Define a 3-chain to be a (not necessarily contiguous) subsequence of three integers, which is either monotonically increasing or monotonically decreasing. We will show here that any sequence of five distinct integers will contain a 3-chain. Write the sequence as a1, a2, a3, a4, a5. Note that a monotonically increasing sequences is one in which each term is greater than or equal to the previous term. Similarly, a monotonically decreasing sequence is one in which each term is less than or equal to the previous term. Lastly, a subsequence is a sequence derived from the original sequence by deleting some elements without changing the location of the remaining elements.
(a) [4 pts] Assume that a1 < a2. Show that if there is no 3-chain in our sequence, then a3 must be less than a1. (Hint: consider a4!)
(b) [2 pts] Using the previous part, show that if a1 < a2 and there is no 3-chain in our sequence, then a3 < a4 < a2.
(c) [2 pts] Assuming that a1 < a2 and a3 < a4 < a2, show that any value of a5 must result in a 3-chain.
(d) [4 pts] Using the previous parts, prove by contradiction that any sequence of five distinct integers must contain a 3-chain.

The required ratio of Quantricide to water is 1 fl oz of Quantricide to 64 fl oz of water.

The ratio can be defined as the proportion of the fraction of one quantity towards others. e.g.- water in milk.

Here,
Since we are mixing 2 fl oz of Quantricide with one gallon of water, the ratio of Quantricide to water is, 2 fl oz : 1 gallon

However, it is common to express ratios in simplified form, so we can convert the units of gallons to fluid ounces to get a ratio in terms of fluid ounces,
1 gallon = 128 fl oz (since 1 gallon contains 128 fluid ounces)

So, the ratio of Quantricide to water in terms of fluid ounces is:

2 fl oz : 128 fl oz

This ratio can be simplified by dividing both sides by 2:

1 fl oz : 64 fl oz

Therefore, the ratio of Quantricide to water is 1 fl oz of Quantricide to 64 fl oz of water.

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Answer:

To find the fourth derivative of f(x), we can use the fact that f(x) can be expressed as:

f(x) = 5x^4 (1 - x)^-1

Then, using the product rule repeatedly, we can find the derivatives of f(x) up to the fourth order:

f'(x) = 20x^3 (1 - x)^-1 - 5x^4 (1 - x)^-2

f''(x) = 60x^2 (1 - x)^-1 + 40x^3 (1 - x)^-2 + 10x^4 (1 - x)^-3

f'''(x) = 120x (1 - x)^-1 + 180x^2 (1 - x)^-2 + 120x^3 (1 - x)^-3 + 20x^4 (1 - x)^-4

f^4(x) = 120 (1 - x)^-1 + 720x (1 - x)^-2 + 1080x^2 (1 - x)^-3 + 480x^3 (1 - x)^-4 + 60x^4 (1 - x)^-5

So, we have found the fourth derivative of f(x) without using the quotient rule four times.

The fourth derivative of the function f(x) = 5x⁴/(1 - x)⁻¹ is,

f''''(x) = 120/(1 - x)⁻¹ + 720x/(1 - x)⁻² + 1080x²/(1 - x)⁻³ + 480x³/(1 - x)⁻⁴

+ 60x⁴/(1 - x)⁻⁵.

A technique for determining a function's derivative is differentiation. Mathematicians use a procedure called differentiation to determine a function's instantaneous rate of change based on one of its variables.

We have to find the fourth derivative of f(x), f(x) = 5x⁴/(1 - x)⁻¹

The derivatives of f(x) up to the fourth order can then be discovered by continually applying the product rule,

f'(x) = 20x³/(1 - x)⁻¹ - 5x⁴/(1 - x)⁻²

f''(x) = 60x²/(1 - x)⁻¹ + 40x³/(1 - x)⁻² + 10x⁴/(1 - x)⁻³

f'''(x) = 120x/(1 - x)⁻¹ + 180x²/(1 - x)⁻² + 120x³/(1 - x)⁻³ + 20x⁴/(1 - x)⁻⁴

f''''(x) = 120/(1 - x)⁻¹ + 720x/(1 - x)⁻² + 1080x²/(1 - x)⁻³ + 480x³/(1 - x)⁻⁴ +

60x⁴/(1 - x)⁻⁵.

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a) The solution to the equation with the initial condition is x = 40[tex]e^{(t/1000)}[/tex].

b) Approximately 40, 42, 45, and 49 rabbits on the island after 1, 5, 10, and 15 months.

a) To find the solution to the equation with the initial condition, do we need to solve the differential equation x? = x ? (1/1000)x² with the initial condition x(0) = 40.

