Write the number for thirteen million thirty thousand

We can conclude that r - s is rational.

Proof: Suppose r and s are rational numbers.

We must show that r - s is rational.

To prove this, we will use the closure property of rational numbers under subtraction.

Starting point: Suppose r and s are rational numbers.

Conclusion to be shown: We must show that r - s is rational.

By definition, a rational number can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero.

Let r = a/b and s = c/d, where a, b, c, and d are integers and b, d are not equal to zero.

Now, we can express r - s as (a/b) - (c/d).

By the closure property of rational numbers under subtraction, the difference of two rational numbers is also a rational number.

Therefore, we can conclude that r - s is rational.

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Given question is incomplete, the complete question is below

Outline a proof of the following statement by writing the "starting point" and the "conclusion to be shown" in a proof of the statement.

∀ real numbers r and s, if r and s are rational then r−s is rational.

That is, complete the sentences below.

Proof: Suppose ___________.

We must show that ______________.

Answer:

True

Step-by-step explanation:

Answer:

Altitude of the right triangle is 4 cm.

Step-by-step explanation:

The altitude divides the right triangle's hypotenuse as 2 cm and 8 cm long.Let's use formula for similar right triangles proportional equation.I'll show you the diagram with steps.

Answer:

3,360 different codes.

Step-by-step explanation:

Here we have a set of 8 numbers:

{0, 0, 1, 5, 8, 9, 9, 9}

Now we want to make an 8th digit code with those numbers (each number can be used only once)

Now let's count the number of options for each digit in the code.

For the first digit, we will have 8 options

For the second digit, we will have 7 options (because one was already taken)

For the third digit, we will have 6 options (because two were already taken)

you already can see the pattern here:

For the fourth digit, we will have 5 options

For the fifth digit, we will have 4 options

For the sixth digit, we will have 3 options

For the seventh digit, we will have 2 options

For the eighth digit, we will have 1 option.

The total number of codes will be equal to the product between the numbers of options for each digit, then we have that the total number of codes is:

N = 8*7*6*5*4*3*2*1 = 8!

But wait, you can see that the 9 is repeated 3 times (then we have 3*2*1 = 3! permutations for the nines), and the 0 is repeated two times (then we have 2*1 = 2! permutations for the zeros).

Then we need to divide the number of different codes that we found above by 3! and 2!.

We get that the total number of different codes is:

C = [tex]\frac{8!}{2!*3!} = \frac{8*7*6*5*4}{2} = 8*7*6*5*2 = 3,360[/tex]

3,360 different codes.

The number of eight digit code she can make is, 3360.

If any number have n digits, then number of ways it can be arranged = [tex]n![/tex]

Given that,  Born date is,  08/05/1999

Total number of digits in birth date =  8

So, number of ways it can be arranged = [tex]8![/tex]

Since, In birth date 9 is three times and 0 is two times.

Therefore, number of arrangements = [tex]\frac{8!}{3!*2!}=3360[/tex]

Therefore, she can make 3360 eight digit codes from given birth date.

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The sample space, in terms of the condition (functional or defective) of each nozzle after a year, can be represented using the symbols "f" and "d" to denote a functional and defective nozzle, respectively.

The possible outcomes in the sample space can be described as a combination of these symbols. For example, if we have three nozzles, the sample space could include outcomes such as "fff" (all three nozzles are functional), "dfd" (the first and third nozzles are functional, while the second one is defective), "ffd" (the first two nozzles are functional, while the third one is defective), and so on.

Each outcome in the sample space corresponds to a particular arrangement or configuration of functional and defective nozzles after a year. The sample space encompasses all the possible combinations and provides a comprehensive representation of the different outcomes that can occur.

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The higher Variability in the math book data, it is not a valid inference to conclude that the grammar book has more words per page solely based on the mean comparison.

Based on the given information, the valid inference would be:

d) No, because the mean is larger in the grammar book.

The mean number of words per page in the grammar book is 49.7, while the mean number of words per page in the math book is 34.5. Since the mean in the grammar book is larger, Macy's claim seems valid at first glance. However, it is important to consider other factors such as the variability in the data.

The mean absolute deviation (MAD) provides a measure of the variability or spread of the data. In this case, the MAD for the grammar book is 18.4, while the MAD for the math book is 41.9. The fact that the MAD for the math book is significantly higher indicates that there is more variability in the number of words on each page in the math book.

