The well ordering property can be used to show that there is a unique greatest common divisor of two positive integers. Let a and b be positive integers, and let S be the set of positive integers of the form as + bt, where s and t are integers.
a) Show that S is nonempty
b) Use the well ordering property to show that S has a smallest element c.
c) Show that if d is a common divisor of a and b, then d is a divisor of c.
d) Show that c | a and c | b [Hint: First, assume that c does not divide a. Then a = qc + r, where 0 < r < c. Show that r E S, contradicting the choice of c.]
e) Conclude from (c ) and (d) that the greatest common divisor of a and b exists. Finish the proof by showing that this greatest common divisor is unique.

The ecologist needs to carefully choose the size of the sampling region based on the pdf of R.

Sampling is the process of selecting a representative subset of individuals or objects from a larger population to study or analyze.

To ensure that their data is representative of the entire ecosystem, it is important to choose a sampling region that is large enough and well-defined.

In this scenario, an ecologist wishes to mark off a circular sampling region with a fixed radius of 13 meters. However, due to natural variation, the actual radius of the sampling region is a random variable, denoted by R. The probability density function (pdf) of R is given as

=> s(r) = { 1-(13-r)] ¹²,

which means that the probability of a particular value of R is proportional to the area of a circle with that radius.

Now, when the ecologist takes a sample, it is important to consider the effects of this random variation in the sampling region. If the radius of the sampling region is too small, it may not capture the full range of variability in the ecosystem.

On the other hand, if the radius is too large, it may include areas that are not representative of the ecosystem as a whole.

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The center and radius of the circle in the equation is given by (-5, 3) and 4 respectively.

A circle is a round-shaped figure that has no corners or edges. In geometry, a circle can be defined as a closed, two-dimensional curved shape.

Given that, an equation of a circle, (x+5)²+(y-2)² = 4, we need to identify the center and radius of the circle,

The standard form for the equation of a circle is (x−h)²+(y−k)² = r². The center is (h, k) and the radius measures r units.

Comparing with the standard form, we get,

(h, k) = (-5, 2) and r² = 4

Therefore, the center is (-5, 2) and the radius is r = 4

Hence, the center and radius of the circle in the equation is given by (-5, 3) and 4 respectively.

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u is the slope of the line passing through the points (1, -3) and (1, 0).

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.

The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is

m=y₂-y₁/x₂-x₁

We have to find the slope of the line passing through the points (1, -3) and (1, 0)

m=0-(-3)/1-1

=3/0

Undefined

Hence, the slope of the line passing through the points (1, -3) and (1, 0) is u.

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Answer:

Step-by-step explanation:

e = 20 * (1+6/100) = 20 *  53 50  = 20 * 1.06 =$ 21.2

Answer:

s=2

Step-by-step explanation:

We do not have strong evidence against the 0.07 claim so, the correct answer is: No, the p-value is over 0.10.

Let's calculate the p-value and compare it to the level of significance (alpha) to determine if there is strong evidence against the 0.07 claim.

Let p be the true proportion of employees arriving late. The null hypothesis is that p = 0.07 and the alternative hypothesis is that p > 0.07.

The sample proportion of employees arriving late is x/n = 23/200 = 0.115. Under the null hypothesis, the sampling distribution of the sample proportion follows a normal distribution with a mean of 0.07 and standard deviation [tex]\sqrt{((0.07 * 0.93) / 200)}[/tex] = 0.0319.

The z-score corresponding to the sample proportion is (0.115 - 0.07) / 0.0319 = 1.41. The p-value is the probability of observing a z-score greater than or equal to 1.41 under the null hypothesis, which can be calculated as P(Z >= 1.41) = 0.078.

Since the p-value is greater than the level of significance alpha (which is typically 0.05 or 0.01), we fail to reject the null hypothesis that the true proportion of employees arriving late is 0.07.

Therefore, we do not have strong evidence against the 0.07 claim so, the correct answer is: No, the p-value is over 0.10.