Separating variables and integrating, we get:

dx/x = (1/1000) dt

Integrating both sides, we get:

ln|x| = (1/1000) t + C

where C is a constant of integration.

Using the initial condition x(0) = 40, we can solve for C:

ln|40| = C

C = ln|40|

Substituting this value of C, we get:

ln|x| = (1/1000) t + ln|40|

Simplifying, we get:

x = [tex]e^{(ln|40|+(1/1000) t)[/tex] = 40[tex]e^{(t/1000)}[/tex]

Therefore, the solution to the differential equation with the initial condition is x = 40[tex]e^{(t/1000)}[/tex].

b) To find the number of rabbits on the island after 1 month, 5 months, 10 months, and 15 months, we simply substitute the given values of t into the solution and round to the nearest integer.

After 1 month, x = 40[tex]e^{(1/1000)}[/tex] ≈ 40.04, so there are approximately 40 rabbits on the island.

After 5 months, x = 40[tex]e^{(5/1000)}[/tex] ≈ 41.67, so there are approximately 42 rabbits on the island.

After 10 months, x = 40[tex]e^{(10/1000)}[/tex] ≈ 45.26, so there are approximately 45 rabbits on the island.

After 15 months, x = 40[tex]e^{(15/1000)}[/tex] ≈ 49.28, so there are approximately 49 rabbits on the island.

Therefore, there are approximately 40, 42, 45, and 49 rabbits on the island after 1, 5, 10, and 15 months, respectively.

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Answer:

Step-by-step explanation:

133 ft

Answer:

C

Step-by-step explanation:

in the triangle with respect to the 50° angle, the side opposite is c and the adjacent side is 14.

thus using the tangent ratio in the right triangle

tan50° = [tex]\frac{opposite}{adjacent}[/tex] = [tex]\frac{c}{14}[/tex] ( multiply both sides by 14 )

14 × tan50° = c , then

c ≈ 16.7  ( to 1 decimal place )

In this problem, the cosine function is used to solve for the length of side C in a right-angled triangle, because the angle and the adjacent side are known. According to trigonometry, the cosine for an angle is the length of the adjacent side divided by the hypotenuse.

In the given right-angled triangle, since the length of side C (hypotenuse) is unknown and one of the acute angles (50 degrees) is known, you should use the cosine function to solve for the length of side C. According to Trigonometry, cosine of an angle (in a right-angled triangle) is equal to the adjacent side divided by the hypotenuse. Hence, cos(50) = 14 / C. You could solve this equation for C.

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Since we rejected the null hypothesis, you can consider opening a restaurant at that location.

In order to determine whether the mean income of the sample is sufficient to support the type of upscale restaurant you wish to open, we can perform a hypothesis test.

The null hypothesis (H₀) is that the mean income of the population near the potential location is $85,000 or less.

The alternative hypothesis (Ha) is that the mean income of the population near the potential location is greater than $85,000.

Using a one-sample t-test to test this hypothesis, since the population standard deviation is unknown and the sample size is relatively small (n = 50).

Let's assume that the sample mean income is $90,000 and the sample standard deviation is $10,000.

The test statistic can be calculated as:

t = (sample mean - hypothesized mean) / (sample standard deviation / √n))

t = ($90,000 - $85,000) / ($10,000 / √50))

t = 2.50

The degrees of freedom for this test are n-1 = 49. Using a significance level of 0.05 and a one-tailed test, the critical t-value is 1.676.

Since the calculated t-value (2.50) is greater than the critical t-value (1.676), we can reject the null hypothesis and conclude that there is sufficient evidence to support the claim that the mean income of the population near the potential location is greater than $85,000.

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The most  important reason to avoid using a volunteer sample is Your conclusions could not be reliably generalized to a larger population.

Then, we want to determine the most important reason to avoid using a volunteering sample.

A levy sample is a slice system where people can choose whether or not they share in the check. While easier, this system can beget issues statistically.

The primary issue that this slice fashion faces is that the sample won't be arbitrary. You'll probably get people who are more opinioned about the subject the sample enterprises. This means that the sample won't truly be arbitrary, which in turn means it'll not be representative of the population. However, also the conclusions from the sample can not be generalized to the population, If the sample isn't representative of the population.

Thus, the answer is B. Your conclusions couldn't be reliably generalized to a larger population.

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The Volume of Pyramid is 512 in³.

Each thing in three dimensions takes up some space. The volume of this area is what is being measured. The space occupied within an object's borders in three dimensions is referred to as its volume.

