Prove or disprove the following statements.
a. If a, b, c and d are integers such that a|b and c|d, then a + d|b + d. b. if a, b, c and d are integers such that a|b and c|d, then ac|bd. e. if a, b, c and d are integers such that a b and b c, then a c.

Hi! To find a set of the smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets, follow these steps:

1. List out the given subsets: {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}.
2. Combine the elements of both subsets without repeating any numbers: {1, 2, 3, 4, 5, 6, 8, 10}.
3. The combined set is {1, 2, 3, 4, 5, 6, 8, 10}, which has a size of 8.

So, the smallest possible set that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets is {1, 2, 3, 4, 5, 6, 8, 10}.

The smallest possible set that includes both  {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}  subsets is {1, 2, 3, 4, 5, 6, 8, 10}.

To find a set of the smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets, follow these steps:
1. Identify the given subsets: {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}.
2. Combine the elements from both subsets without repeating any numbers.
3. Organize the combined elements in ascending order.
Your answer: The smallest possible set that includes both subsets is {1, 2, 3, 4, 5, 6, 8, 10}.

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Hi! To find a set of the smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets, follow these steps:

1. List out the given subsets: {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}.
2. Combine the elements of both subsets without repeating any numbers: {1, 2, 3, 4, 5, 6, 8, 10}.
3. The combined set is {1, 2, 3, 4, 5, 6, 8, 10}, which has a size of 8.

So, the smallest possible set that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets is {1, 2, 3, 4, 5, 6, 8, 10}.

The smallest possible set that includes both  {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}  subsets is {1, 2, 3, 4, 5, 6, 8, 10}.

To find a set of the smallest possible size that has both {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10} as subsets, follow these steps:
1. Identify the given subsets: {1, 2, 3, 4, 5} and {2, 4, 6, 8, 10}.
2. Combine the elements from both subsets without repeating any numbers.
3. Organize the combined elements in ascending order.
Your answer: The smallest possible set that includes both subsets is {1, 2, 3, 4, 5, 6, 8, 10}.

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The ways in which we can shade the shape to get the total is: Shape 1: 3/5 shaded. Shape 2: 2/5 shaded. Shape 3: 2/5 shaded. To decompose g we use 7/5 = 4/5 + 3/5.

To express a quantity that is a portion of a whole, use fractions. They are made up of a denominator and a numerator, two integers separated by a horizontal line. The denominator is the total number of equally sized components that make up the whole, while the numerator is the portion of the whole that is being taken into account.

Because they are different ways of expressing the same quantity, decimals and percentages have a relationship to fractions. With a base of 10, decimals can be used to express fractions, with each digit standing for a different power of 10.

Given that, the total shaded area must be 7/5.

The ways in which we can shade the shape to get the total is:

Shape 1: 3/5 shaded

Shape 2: 2/5 shaded

Shape 3: 2/5 shaded

Now, for two shapes we can use the addition of two fractions as follows:

7/5 = 4/5 + 3/5

The example of the form is:

7/5 = 4/5 + 3/5

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The following parts can be answered by the concept of standard deviation.

a.  The population mean is $55.
b.  The standard deviation of the sampling distribution is approximately 2.688 (rounded to three decimal places).

(a) The mean of the sampling distribution of the mean amount spent per sale for samples of size 50 is equal to the population mean. In this case, the population mean is $55.

(b) To find the standard deviation of the sampling distribution, we use the formula:

Standard deviation of the sampling distribution = (Population standard deviation) / sqrt(sample size)

In this case, the population standard deviation is $19 and the sample size is 50. Plugging these values into the formula, we get:

Standard deviation of the sampling distribution = 19 / sqrt(50) ≈ 19 / 7.071 ≈ 2.688

So, the standard deviation of the sampling distribution is approximately 2.688 (rounded to three decimal places).

Therefore,

a.  The population mean is $55.
b.  The standard deviation of the sampling distribution is approximately 2.688 (rounded to three decimal places).