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The illustration of the types of solutions of systems of equations is explained below.

An equation is written in the form of variables and constants separated by the operation of multiplication and division,

An equation states that terms in different forms on both sides of the equality sign are equal.

Multiplication and division do not separate the terms of an equation.

Given, 5 - 4 + 7x + 1 = ax + b.

7x + 2 = ax = b.

Now, If we put b ≠ 2 and x = 7 we have,

7x + 2 = 7x + b.

2 = b, but b ≠ 2 so no solution.

5 - 4 + 7x + 1 = ax + b

7x + 2 = ax + b.

To have one solution we can assume any x value except 7 and any b value except 2,

7x + 2 = 6x + 3.

x = - 1.

5 - 4 + 7x + 1 = ax + b

To have infinitely many solutions both sides must be some constant multiple of each other.

7x + 2 = k(ax + b).

7x + 2 = 2(7x + 2).

7x + 2 = 14x + 4.

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Answer:

1 - 3i, 4

Step-by-step explanation:

The polynomial has real coefficients, so if it has a complex root, it must have at least two complex roots which are complex conjugates.

Since 1 + 3i is a root, then 1 - 3i is also a root.

Two of the factors of the polynomial are:

x - (1 + 3i) and x - (1 - 3i)

Simplify the factors above:

x - 1 - 3i and x - 1 + 3i

Find their product:

(x - 1 - 3i)(x - 1 + 3i) =

Rewrite them showing we have the product of  a sum and a difference.

= [(x - 1) - 3i][(x - 1) + 3i]

Multiply the factors above noticing they are a sum and a difference which follows the pattern (a + b)(a - b) = a² - b².

= (x - 1)² - (3i)²

= x² - 2x + 1 - 9(-1)

= x² - 2x + 10

Now we divide the original polynomial by the product we just found using long division.

                                        x - 4

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

         x² - 2x + 10      |    x³ - 6x² + 18x - 40

                                  -   x³ - 2x² + 10x

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

                                            -4x² + 8x - 40

                                      -     -4x² + 8x - 40

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

                                                0 +    0  + 0

Now we know that x³ - 6x² + 18x - 40 factors into (x² - 2x + 10)(x - 4).

(x² - 2x + 10)(x - 4) = 0

From x² - 2x + 10, we have roots x = 1 + 3i and x = 1 - 3i.

x - 4 = 0

x = 4

From x - 4, we have root x = 4.

Answer: 1 - 3i, 4

The area of a rectangle with the given length breadth is 1.5 square feet.

The area occupied by a rectangle within its boundary is called the area of the rectangle. The formula to find the area of a rectangle is Area = Length × Breadth.

Given that, length of a rectangle= 3 ft and width of a rectangle= 1/2 ft.

Area is defined as the total space taken up by a flat surface or shape of an object.

Here, area of a rectangle =Length × Breadth

= 3×1/2

= 3×0.5

= 1.5 square feet

Therefore, the area of a rectangle is 1.5 square feet.

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The fourth number ​is equal to -6.

Mathematically, the arithmetic average for a given data set can be calculated by using this formula:

Average = [F(x)]/n

Next, we would calculate the arithmetic average for the four numbers (data set) as follows:

Let the fourth number be represented by the variable n.

Total numbers = 17 + 13 + 16 + n

Total numbers = 46 + n

10 = 46 + n/4

40 = 46 + n

n = 40 - 46

n = -6

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Answer:

13.3 miles

Step-by-step explanation:

Picture a right triangle with legs of 3 and 13.  The hypotenuse is the straight line distance back to the starting point.  Use the Pythagorean theorem to find the hypotenuse (c):

c² = 13² + 3² = 178

c = √178 = 13.3 miles

"79 % of all students at the local university voted in the last election" the numerical value describes a population parameter.

Population Parameter is a quantity/measurement that can be used as material for follow-up on population management activities.

 Population is a generalized area consisting of: objects/subjects that have certain qualities and characteristics determined by the researcher to be studied and then drawn conclusions.