Given:

Hypotenuse= 10 inch

leg length= 8 inch

Using Pythagorean Theorem

h² = 10² - 8²

h²= 100 - 64

h = 6 inch

So, the Volume of Pyramid

= lwh/3

= 6 x 16 x 16 /3

= 2 x 196

= 512 in³

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Given two lines with slopes m<1 and m<2, the slope of a third line that is perpendicular to these lines would be the negative reciprocal of either slope. The negative reciprocal of m<1 is -1/m<1 and the negative reciprocal of m<2 is -1/m<2.

Since m<1 = 38 degrees, the negative reciprocal of m<1 would be -1/38.

And since m<2 = 78 degrees, the negative reciprocal of m<2 would be -1/78.

So, if m<3 is the slope of a line perpendicular to the lines with slopes m<1 and m<2, then m<3 could be either -1/38 or -1/78, depending on which slope is chosen.

If  −2 ≤ x ≤ 5 and a ≤ 4x + 1 ≤ b, then the inequality can be written as:

−7 ≤ 4x + 1 ≤ 21

What is expression ?

An expression in mathematics is a combination of numbers, variables, and mathematical operations, such as addition, subtraction, multiplication, and division, that represents a value or a quantity. Expressions can be simple or complex, and they can be written in various forms, such as using variables, exponents, radicals, logarithms, trigonometric functions, and more.

For example, 2x + 5 is an expression that contains a variable x and represents a value that depends on the value of x. Another example is √(a^2 + b^2), which is an expression that contains two variables a and b and represents the square root of the sum of their squares.

According to given condition :

We know that −2 ≤ x ≤ 5. Therefore, the smallest possible value of 4x is −8 (when x = −2) and the largest possible value of 4x is 20 (when x = 5). Adding 1 to both sides of the inequality, we get:

−8 + 1 ≤ 4x + 1 ≤ 20 + 1

−7 ≤ 4x + 1 ≤ 21

So, a = −7 and b = 21.

Therefore, if  −2 ≤ x ≤ 5 and a ≤ 4x + 1 ≤ b, then the inequality can be written as:

−7 ≤ 4x + 1 ≤ 21

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Answer: 85 miles left

Step-by-step explanation:

65 miles x 7 hours = 455 miles

540 miles - 455 miles = 85 miles

Answer:

Step-by-step explanation:

You want to identify the rate of change of a graph that has points 1 week apart horizontally and 4 cm apart vertically.

The rate of change (m) of a curve on a graph is the ratio of rise to run.

Here, the graphed points differ by 4 cm vertically, and 1 cm horizontally. The rate of change is ...

  m = rise/run = (4 cm)/(1 week) = 4 cm in 1 week

  = 4/1 = 4 . . . . . centimeters per week

The rate of change can be described as ...

Part A: Let x be the number of rides you can enjoy at the South Florida Fair. The total cost of admission and rides cannot exceed the $52 you have, so we can write the following inequality:

32.50 + 0.60x ≤ 52

Part B: We can solve the inequality by first subtracting 32.50 from both sides:

0.60x ≤ 19.50

Then, we can divide both sides by 0.60 to isolate x:

x ≤ 32.5

Therefore, the maximum number of rides you can enjoy at the South Florida Fair is 32 rides, since you need to pay for admission and cannot exceed the $52 you have to spend.

The function that models the data is y = 42 ([tex]\frac{1}{2}[/tex])ˣ.

Exponential functions are functions where the independent variable, x is in the exponent.

From the graph, it is clear that it is an exponential function.

Exponential functions will be of the form y = k bˣ.

We have the point (0, 42).

Substituting the point,

k b⁰ = 42

k = 42, since any number raised to 0 is 1, b⁰ = 1.

So the function is y = 42 bˣ.

Substituting the point (1, 21),

42 b¹ = 21

b = 21 / 42

b = 1/2

So the function is y = 42([tex]\frac{1}{2}[/tex])ˣ.

Hence the function is y = 42([tex]\frac{1}{2}[/tex])ˣ.

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Answer:

The recursive formula for this sequence is y(n) = 2 * y(n-1). Starting with y(1) = -3, this formula produces the sequence of -3, -6, -12, -24, -48, and -96.

The ratio of the volume of a (right circular) cone to the volume of a cylinder with the same height and base radius is 1/3 or 1:3.

The volume of a cone and a cylinder are formulas that are commonly used in geometry. The volume of a right circular cone with height h and base radius r is given by:

V₁ = (1/3)πr^2h

The formula for the volume of a right circular cylinder with height h and base radius r is given by:

V₂ = πr^2h

To find the ratio of the volume of a cone to the volume of a cylinder with the same height and base radius, we can divide the volume of the cone by the volume of the cylinder:

V₁/V₂ = [(1/3)πr^2h]/[πr^2h]

We can simplify this expression by canceling out the factors of π, r^2, and h:

V₁/V₂ = 1/3

Therefore, the ratio of the volume of a cone to the volume of a cylinder with the same height and base radius is 1/3. This means that a cone with the same height and base radius as a cylinder has a volume that is one-third that of the cylinder.

