative f'(x) = 0 for every x in its domain. Alternatively, a function that has every point as a critical point can be expressed as a function with a derivative that is always equal to zero. For example, the constant function f(x) = c has a derivative f'(x) = 0 for every x in its domain, and thus every point in its domain is a critical point.

In addition, an example of a function with every point as a critical point is the constant function, f(x) = c, where c is a constant. The derivative of this function is f'(x) = 0, which means that the derivative is zero for all values of x. Therefore, every point of the constant function is a critical point.

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Answer: 28:35

Step-by-step explanation: The missing number is 28, here's how to solve it:

So, if we know both components of the first ratio AND the second component of the 2nd ratio, we need to divide the 2nd components from each ratio. So, 35 / 5 = 7. Now, we need to multiply 7 by 4. We get 28. So, the 2nd ratio is 28:35. I hope this helps!

(Look at the attachment for a better idea)

The depth of the water RF is, 18 ft.

Any relationship that is always in the same ratio and quantity which vary directly with each other is called the proportional.

We have to given that;

⇒ ΔSML ≅ ΔSRF

⇒ SM = 10

⇒ ML = 6

⇒ SR = 30

Since, Both triangles SML and SRF are similar.

Hence, We get;

⇒ SM / ML = SR / RF

Substitute all the values, we get;

⇒ 10 / 6 = 30 / RF

⇒ RF = 30 × 6 / 10

⇒ RF = 3 × 6

⇒ RF = 18 ft

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The smaller number is 15 and the larger number is 36.

Let x and y be the smaller and larger numbers, respectively.

From the given information, we know:

y = x + 21 (because the larger number is 21 more than the smaller number)

x + y = 51 (because the sum of the two numbers is 51)

We can substitute the first equation into the second equation to find the value of x:

x + (x + 21) = 51

x + x + 21 = 51

2x = 30

x = 15

Now that we know the value of x, we can use the first equation to find the value of y:

y = x + 21

y = 15 + 21

y = 36

So the smaller number is 15 and the larger number is 36.

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