The population or universe is the total number of units or individuals whose characteristics are to be studied. And these units are called units of analysis, and can be people, institutions, things, etc.

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The expression is not true.

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

Given that;

The expression is,

⇒ 3 (6 - 4) = - 3

Now, We can check as;

⇒ 3 (6 - 4) = - 3

LHS;

⇒ 3 (6 - 4)

⇒ 3 × 2

⇒ 6

RHS;

⇒ - 3

Hence, The expression is not true.

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Problem 4

Explanation:

Draw a horizontal divider line to split the figure into two rectangles.

The rectangle up top is 30 by 30, so it has area 30*30 = 900 square inches.

The rectangle on the bottom has area 50*30 = 1500 square inches.

total = 900+1500 = 2400 square inches

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

Problem 5

Explanation:

Draw a horizontal divider line to split the figure into two rectangles.

A = area of rectangle on top = 2*2 = 4 square inches

B = area of rectangle on bottom = 4*2 = 8 square inches

C = total = A+B = 4+8 = 12 square inches

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

Problem 6

Explanation:

Draw a horizontal divider line to get two rectangles.

A = area of rectangle on top = 36*36 = 1296 square yards

B = area of rectangle on bottom = 72*18 = 1296 square yards

C = total = A+B = 1296+1296 = 2592 square yards.

You have the correct answer. Nice work.

Answer:

4)  2400 in²

5)  12 in²

6)  2592 yd²

Step-by-step explanation:

To calculate the area of each given composite figure, divide the figure into two rectangles and sum the area of the two rectangles.

[tex]\boxed{\begin{minipage}{5cm}\underline{Area of a rectangle}\\\\$A=w\cdot l$\\\\where:\\ \phantom{ww} $\bullet$ \quad $w$ is the width.\\ \phantom{ww} $\bullet$ \quad $l$ is the length.\\\end{minipage}}[/tex]

Separate the figure into two rectangles by drawing a horizontal line (see attachment).

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&= 30\cdot 30+50\cdot 30\\&=900+1500\\&=2400\; \sf in^2\end{aligned}[/tex]

Separate the figure into two rectangles by drawing a horizontal line (see attachment).

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&=2 \cdot 2+ 4\cdot2 \\&=4+8\\&=12\; \sf in^2\end{aligned}[/tex]

Separate the figure into two rectangles by drawing a horizontal line (see attachment).

[tex]\begin{aligned}\textsf{Total Area}&=\textsf{Area 1}+\textsf{Area 2}\\&=36 \cdot36 +72 \cdot18 \\&=1296+1296\\&=2592 \; \sf yd^2\end{aligned}[/tex]

Answer:

1 ticket - $4.25

5 tickets - $21.25

Step-by-step explanation:

12.75/3=4.25 which is one ticket then take the price of one ticket and multiply it by 5, 4.25*5=21.25.

Without knowing the specific ingredients and proportions used in a smoothie, it is impossible to definitively say whether any ingredients make up an equal amount. However, it is likely that the proportions of ingredients in a smoothie vary based on the desired taste, texture, and nutritional content.

For example, a smoothie may have more fruit than liquid to create a thicker consistency, or more liquid than fruit to create a thinner consistency. Additionally, the amount of sweeteners, such as sugar or honey, may vary based on personal preference and the natural sweetness of the fruit used.

In conclusion, without more specific information about the ingredients and proportions used in a particular smoothie, it is not possible to determine whether any ingredients make up an equal amount. The proportions of ingredients in a smoothie are likely to vary based on various factors and personal preference.

If the chute does not open until 1 min after he leaves the helicopter, then the equation to calculate the seconds it will he hit the ground is (1/2)mv² + ∫ F(v) dv = 0

The force due to air resistance is assumed to be proportional to the velocity of the parachutist, and we will use this information to determine the time it takes for the parachutist to reach the ground with and without a chute.