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Answer:

A radius is half of a diameter

To find the radius, divide the diameter by 2

If you don't have a π button on your calculator, you should use 3.14159 (for accuracy) instead.

The segment that goes all the way across the circle through the center point is called the diameter.

A radius is half of a diameter.

To find the radius, divide the diameter by 2.

If you don't have a π button on your calculator, you should use an approximation, such as 3.14, instead (for accuracy).

The required for a. Genevieve must multiply the original dimensions with a lower scale factor lower than 1 while for b. Genevieve must multiply the original dimensions with a lower scale factor of more than 1.

The scale factor is defined as the ratio of the modified change in length to the original length.

Here,
To create a smaller model that fits in a scale model of the entire city, Genevieve will need to use a smaller scale. She can calculate the new measurements by dividing the dimensions of the original model by the desired scale factor. For example, if she wants the new model to be half the size of the original, she would divide all dimensions by 2.

To create a slightly larger model for the entrance of the old library building, Genevieve can use a larger scale. She can calculate the new measurements by multiplying the dimensions of the original model by the desired scale factor. For example, if she wants the new model to be 10% larger than the original, she would multiply all dimensions by 1.1.

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Answer:    7:17

Step-by-step explanation:

Ate breakfast 7:05

Clock stopped 12 minutes before

7:05 + 12 = 7:17

Answer:

Step-by-step explanation:

The inequality to determine the number of gigabytes Paisley can use while staying within her budget can be written as:

60 + 3g < 75

Solving for g:

60 + 3g < 75

-60 -60

0 + 3g < 15

/3 /3

g < 5

So Paisley can use up to 5 gigabytes while staying within her budget.

The area of polygon ABCD is 31 square units.

Coordinating refers to the process of organizing or synchronizing different elements or parts to work together in a harmonious and efficient manner towards a common goal or objective. In other words, coordinating involves bringing together various people, resources, activities, or tasks in a way that maximizes their effectiveness and minimizes any conflicts or inefficiencies.

We can find the area of polygon ABCD using the formula for the area of a quadrilateral:

Area = 1/2 * |(x1y2 + x2y3 + x3y4 + x4y1) - (y1x2 + y2x3 + y3x4 + y4x1)|where (x1, y1), (x2, y2), (x3, y3), and (x4, y4) are the coordinates of the vertices of the quadrilateral in order.

Substituting the coordinates of the given vertices, we get:

Area = 1/2 * |(5*(-3) + (-1)2 + (-1)6 + 5(-3)) - (-3(-1) + (-1)5 + 2(-1) + 6*5)|

Simplifying this expression, we get:

Area = 1/2 * |(-15 - 2 - 6 - 15) - (3 + 5 - 2 + 30)|

Area = 1/2 * |-38 - 24|

Area = 1/2 * |-62|

Area = 31

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Answer:

Step-by-step explanation:

Just divide the amount he expects to earn between the payment for each working hour.

[tex]Hours -to-babysit=\frac{168}{6}=28[/tex]

Hermes must babysit 28 hours to earn $168 for his new videogame

The coordinates of D on the given line segment is  (8/3, 0).

Suppose we've two endpoints of a line segment as:

A(p,q), and B(m,n)

Then let the midpoint be M(x,y) on that line segment. Then, its coordinates are:

[tex]x = \dfrac{p+m}{2}[/tex]

and

[tex]y = \dfrac{q+n}{2}[/tex]

We are given that;

y = 3x – 8

Now,

When a line cuts the x-axis, its y-coordinate is always 0. Therefore, we can substitute y = 0 into the equation y = 3x - 8 and solve for x:

0 = 3x - 8

3x = 8

x = 8/3

Therefore, the point D has coordinates (8/3, 0).

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The graph of the equation y = -0.5x + 12 will be a straight line

The equation y = -0.5x + 12 represents a linear function

Based on the slope -0.5 this means that as the value of x increases, the value of y will decrease.

Additionally, since the y-intercept is 12, the line will cross the y-axis at the point (0, 12).

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The missing side length is 7 yd.

The object given is made up of two rectangles. The width of the upright rectangle is to be determined. In order to determine this value, the mathematical operation that would be used is subtraction.

Subtraction is the process of determining the difference between two or more numbers. The sign that is used to represent subtraction is -.