This high variability in the math book data suggests that there could be pages with a significantly higher number of words, even though the mean is lower. On the other hand, the lower MAD for the grammar book suggests that the number of words per page in the grammar book is more consistent.

Therefore, considering the higher variability in the math book data, it is not a valid inference to conclude that the grammar book has more words per page solely based on the mean comparison.

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Incorrect. We are 95% confident that the interval from 26.2 and 27.4 captures the BMI of all young American women is the correct interpretation of the confidence interval.

A 95% confidence interval is a range of values that we can be 95% sure contains the true mean of the population. The 95% confidence interval for the mean body mass index (BMI) of young American women is 26.8 ± 0.6.

This means that we are 95% confident that the true mean BMI of young American women is between 26.2 and 27.4.

Consequently, the statement, "The mean BMI of young American women cannot be 28" is incorrect as the confidence interval does not include the value 28.

However, this does not imply that the true mean BMI of young American women cannot be 28.

A confidence interval is a statistical range that provides an estimate of the possible values for an unknown population parameter, such as a mean or proportion, based on a sample from that population. It provides a range of values within which the true population parameter is likely to fall, along with a specified level of confidence.

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a) John withdraws $1,300.

b) The annual effective yield rate for John's six-year investment is 2.09%.

a) To find the amount that John withdraws, we need to calculate the future value of his deposits after six years.

For the first three years, the deposits gain interest at a force of interest of 0.06. So after three years, the balance becomes $500 * (1 + 0.06)^3 = $595.44.

After three years, the interest rate changes to an effective rate of discount of 8%. Using the formula for the future value of a single sum with a discount rate, we can calculate the balance after six years:

$595.44 * (1 - 0.08)^3 = $429.97.

Therefore, John withdraws $429.97.

b) The annual effective yield rate can be found by calculating the rate of return on John's initial deposit over six years.

Let's assume John's initial deposit is $D. After three years, it grows to $D * (1 + 0.06)^3 = $1.191D. After six years, it becomes $1.191D * (1 - 0.08)^3 = $0.924D.

To find the annual effective yield rate, we need to solve the equation:

$D * (1 + r)^6 = $0.924D,

where r is the annual effective yield rate.

Simplifying the equation:

(1 + r)^6 = 0.924,

Taking the sixth root of both sides:

1 + r = 0.924^(1/6),

r = 0.0209.

Therefore, the annual effective yield rate for John's six-year investment is 2.09%.

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The given boundary value problem, y'' + 4y = 0, with boundary conditions y(0) = 0 and y(L) = 0, has a unique solution. Therefore, the solution to the given boundary value problem is y(x) = c2 sin(2x), where c2 is a constant and L = nπ/2.

To solve the given boundary value problem, we start by finding the general solution to the homogeneous differential equation y'' + 4y = 0. The characteristic equation associated with this differential equation is r^2 + 4 = 0, which has complex roots: r1 = 2i and r2 = -2i.

The general solution to the homogeneous equation is y(x) = c1 cos(2x) + c2 sin(2x), where c1 and c2 are constants. Now, we apply the boundary conditions to determine the specific solution.

Using the first boundary condition y(0) = 0, we have 0 = c1 cos(0) + c2 sin(0), which simplifies to c1 = 0. Therefore, the solution becomes y(x) = c2 sin(2x).

Now, we use the second boundary condition y(L) = 0. Substituting L for x in the solution, we get 0 = c2 sin(2L). For this equation to hold for all L, sin(2L) must be equal to zero, which means 2L = nπ, where n is an integer. Solving for L, we have L = nπ/2.

Therefore, the solution to the given boundary value problem is y(x) = c2 sin(2x), where c2 is a constant and L = nπ/2. Since both boundary conditions are satisfied for y(x) = 0, we conclude that the only solution to the problem is y(x) = 0.

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There is no region in the complex plane to shade for the given inequality.

To shade the region in the complex plane defined by {z ∈ C: 2+2+2i| ≤ 2}, let's break down the problem step by step.