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The required ‘test statistic’ is 36.25

To solve this problem, we'll use the Chi-squared test statistic for testing the variance of a population. Here are the steps:

Identify the given information:
  - Sample size (n) = 30
  - Sample variance (s²) = 625
  - Null hypothesis (H₀): σ² = 500
  - Alternative hypothesis (Hₐ): σ² ≠ 500

Calculate the degrees of freedom (df) using the formula: df = n - 1
  - df = 30 - 1 = 29

Calculate the Chi-squared test statistic (χ²) using the formula: χ² = (n - 1) * (s² / σ²)
  - χ² = (29) * (625 / 500)

Compute the test statistic value:
  - χ² = 29 * (1.25) = 36.25

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The compound statement is not entirely clear as it contains several separate expressions. However, I can break down and analyze each of the parts:

7 + 1 2 7: This is a mathematical expression that represents the sum of 7, 1, and 27, which evaluates to 35.

7,1,7: This is a list of three individual numbers.

7+1: This is a mathematical expression that represents the sum of 7 and 1, which evaluates to 8.

74157: This is a five-digit number with no apparent mathematical relationship to the other expressions.

7+1=7: This is a mathematical equation that tests the equality of the expressions 7+1 and 7. Since 7+1 is not equal to 7, this equation is false.

27: This is a single number that is not obviously related to the other expressions.

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Answer: 108 cups of cranberry juice. Brainliest?

Step-by-step explanation:

mixed juice. How many cups of cranberry juice did Jetia use?

Let's start by assuming that Jetia used x cups of cranberry juice to make the mixed juice. Then, since the ratio of cranberry juice to apple juice is 5:8, she must have used (5/8)x cups of apple juice.

We know that the total amount of mixed juice is 177 cups, so we can set up an equation based on the total amount of juice:

x + (5/8)x = 177

Simplifying this equation, we get:

(13/8)x = 177

Multiplying both sides by 8/13, we get:

x = 108

Therefore, Jetia used 108 cups of cranberry juice to make the mixed juice.

Having drawn and attached the scatter plot for the given data, the equation for the line of best fit is y = 0.083x + 97.55

After plotting the scatter point, we choose two points on the line of best fit that are far apart.

The points chosen are:

(150, 110) And (1240, 200)

Using the slope-intercept form we proceed to find the equation

y = mx + b

y =  cost of plane tickets

x = distance

m = slope

m = (y2-y1)/(x2-x1)
m = (200-110) / (1240 - 150)
m = 0.083

So, we can state

y = mx + b

110 = 0.083 * 150 + b
b = 97.55

hence,

the line of best fit is:

y = 0.083x + 97.55


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The exact value of the cos 105 degrees by the using the half angle formula is -0.2588.

To find the exact value of cos(105°) using the half-angle formula, we first need to express 105° as half of another angle. Since 105° is equal to 210°/2,

we can use the half-angle formula for cosine:

cos(x/2) = ±√[(1 + cos(x))/2]

In our case, x = 210°. Now, we need to find the value of cos(210°):

210° lies in the third quadrant, where both sine and cosine are negative. To find the reference angle, subtract 180°:

210° - 180° = 30°

So, cos(210°) = -cos(30°) = -√3/2.

Now, let's plug the value of cos(210°) into the half-angle formula:

cos(105°) = ±√[(1 - √3/2)/2]

Since 105° lies in the second quadrant, where cosine is negative, we choose the negative root:

cos(105°) = -√[(1 - √3/2)/2]

               =-0.2588

Explanation:- Here to find the value of the cosine 105 degree by using the half angle formula first we write the half angle formula of the cosine and substituted x=210 degree and simplify.

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The measure of the missing part of proportion is 17.50

Since the relations between variables, either direct or inverse proportional, can be built to find the desired measures in the problem.

Given that Weight Price 2 -lb. $7.00

5 - lb. $

1: 3.50 dollars

Now multiply by 5 on each side

5 : 17.50

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The power (P) when I = 64.1 A and V = 12.8 V is 819.68 watts (W).

Hello! I understand that you need help in finding the power (P) when the current (I) is 64.1 A and the voltage (V) is 12.8 V.
Understand the relationship between power, current, and voltage.
The power (P) in an electrical circuit can be calculated using the formula P = I × V,

where I is the current in amperes (A) and V is the voltage in volts (V).
Plug in the given values.
In this case, we are given I = 64.1 A

and V = 12.8 V.

Plug these values into the formula:
P = 64.1 A × 12.8 V
Calculate the power.
Multiply the current and voltage to find the power:
P = 819.68 W.

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7 of the 25 measures are ratings of at most 6, that is, 6 is less than both the mean and the median of the distribution, hence it is not a good description of the center of this data set.

What is a data set?