Width of the upright rectangle = 15 - 8 = 7 yd

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Answer:

a. Never

b. Never

c. Sometimes

Step-by-step explanation:

a. The difference of a binomial and a binomial is never a binomial, as the result will always be another type of expression.

b. When a 6th-degree polynomial with 5 terms is written in standard form, the second term will always have a degree that is less than the 6th degree, since there are only 5 terms and the degree of each term decreases as its position increases.

c. The sum of a 4th-degree polynomial and a 2nd-degree polynomial is sometimes a 2nd-degree polynomial, depending on the coefficients of the terms. It is possible for the degree of the sum to be lower than the degree of both individual polynomials if the coefficients of the higher degree terms cancel out.

The difference of a binomial and a binomial always yields a binomial. A 6th-degree polynomial with 5 terms in standard form always has a degree of 5 for the second term. The sum of a 4th-degree polynomial and a 2nd-degree polynomial never results in a 2nd-degree polynomial.

1. For the operation a. The difference of a binomial and a binomial is a binomial.

Answer: Always

Reason: This is true because when you subtract a binomial from another, the result is always a binomial as well. For instance, (x+y)-(x-y) would result in 2y, a binomial.

2. For the operation b. When a 6th-degree polynomial with 5 terms is written in standard form, the second term has a degree of 5.

Answer: Always

Reason: This is always the case because in ordered or standard form, the terms decrease by degree in sequence

3. For the operation c. The sum of a 4th-degree polynomial and a 2nd-degree polynomial is a 2nd-degree polynomial.

Answer: Never

Reason: The highest degree of the sum of two polynomials is the higher degree of the individual polynomials, in this case, it would be a 4th degree polynomial.

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The constraint 3X₁²  + 7X₂ ≤ 45 has a degree of two and it cannot be used as constraint in a linear programming problem.

The correct answer is an option (e)

We know that a linear constraint in linear programming occurs when linear components are added or subtracted. Also the resulting expression must either increase, decrease, or be exactly equal to a right-hand side value.

Any constraint of a linear programming problem is referred to as linear if all of its terms are of the first order.

We know that the degree of the linear constraint is one, hence the constraint that does not have a degree of one is not a linear constraint.

Here we can observe that the constraint 3X₁²  + 7X₂ ≤ 45 has a degree of two and it cannot be used as constraint in a linear programming problem.

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The complete question is:

Which of the following constraints are not linear or cannot be included as a constraint in a linear programming problem?

a. 2X₁ + X₂ − 3X₃ ≥ 50

b. 2X₁ + √X₂ ≥ 60

c. 4X₁ - 1/3 X₂ = 75

d. 3X₁ + 2X₂ - 3x₃ / X₁ + X₂ + X₃ ≤ 0.9

e. 3X₁²  + 7X₂ ≤ 45

After divide, the value of expression (-9∠30 ) ÷ (-3∠-30) is

⇒ (-9∠30 ) ÷ (-3∠-30) = 3

Division method is used to distributing a group of things into equal parts. Division is just opposite of multiplications. For example, dividing 20 by 2 means splitting 20 into 2 equal groups of 10.

Given that;

The expression is,

⇒ (-9∠30 ) ÷ (-3∠-30)

Now, We can divide the expression as;

⇒ (-9∠30 ) ÷ (-3∠-30)

⇒ (-9∠30 ) / (-3∠-30)

⇒ - 9/- 3

⇒ 3

Thus, After divide, the value of expression (-9∠30 ) ÷ (-3∠-30) is

⇒ (-9∠30 ) ÷ (-3∠-30) = 3

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From the given information, the mean is 0.3025, the median is 0.3065, and the mode is 0.277 and 0.321.

To find the mean, we need to add up all the values and divide by the total number of values:

Mean = $\frac{0.302 + 0.278 + 0.321 + 0.283 + 0.311 + 0.312 + 0.320 + 0.276 + 0.275 + 0.281 + 0.305 + 0.277 + 0.308 + 0.303 + 0.322 + 0.317 + 0.305 + 0.291 + 0.321 + 0.277}{20} \approx 0.3025$

To find the median, we need to arrange data in the order from smallest to largest:

0.275, 0.276, 0.277, 0.277, 0.278, 0.281, 0.283, 0.291, 0.305, 0.305, 0.308, 0.311, 0.312, 0.317, 0.320, 0.321, 0.321, 0.322, 0.303, 0.302

There are 20 values, so median is average of the 10th and 11th values:

Median = $\frac{0.305 + 0.308}{2} = 0.3065$

To find the mode, we look for the value that appears most frequently. In this case, there are two modes: 0.277 and 0.321.

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