The inequality given is: |2+2+2i| ≤ 2

First, let's simplify the expression within the absolute value:

2 + 2 + 2i = 4 + 2i

The inequality now becomes: |4 + 2i| ≤ 2

To find the absolute value of a complex number z = a + bi, we use the formula: |z| = √(a² + b²)

Applying this formula to our complex number, we have:

|4 + 2i| = √(4² + 2²) = √(16 + 4) = √20 = 2√5

Now the inequality becomes: 2√5 ≤ 2

To solve for √5, we divide both sides of the inequality by 2:

√5 ≤ 1

Since the square root of 5 is approximately 2.236, and it is not less than or equal to 1, the inequality is not satisfied.

Therefore, there is no region in the complex plane to shade for the given inequality.

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(a) The Y-intercept, bo = 6, implies that when the value of X is 0, the mean value of Y is 6.

(b) The slope, by = 4, implies that for each increase of 1 unit in X, the value of Y is expected to increase by 4 units.

(c) The mean value of Y for X = 4 is predicted to be 22.

The Y-intercept, bo, represents the starting point of the prediction line. In this case, when X is 0, the mean value of Y is expected to be 6. This implies that before any increase or decrease in X, the average value of Y starts at 6.

The slope, by = 4, indicates the rate at which Y is expected to change for each unit increase in X. Therefore, for every 1 unit increase in X, the value of Y is expected to increase by 4 units. This implies a linear relationship between X and Y, where the increase or decrease in X directly influences the corresponding change in Y.

To predict the mean value of Y for X = 4, we can use the prediction line equation: Y = 6 + 4x. Substituting X = 4 into the equation, we get: Y = 6 + 4(4) = 6 + 16 = 22. Therefore, the mean value of Y for X = 4 is predicted to be 22.

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The maximum value of the objective function Z is 1800. The optimal values for the decision variables are X1 = 10, X2 = 0, and X3 = 0. The constraints are satisfied, and the optimal solution has been reached using the Simplex Method.

To compute the given problem using the Simplex Method, we need to convert it into a standard form.

The standard form of a linear programming problem consists of maximizing or minimizing a linear objective function subject to linear inequality constraints and non-negativity constraints.

Let's rewrite the problem in standard form:

Maximize:

Z = 50X1 + 20X2 + 10X3

Subject to the constraints:

2X1 + 4X2 + 5X3 <= 200

X1 + X3 <= 90

X1 + 2X2 <= 30

X1, X2, X3 >= 0

To convert the problem into standard form, we introduce slack variables (S1, S2, S3) for each constraint and rewrite the constraints as equalities:

2X1 + 4X2 + 5X3 + S1 = 200

X1 + X3 + S2 = 90

X1 + 2X2 + S3 = 30

Now, we have the following equations:

Objective function:

Z = 50X1 + 20X2 + 10X3 + 0S1 + 0S2 + 0S3

Constraints:

2X1 + 4X2 + 5X3 + S1 = 200

X1 + X3 + S2 = 90

X1 + 2X2 + S3 = 30

X1, X2, X3, S1, S2, S3 >= 0

Next, we will create a table representing the initial simplex tableau:

  | X1 | X2 | X3 | S1 | S2 | S3 | RHS |

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

Z  | 50 | 20 | 10 | 0  | 0  | 0  | 0   |

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

S1 | 2  | 4  | 5  | 1  | 0  | 0  | 200 |

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

S2 | 1  | 0  | 1  | 0  | 1  | 0  | 90  |

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

S3 | 1  | 2  | 0  | 0  | 0  | 1  | 30  |

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

To compute the optimal solution using the Simplex Method, we'll perform iterations by applying the simplex pivot operations until we reach an optimal solution.

Iterating through the simplex method steps, we can find the following tableau:

  | X1 | X2 | X3 | S1 | S2 | S3 | RHS |

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

Z  | 0  | 40 | 10 | 0  | 0  | -500| 1800|

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

S1 | 0  | 3  | 5  | 1  | 0  | -40 | 120  |

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

S2 | 1  | 0  | 1  | 0  | 1  | 0   | 90   |

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

X1 | 0  | 2  | 0  | 0  | 0  | -1  | 10   |

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

The optimal solution is Z = 1800, X1 = 10, X2 = 0, X3 = 0, S1 = 120, S2 = 90, S3 = 0.