A data set is a group of related data. In the case of tabular data, a data set relates to one or more database tables, where each row refers to a specific record in the corresponding data set and each column to a specific variable.

Here, we have

Given: Twenty-five students were asked to rate—on a scale of 0 to 10—how important it is to reduce pollution. A rating of 0 means “not at all important” and a rating of 10 means “very important.”

From the plot given in this exercise, it is found that only 7 of the 25 measures are ratings of at most 6, that is, 6 is less than both the mean and the median of the distribution.

Hence it is not a good description of the center of this data set.

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The sum in expanded form is 23 / 24 + 23 / 25 + 23 / 26 + 23 / 27 + 23 / 28 + 23 / 29.

The sum in expanded form is given by the expression 23 / (i + 23), where i varies from 1 to 6.

The sum in expanded form can be calculated by substituting the values of i from 1 to 6 into the expression 23 / (i + 23) and summing them up.

When i = 1, the expression becomes 23 / (1 + 23) = 23 / 24.

When i = 2, the expression becomes 23 / (2 + 23) = 23 / 25.

When i = 3, the expression becomes 23 / (3 + 23) = 23 / 26.

When i = 4, the expression becomes 23 / (4 + 23) = 23 / 27.

When i = 5, the expression becomes 23 / (5 + 23) = 23 / 28.

When i = 6, the expression becomes 23 / (6 + 23) = 23 / 29.

Therefore, the sum in expanded form is 23 / 24 + 23 / 25 + 23 / 26 + 23 / 27 + 23 / 28 + 23 / 29.

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The sum in expanded form is 23 / 24 + 23 / 25 + 23 / 26 + 23 / 27 + 23 / 28 + 23 / 29.

The sum in expanded form is given by the expression 23 / (i + 23), where i varies from 1 to 6.

The sum in expanded form can be calculated by substituting the values of i from 1 to 6 into the expression 23 / (i + 23) and summing them up.

When i = 1, the expression becomes 23 / (1 + 23) = 23 / 24.

When i = 2, the expression becomes 23 / (2 + 23) = 23 / 25.

When i = 3, the expression becomes 23 / (3 + 23) = 23 / 26.

When i = 4, the expression becomes 23 / (4 + 23) = 23 / 27.

When i = 5, the expression becomes 23 / (5 + 23) = 23 / 28.

When i = 6, the expression becomes 23 / (6 + 23) = 23 / 29.

Therefore, the sum in expanded form is 23 / 24 + 23 / 25 + 23 / 26 + 23 / 27 + 23 / 28 + 23 / 29.

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The argument is valid. According to modus tollens, the conclusion "I will not eat" logically follows

The argument follows a valid logical form known as modus tollens, which is a valid deductive argument form. Modus tollens states that if a conditional statement (e.g., "if A, then B") is true and the consequent (B) is false, then the antecedent (A) must also be false.

In this argument, the conditional statement is "If I'm hungry, then I will eat" (A = I'm hungry, B = I will eat), and the premise "I'm not hungry" establishes that the consequent (B) is false.

Therefore, according to modus tollens, the conclusion "I will not eat" logically follows

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N(e ∪ f) = 194.

Using the inclusion-exclusion principle, we have:

n(e ∪ f) = n(e) + n(f) - n(e ∩ f)

We are given n(e ∩ f) = 73 and n(e ∪ f) = 194, so we can rearrange to solve for n(f):

n(f) = n(e ∪ f) - n(e) + n(e ∩ f)

n(f) = 194 - 106 + 73

n(f) = 161

Finally, to find n(e ∪ f), we can substitute the values we have found into the first equation:

n(e ∪ f) = n(e) + n(f) - n(e ∩ f)

n(e ∪ f) = 106 + 161 - 73

n(e ∪ f) = 194

Therefore, n(e ∪ f) = 194.

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The most general antiderivative of f(x) = 1/2 + 5/6x² − 4/5x³ is F(x) = 1/2x + 5/18x³ − 1/5x⁴ + C, where C is the constant of the antiderivative.

To check this answer, we can differentiate F(x) and see if it gives us back f(x). Taking the derivative of F(x), we get f(x) = d/dx (1/2x + 5/18x³ − 1/5x⁴ + C) = 1/2 + 5/6x² − 4/5x³, which matches the original function f(x). Therefore, F(x) is the most general antiderivative of f(x).