The force of air resistance is proportional to the velocity of the parachutist. This means that the larger the velocity, the larger the force of air resistance.

As the velocity increases, the force of air resistance also increases, which reduces the kinetic energy. When the chute opens, the force of air resistance increases even more, which further reduces the kinetic energy and slows down the parachutist.

To determine the time it takes for the parachutist to hit the ground, we can use the following equation:

PE + KE + work done by air resistance = PE + KE

where PE is the gravitational potential energy, KE is the kinetic energy, and work done by air resistance is the work done by the force of air resistance.

When the chute is closed, we can write:

mgh = (1/2)mv² + ∫ F(v) dv

where m is the mass of the parachutist, g is the acceleration due to gravity, h is the height of the helicopter above the ground, and the integral is taken over the velocity range from 0 to v.

Similarly, when the chute is open, we can write:

mgh = (1/2)mv² + ∫ F(v) dv

where b = b4 and the integral is taken over the velocity range from 0 to v.

To solve for the time it takes for the parachutist to hit the ground, we need to determine the velocity at which he hits the ground. This is the velocity where KE = 0, and we can find it by equating the expressions for KE and work done by air resistance when the parachutist hits the ground:

(1/2)mv² + ∫ F(v) dv = 0

We can solve these equations numerically to find the time it takes for the parachutist to hit the ground with and without the chute.

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This function is continuous everywhere, as the limit of sinx/x as x approaches 0 is 1, and f(0) = 1. It cannot be extended to a larger domain, as it is already continuous everywhere.

For example, when x = 0.1, f(x) = 0.998334166, and when x = 0.001, f(x) = 0.998999667.This function is continuous everywhere, as the limit of sinx/x as x approaches 0 is 1, and f(0) = 1. This means that there are no breaks or discontinuities in the function, and it is continuous everywhere. To demonstrate this, we can look at specific values of x. For example, when x = 0.1, f(x) = 0.998334166, and when x = 0.001, f(x) = 0.998999667. This shows that the function is smoothly transitioning from one value to the next, and is therefore continuous everywhere. The function cannot be extended to a larger domain, as it is already continuous everywhere. This means that the function is defined everywhere on its domain, and there is no need to extend it.

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a. 1

b. -1

?

90

10

c. 0

d. -2

The percent of increase in the height of the stalks is given as follows:

0%.

The percent of increase is obtained applying the proportions in the context of this problem.

A proportion is applied because the percent increase is given by the division of the increase by the initial amount, and multiplied by 100%.

The parameters for this problem are given as follows:

Hence the percent increase is given as follows:

0/52.6 x 100% = 0%.

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The total number of golf balls with Jack as -

{n} = 3b + 6.

Given is that Jack had 3 bags of golf balls with {b} balls in each bag; then his friend gave him 6 golf balls more.

We can write the total number of golf balls with Jack as -

{n} = 3b + 6

Therefore, the total number of golf balls with Jack as -

{n} = 3b + 6.

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{Question in english -

Jack had 3 bags of golf balls with b balls in each bag; then his friend gave him 6 golf balls more. How many golf balls does Jack have now?}

The missing step is 9(2p+ 1)/3 = 6p/3 + 15/3.

What is division?

The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.

Given:
An original equation,

9(2p+ 1) = 6p + 15.

Result of the first step:

3(2p + 1) = 2p + 5

To find the missing step:

Divide the original equation by 3 into both sides,

we get,

9(2p+ 1)/3 = 6p/3 + 15/3.

Simplifying further,

3(2p + 1) = 2p + 5

Therefore, the required expression is 9(2p+ 1)/3 = 6p/3 + 15/3.

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Answer:

Step-by-step explanation:

l

Answer:

15+1

Step-by-step explanation:

Company A will charge $7 more than company B.

Expression in maths is defined as the relation of numbers variables and functions by using mathematical signs like addition, subtraction, multiplication and division.

Given that a family wants to rent a car to go on vacation. Company A charges ​$55.50 and 16 cents per mile. Company B charges ​$50.50 and 9 cents per mile.