Therefore, the maximum value of Z is 1800, and the values of X1, X2, and X3 that maximize Z are 10, 0, and 0, respectively.

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the volume of the solid within the cone is ρ²(12sin⁴(φ) - 11sin²(φ) + 3) = 0

To find the volume of the solid within the cone and between the spheres using spherical coordinates, we need to determine the limits of integration for the variables ρ, θ, and φ.

In spherical coordinates, we have the following relationships:

x = ρsin(φ)cos(θ)

y = ρsin(φ)sin(θ)

z = ρcos(φ)

Given:

Cone equation: z = 13x² + 3y²

Sphere equation: x² + y² + z² = 4

Plane equation: x + y + z = 25

First, let's determine the limits for the variable ρ:

Since we are dealing with spheres, we can set ρ to range from 0 to the radius of the larger sphere, which is 2.

0 ≤ ρ ≤ 2

Next, let's determine the limits for the variable θ:

The solid lies within the entire range of θ, which is from 0 to 2π.

0 ≤ θ ≤ 2π

Finally, let's determine the limits for the variable φ:

To find the limits for φ, we need to consider the intersection between the cone and the spheres.

1. Intersection of the cone and the larger sphere:

Substituting the equations of the cone and the larger sphere, we get:

13x² + 3y² = 4 - x² - y² - z²

12x² + 4y² + z² = 4

12(ρsin(φ)cos(θ))² + 4(ρsin(φ)sin(θ))² + (ρcos(φ))² = 4

12ρ²sin²(φ)cos²(θ) + 4ρ²sin²(φ)sin²(θ) + ρ²cos²(φ) = 4

ρ²(12sin²(φ)cos²(θ) + 4sin²(φ)sin²(θ) + cos²(φ)) = 4

ρ²(12sin²(φ)(1 - sin²(θ)) + cos²(φ)) = 4

ρ²(12sin²(φ) - 12sin²(φ)sin²(θ) + cos²(φ)) = 4

ρ²(12sin²(φ) - 12sin²(φ)(1 - cos²(θ)) + cos²(φ)) = 4

ρ²(12sin²(φ) - 12sin²(φ) + 12sin²(φ)cos²(θ) + cos²(φ)) = 4

ρ²(12sin²(φ)cos²(θ) + cos²(φ)) = 4

We need to solve this equation for ρ. Simplifying further:

ρ²(12sin²(φ)cos²(θ) + cos²(φ)) = 4

ρ²(12sin²(φ)(1 - sin²(φ)) + cos²(φ)) = 4

ρ²(12sin²(φ) - 12sin⁴(φ) + cos²(φ)) = 4

ρ²(12sin²(φ) - 12sin⁴(φ) + 1 - sin²(φ)) = 4

ρ²(11sin²(φ) - 12sin⁴(φ) + 1) = 4

ρ²(12sin⁴(φ) - 11sin²(φ) + 3) = 0

Since ρ cannot be negative

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Answer:

axis of symm:  x = -1

vertex:  (-1, -4)

domain:  all real numbers

range:  y ≥ -4

x-intercepts:  1, 3

y-intercept:  -3

minimum at (-1, -4)

Step-by-step explanation:

Answer:

f(3) = 3

Step-by-step explanation:

f(1) = 1

f(2) = 2

f(n) = f(n − 2) + f(n − 1)

f(3) = f(3 - 2) + f(3 - 1)

     = f(1) + f(2) = 1 + 2 = 3

Special Note:  Have you heard of the Fibonacci sequence?  

The formula f(n) = f(n − 2) + f(n − 1) is used to find the terms of the of the  Fibonacci sequence

Answer:

Step-by-step explanation:

Hi

Answer:

15

Step-by-step explanation:

The ratio of (the foot of the ladder to the person) : (the person's height) is equal to (the wall to the foot of the ladder) : (the height of the wall) because the triangles are similar. So, if you solve the problem you will get:

(x= the height of the wall)

6:6 = 6+9:x

6:6 = 15:x

1:1=15:x

x=15

The height of the wall is 15 feet.