The constant of integration, denoted by C, is added because when taking the derivative of a constant, it is equal to zero. Thus, the constant of integration can be any real number, and it is included in the antiderivative to account for all possible functions that have f(x) as their derivative.

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Answer:

[tex]26m^{2}[/tex]

Step-by-step explanation:

To solve this problem you find the total area of the entire rectangle and subtract the area of the sand from it. That will give you the area of the grass.

To find the total area you need to do [tex]b*h[/tex], in this case, the base is [tex]2+3+2[/tex] or 7. The height is 5. So to find the area, you have to multiply  [tex]7*5[/tex] to get[tex]35m^{2}[/tex].

To find the area of the grass you multiply the [tex]b*h[/tex] or [tex]3*3[/tex] to get the area of 9.

Now the last step is to subtract [tex]35 - 9[/tex], doing so gives you your answer of 26 m

I would love to get Brainliest. I'm trying to get to the Genius status on Brainly.

The value of f ⁻¹ (x) would be,

⇒ f⁻¹(x) = - x/2 - 1

We have to given that;

Function is,

⇒ f (x) = - 2x - 2

Now, We can find the inverse of function as;

⇒ f (x) = - 2x - 2

⇒ y = - 2x - 2

Solve for x;

⇒ y + 2 = - 2x

⇒ x = - (y + 2)/2

⇒ x = - y/2 - 1

⇒ f⁻¹(x) = - x/2 - 1

Thus, The value of f ⁻¹ (x) would be,

⇒ f⁻¹(x) = - x/2 - 1

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The area of the shaded area is 0.314m²

First let's analyze the shape.

It comprises:

A square whose side length is 1m

Then there is the circle that passes through a corner of the square and is tangent to the opposite two sides of the square. So the region shaded in blue is the region contained by the circle and the square but not common to both shapes.

To get the area of this portion shaded in blue, here is what we must do:

So the Area of the region shaded blue is:

Area of the circle + area of the square - 2 x Area of the overlapping area.

To further help our analysis, we must deconstruct this shape into familiar ones by creating a composite right triangle as shown in the attached image.


Since we have a right triangle, the arc that it subtends will be  a 180° arc thus, the hypotenuse of the triangle is the diameter of the circle.

Thus, when we created a right triangle from the shape, we also carved out a semi-circle.

This means that the overlapping areas are:

Area of the composite right triangle + the Area of the Semi Circle.

Since we now have two semi circles, we can cancel out the area of the circle above.

Our new equation for the area shaded in blue becomes:

The area of the square  - the area of 2 Right Triangles

Now we can being to compute for the area of the shapes:


Taking the right triangle firs, create a line from the radius bisecting the triangle at 90 degrees. This gives us two equal smaller right triaangles.

Since the ∠90 is bisected, this means that the sides will be r√2 and the height of both right triangles = r (radius)

extending the radius into the opposite direcitn, we now have the diagonal of the square.

Diagonal = r + r√2 = √2
Solving for r we say:

r(1+√2) = √2
r = √2/(1+√2)

To further simplify the above, we can multiply it by the conjugate:

r = √2/(1+√2) x (1-√2)/(1-√2)

Thus,

r = 2-√2


So since the length of the square is 1m, then the area = 1x1 = 1m

The area of the composite triangle =( r√2 x r√2)/ 2 = r²

Since the area of the shaded region as given above is
The area of the square  - the area of 2 Right Triangles

Then it's area = 1 - 2r²

We can simplify this further to get:

1-2(2-√2)²

= 8 √2 - 11m²
area of the shaded region ≈ 0.314m²

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Answer: (-8, -6, -3, 7, 9, 10)

Step-by-step explanation:

The domain of a function is the set of all possible x-values that correspond to the given ordered pairs.

Therefore, the domain of the function is:  (-8, -6, -3, 7, 9, 10)

The Euclidean algorithm gives gcd(a,b) = gcd(b,r) and gcd(b,r) = gcd(r,s).

The Euclidean algorithm is a recursive algorithm to find the greatest common divisor (gcd) of two integers a and b. It works by repeatedly finding the remainder of the division of the larger number by the smaller number, until the remainder is 0, at which point the gcd is the last non-zero remainder.

When applying the algorithm to a and b with a > b, we can write a = qb + r, where q is the quotient and r is the remainder. Then, we apply the algorithm to b and r to find the gcd(b,r), and so on until we reach 0. At each step, the gcd of the current pair of numbers is equal to the gcd of the previous pair, since any common divisor of the current pair must also divide the previous pair. Therefore, we have gcd(a,b) = gcd(b,r) = gcd(r,s), where s is the final non-zero remainder.