The expression for the charges by company A is,

C = 16x + 55.50

The expression for the charges by company B,

C = 9x + 55.50

The amount of company A more than company B is calculated as:-

Amount = ( 16 + 55.50) - ( 9 + 55.50 )

Amount = 71.50 - 64.50

Amount = $7

Hence, company A will charge $7 more.

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we can conclude that the value of n for which [tex]t_n[/tex] = 0.3325 is between 3 and 4 and the value of n for which [tex]S_n[/tex]= -6 is 6.

A) To find the value of n for which [tex]t_n[/tex] = 0.3325, we can use trial and error method.

Starting with n=1, we have:

[tex]t_1 = (1^2 - 1)/(3 * 1^2) = 0[/tex]

For n=2, we have:

[tex]t_2 = (2^2 - 1)/(3 * 2^2) = (3)/(12) = 0.25[/tex]

For n=3, we have:

[tex]t_3 = (3^2 - 1)/(3 * 3^2) = (8)/(27) = 0.296296...[/tex]

Continuing in this way, we find that t_3 < 0.3325 < t_4, where

[tex]t_4 = (4^2 - 1)/(3 * 4^2) = (15)/(48) = 0.3125[/tex]

Thus, we can conclude that the value of n for which [tex]t_n[/tex] = 0.3325 is between 3 and 4, and it can be found more accurately by using numerical methods such as bisection or Newton's method.

B) To find the sum of the arithmetic series 16 + 13 + 10 + ..., we can use the formula for the sum of an arithmetic series:

[tex]S_n = n/2 * (a_1 + a_n),[/tex]

where n is the number of terms in the series, [tex]a_1[/tex] is the first term (16 in this case), and a_n is the nth term.

Let's assume that the nth term is represented by [tex]a_n = 16 - 3 * (n - 1),[/tex] so that a_1 = 16 and a_n = 16 - 3 * (n - 1).

Then, we have:

S_n = n/2 * (16 + 16 - 3 * (n - 1)) = n/2 * (16 + 16 - 3n + 3) = n/2 * (16 + 16 - 3n + 3) = n/2 * (32 - 3n + 3) = 16n - (3/2)n^2 + (3/2)n

We want to find the value of n for which S_n = -6, so we can substitute the value of S_n into the equation and solve for n:

[tex]S_n = n/2 * (16 + 16 - 3 * (n - 1)) = n/2 * (16 + 16 - 3n + 3) = n/2 * (16 + 16 - 3n + 3) = n/2 * (32 - 3n + 3) = 16n - (3/2)n^2 + (3/2)n[/tex]

[tex]-6 = 16n - (3/2)n^2 + (3/2)n[/tex]

Expanding the equation and solving for n, we find that n = 6.

Thus, the value of n for which [tex]S_n[/tex]= -6 is 6.

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Answer:

the cost of renting a truck from both companies will be the same for 100 miles.

Step-by-step explanation:

Let's assume that the cost of renting a truck from Company A for x miles is C_A(x) and the cost of renting a truck from Company B for x miles is C_B(x).

From the given information, we know that:

C_A(x) = 45 + 0.10x

C_B(x) = 25 + 0.30x

We need to find the value of x for which C_A(x) = C_B(x). So we can set the two equations equal to each other and solve for x:

45 + 0.10x = 25 + 0.30x

20 = 0.20x

x = 100

Therefore, the cost of renting a truck from both companies will be the same for 100 miles.

27 inches is the radius of a sphere with a volume of 26, 244π cubic inches.

a three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.

Given that volume of a sphere is  26, 244π cubic inches.

We need to find the radius of sphere.

V=4/3πr³

26, 244π=4/3πr³

26, 244=1.333r³

Divide both sides by 1.333

26,244/1.333=r³

19688=r³

Cube root on both sides

∛19688 = r

27=r

Hence, 27 inches is the radius of a sphere with a volume of 26, 244π cubic inches.

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