Answer:

Every week, the radioactivity decreases by a factor of 5

After 1 week, it goes from 5,000 to 1,000

2 weeks  1,000 / 5 =200

3 weeks 200 / 5 = 40

4 weeks 40 / 5 = 8

5 weeks 8 / 5 = 1.6

6 weeks 1.6 / 5 = 0.064

7 weeks 0.064 / 5 = 0.0128

8 weeks .0128 / 5 = 0.00256

9 weeks 0.00256 / 5 = 0.000512

10 weeks 0.000512 / 5 = 0.0001024

Step-by-step explanation:

Answer:

It is 8/3125

      8

   3125

Step-by-step explanation:

It is the second option

Answer:

V = 314

Step-by-step explanation:

Volume of a cone:

V = πr²(h/3)

Given:

r = 5

h = 12

Work:

V = πr²(h/3)

V = 3.14(5²)(12/3)

V = 3.14(25)(4)

V = 314

Answer:

The volume of the cone is 311.1in³.

Step-by-step explanation:

V = ⅓Bh where B = πr²

V = ⅓πr²h

V = ⅓(22/7)(5²)12

V = 0.33(3.142)(25)12

V = 311.1in³

By using the Pythagorean theorem, we calculate that we need 38 feet of line You would need an extra foot on either end for attaching the line to the fence.

To find the length of the line needed, you can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the line) is equal to the sum of the squares of the other two sides.

In this case, the two sides are the length (20 feet) and the width (30 feet) of the backyard. The line represents the hypotenuse.

Using the Pythagorean theorem, we can calculate the length of the line:

Line = √(Length² + Width²)

Line = √(20² + 30²)

Line = √(400 + 900)

Line = √1300

Line ≈ 36.06 feet

Since you need to add an extra foot on either end, the total length of the line needed would be:

Total length = Line + 2 feet

Total length ≈ 36.06 feet + 2 feet ≈ 38.06 feet

Rounding to the most appropriate whole number, you would need approximately 38 feet of line.

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Using the identities to find values of the sine and cosine functions of the function the angle measure,

cos 2x = 1

sin 2x = -8√17/17.

Given that tan x = -4, we can determine the values of cos 2x and sin 2x.

Using the identity tan x = sin x / cos x, we have sin x = -4 cos x.

Now, we can use the Pythagorean identity sin² x + cos² x = 1 to solve for cos x:

(-4 cos x)² + cos² x = 1

16 cos² x + cos^2 x = 1

17 cos² x = 1

cos² x = 1/17

cos x = ± √(1/17)

Since we know that cos x > 0, we take cos x = √(1/17).

Next, we can find sin x using sin x = -4 cos x:

sin x = -4 × √(1/17) = -4/√17 = -4√17/17.

Now, we can find cos 2x and sin 2x using the double angle identities:

cos 2x = cos² x - sin² x = (1/17) - (-16/17) = 17/17 = 1

sin 2x = 2 sin x cos x = 2 × (-4√17/17) × √(1/17) = -8√17/17.

Therefore, cos 2x = 1 and sin 2x = -8√17/17.

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Answer:

No x-intercepts

y-intercepts: (0,2)

Step-by-step explanation:

Answer:

Angle K is 55 degrees

Step-by-step explanation:

Angle K corresponds to Angle R

Jennifer’s gross pay for the year was $32,854. B: She paid $3,982.48 in federal income tax. C:$1,971.24, D:$476.38, E: $2,447.62.

We have given the images

We have to determine the statements a,b,c,d e, and f.

A tax is a compulsory financial charge or some other type of levy imposed on a taxpayer by a governmental organization in order to fund government spending and various public expenditures.

A: Jennifer’s gross pay for the year was $32,854.

B: She paid $3,982.48 in federal income tax.

C:$1,971.24

D:$476.38

E: $2,447.62

F:Jennifer will receive a refund. She will receive $231.48 in her tax refund because her income is $32,854 and she filed for taxes under the status that she is single so she only needs to pay $3,751 in Taxes but she ended up paying $3,982.48 which is $231.48 over how much she should’ve paid.

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Answer:

C

24Step-by-step explanation:

Trust me on this one.

Answer:

I believe the answer is y=4x+4

Step-by-step explanation:

because the 4 that is on the  Y axis is the Y axis

and the 4 on the x axis will be the one with the variable

Hopefully this helps.

Answer:

false

it can have more

Answer:

Yes

What solid shape do the two nets make?

1. Rectangular Prism

2. Triangular Prism

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