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The period of this function is 6 seconds and it means the time it takes for a person's lung to inhale and exhale in a full cycle.

The function's period alludes to the duration required for one complete cycle, culminating in its initial point. Furthermore, this term represents the length of time necessary for a person's lungs to perform a full inhalation and exhalation sequence.

To determine the period, it is integral to recognize the temporal gap between two successive peaks (or troughs) on the chart. This interval deviation precludes:

8. 5 - 2. 5 = 6 seconds

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The graph of the function f(x) = 2x + 26 will be a line graph.

Some key points about the graph:

• The slope of the line will be 2.

• The y-intercept will be 26, since f(0) = 26.

• The line will pass through the points (0, 26) and (x, 2x + 26).

• As x increases, the value of f(x) also increases but at a increasing rate.

• The graph will be a positively sloped line, increasing from left to right.

A rough sketch of the graph would be:

y

26

24

22

20

18

16

14

12

10

8

6

4

2

-6 -4 -2 0 2 4 6 8 10 12 14 x

Does this help explain the graph? Let me know if you have any other questions!

Using a table of sample sizes and factors for ratio estimation, we find that a sample size of 90 items is appropriate and based on the information provided, we can project a total misstatement of $405,480.67 using ratio projection.

1. To determine the sample size given the information provided, we can use the formula:

Sample size = (Tolerable misstatement / Expected misstatement)² x (Total gross balance in inventory / Sampled balance)

Plugging in the values, we get:

Sample size = ($155,000 / $55,000)² x ($5,500,000 / Sampled balance)
Sample size = 6.73 x ($5,500,000 / Sampled balance)

Assuming a moderate risk of material misstatement, we can use a confidence level of 95%, which corresponds to a Z-score of 1.96. Using a table of sample sizes and factors for ratio estimation, we find that a sample size of 90 items is appropriate.

2. To calculate the total projected misstatement using ratio projection, we first need to determine the ratio of misstatement in the sample to the total inventory. We can do this using the formula:

Ratio of misstatement = Sample misstatement / Sampled balance

Assuming the expected misstatement of $55,000 and a sample size of 90 items, we can set a sampling interval of:

Sampling interval = Total gross balance in inventory / Sample size
Sampling interval = $5,500,000 / 90
Sampling interval = $61,111.11

Using systematic sampling, we can select every 61,111th item from the inventory. Let's say our sample includes 3 items with misstatements totaling $4,500. Then the ratio of misstatement would be:

Ratio of misstatement = $4,500 / Sampled balance

To project the total misstatement, we can use the formula:

Total projected misstatement = Ratio of misstatement x Total gross balance in inventory

Plugging in the values, we get:

Total projected misstatement = ($4,500 / Sampled balance) x $5,500,000

Since we don't know the actual sampled balance, we can use the average sampled balance as an estimate. Assuming an equal distribution of items, the average sampled balance would be:

Average sampled balance = Total gross balance in inventory / Sample size
Average sampled balance = $5,500,000 / 90
Average sampled balance = $61,111.11

Plugging this value in, we get:

Total projected misstatement = ($4,500 / $61,111.11) x $5,500,000
Total projected misstatement = 0.0736 x $5,500,000
Total projected misstatement = $405,480.67

Therefore, based on the information provided, we can project a total misstatement of $405,480.67 using ratio projection.

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The equation that best models the population P(t) of the country, given the constant relative birth and death rates and the number of people emigrating every year, is dP/dt = 0.5P - 30,000.

The rate of population growth is determined by the difference between the birth rate and the death rate, which is (97 - 47) per thousand people per year, or 0.05. This means that the population will grow by 0.05 times the current population each year if there is no emigration. However, since 30,000 people emigrate every year, we need to subtract this number from the population growth rate. Therefore, the rate of population growth can be expressed as 0.05P - 30,000.

To get the population at any given time t, we need to integrate this rate equation concerning time t. The solution to this differential equation is P(t) = (P0 - 60,000)e^(0.05t) + 60,000, where P0 is the initial population at time t=0.

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Answer:

32

Step-by-step explanation:

43-11=32

Answer:

6/8, 9/12, 12/16, 15/20

Step-by-step explanation:

If you simplify the fractions it will all equal 3/4.

Simplify by finding the Greatest Common Factor and dividing both numbers by the GCF.

Th solution of the given seperable differential equation is :

u = (1/5)ln(25t^2 + e^75)

To solve the separable differential equation for u, we need to separate the variables and integrate both sides.

First, we can write the equation as:
(1/e^5u)du = 5t dt

Now we can integrate both sides:
∫(1/e^5u)du = ∫5t dt

To integrate the left side, we can use u-substitution:
Let u = 5u
Then du = 5e^5u du

Substituting into the integral, we get:
(1/5)∫e^5u du = ∫5t dt
(1/5)e^5u = 5t^2/2 + C
Where C is the constant of integration.

Now we can solve for u:
e^5u = 25t^2 + 2C

Taking the natural logarithm of both sides:
5u = ln(25t^2 + 2C)
u = (1/5)ln(25t^2 + 2C)

Using the initial condition u(0) = 15, we can solve for C:
15 = (1/5)ln(2C)
ln(2C) = 75
2C = e^75
C = (1/2)e^75

Substituting this value of C into our solution for u, we get:
u = (1/5)ln(25t^2 + e^75)

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We would need a sample size of at least 1073 children under age 6 in Washington to estimate the true proportion of children living in poverty with 95% confidence and a margin of error of 2%.

To estimate the true proportion of children under age 6 living in poverty in Washington with 95% confidence and a margin of error of 2%, we can use the formula:

n = (Z² * p * q) / E²

where:

Z = the Z-score corresponding to the desired confidence level (1.96 for 95% confidence)

p = the estimated proportion (0.17 based on the federal report)

q = 1 - p

E = the desired margin of error (0.02)

Plugging in these values, we get:

n = (1.96² * 0.17 * 0.83) / 0.02²

n = 1072.45

Rounding up to the next whole number, we would need a sample size of at least 1073 children under age 6 in Washington to estimate the true proportion of children living in poverty with 95% confidence and a margin of error of 2%.

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Answer:

624 rounded 624.3 not rounded

Step-by-step explanation:

hope this helps <3

Using the ADT list, we can implement the function f(n) to compute the values of the recurrence relation F(n) = F(n-1) + 3 x F(n-5), for n > 5, where F(1) = 1, F(2) = 1, F(3) = 1, F(4) = 3, and F(5) = 5.

By computing the values of f(n) in an ordered manner, we can avoid computing the same value more than once. We can use this approach to compute f(6), f(7), f(12), and f(15).

We will use the ADT list to implement the function f(n) to compute the values of the recurrence relation F(n) = F(n-1) + 3 x F(n-5), for n > 5, where F(1) = 1, F(2) = 1, F(3) = 1, F(4) = 3, and F(5) = 5. We will compute the values of f(n) in an ordered manner to avoid computing the same value more than once.

First, we initialize an empty list to store the values of f(n). Then, we add the initial values of f(1), f(2), f(3), f(4), and f(5) to the list.

Next, we use a for loop to compute the values of f(n) for n = 6 to 15. Inside the for loop, we use the recurrence relation F(n) = F(n-1) + 3 x F(n-5) to compute the value of f(n). We check if the value of f(n) has already been computed by checking if the length of the list is greater than or equal to n.

If the value has already been computed, we skip the computation and move on to the next value of n. Otherwise, we add the computed value of f(n) to the end of the list.

After the for loop, the list contains the values of f(1) to f(15). We can extract the values of f(6), f(7), f(12), and f(15) from the list and print them out.

For example, the Python code to implement the above approach is:

# Initialize an empty list to store the values of f(n)

f_list = []

# Add the initial values of f(1), f(2), f(3), f(4), and f(5) to the list

f_list.extend([1, 1, 1, 3, 5])

# Compute the values of f(n) for n = 6 to 15

for n in range(6, 16):

   if len(f_list) >= n:

       # Value of f(n) has already been computed

       continue

   else:

       # Compute the value of f(n) using the recurrence relation

       f_n = f_list[n-2] + 3 * f_list[n-6]

       # Add the computed value to the end of the list

       f_list.append(f_n)

# Extract the values of f(6), f(7), f(12), and f(15) from the list

f_6 = f_list[5]

f_7 = f_list[6]

f_12 = f_list[11]

f_15 = f_list[14]

Print out the values of f(6), f(7), f(12), and f(15)

print("f(6)).

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