the height and age of each child in a random sample of children was recorded. the value of the correlation coefficient between height and age for the children in the sample was 0.8 . based on the least-squares regression line created from the data to predict the height of a child based on age, which of the following is a correct statement? responses on average, the height of a child is 80% of the age of the child. on average, the height of a child is 80% of the age of the child. the least-squares regression line of height versus age will have a slope of 0.8 . the least-squares regression line of height versus age will have a slope of 0.8 . the proportion of the variation in height that is explained by a regression on age is 0.64 . the proportion of the variation in height that is explained by a regression on age is 0.64 . the least-squares regression line will correctly predict height based on age 80% of the time. the least-squares regression line will correctly predict height based on age 80% of the time. the least-squares regression line will correctly predict height based on age 64% of the time.

The statistics indicate that the assumption that the four populations have the same standard deviation has been broken.

a. This research was done via observation.

False, as this is a test. (unlike what Alex89 says). The response variable (minutes of rest) is then recorded because the researcher randomly assigns the fruit flies to one of the treatments, using a TREATMENT, and records the response variable.

b. The statistics indicate that the assumption that the four populations have the same standard deviation has been broken.

False: There is no deviation from the equality of standard deviations. We DO NOT need to test this assumption when the sample sizes are equal. Even if we did, 29.61/19.08 = 1.55 2 means that the greatest standard deviation is less than twice the lowest standard deviation. We are therefore okay.

c. Because ANOVA needs sample sizes, it may be applied to this data are equal.

False - The ANOVA does NOT require equal sample sizes.

3. The correct option is D, For this example, we notice that 3) None of the above

(4) The correct is option A, which states that 4) the population mean rest is the same for all four caffeine concentrations.

Statistics provides a framework for making informed decisions based on data by using methods such as probability theory, hypothesis testing, regression analysis, and statistical modeling. The importance of statistics lies in its ability to transform raw data into meaningful information that can be used to make decisions, develop policies, and solve problems in a wide range of fields, including business, healthcare, government, and social sciences.

Statisticians use various tools and techniques to summarize and describe data, identify patterns and relationships, and make predictions and inferences. They also play a critical role in designing experiments, surveys, and observational studies, ensuring that they are statistically sound and produce reliable results.

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Complete Question:-

An experiment to help determine if insects sleep gave caffeine to fruit flies to see if it affected their rest. The three treatments were a control, a low caffeine dose of 1 mg/ml of blood, a medium dose of 3 mg/ml of blood, and a higher caffeine dose of 5 mg/ml of blood. Twelve fruit flies were assigned at random to the four treatments, three to each treatment, and the minutes of rest measured over a 24-hour period was recorded. The data follow.

Treatment Minutes of rest

Control 450 413 418

Low dose 466 422 435

Medium dose 421 453 419

High dose 364 330 389

Assume the data are three independent SRSs, one from each of the four populations of caffeine levels, and that the distribution of the yields is Normal.

A partial ANOVA table produced by Minitab follows, along with the means and standard deviation of the yields for the four groups.

One-way ANOVA: Rest versus Caffeine

Source DF SS MS F P

Caffeine 11976

Error 538.75

Total

Level N Mean StDev

Control 3 427.00 20.07

Low 3 441.00 22.61

Medium 3 431.00 19.08

High 3 361.00 29.61

3. For this example, we notice that 3) __________

a. this is an observational study.

b. the data show evidence of a violation of the assumption that the four populations have the same standard deviation.

c. ANOVA can be used on these data because ANOVA requires the sample sizes are equal.

d. None of the above

4. The null hypothesis for the ANOVA F test is that 4) __________

a. the population mean rest is the same for all four levels of caffeine.

b. the population mean rest is increasing as the caffeine level gets larger.

c. the population mean rest is decreasing as the caffeine level gets larger.

d. the population mean rest is largest for the high level of caffeine.

Derivative of y w.r.t t is dy/dt = 16 at x = -1.

We need to find dy/dt at x = -1, given y = -2x² + 4 and dx/dt = 4.

Step 1: Differentiate y with respect to x.
Since y = -2x² + 4, we have:

dy/dx = -4x

Step 2: Substitute x = -1 into the dy/dx equation.
When x = -1, we get:

dy/dx = -4(-1) = 4

Step 3: Use the Chain Rule to find dy/dt.
The Chain Rule states that dy/dt = (dy/dx)(dx/dt). We know dy/dx = 4 and dx/dt = 4, so we have:

dy/dt = (4)(4) = 16

Thus, dy/dt = 16 at x = -1.

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The function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet is y = 12x - 4

Let's start by considering the initial amount of printer paper in the closet before any boxes are used. Since each box contains 12 reams of printer paper, if there are x boxes in the closet, then the total number of reams of paper is given by 12x.

Now, if a worker uses 4 reams from one of the boxes, then the total number of reams of paper remaining in the closet is (12x - 4). If we define y as the total number of reams of paper remaining in the closet, then we have:

y = 12x - 4

This function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet.

As x increases, the total number of reams of paper in the closet increases as well. However, each time a worker uses 4 reams of paper from a box, the total number of reams of paper in the closet decreases by 4.

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We can estimate that about 9 grams of water vapor per kg of air would have to condense out to maintain saturation and a relative humidity of 100%.

When air is saturated, it holds the maximum amount of water vapor it can at a given temperature and pressure. Any decrease in temperature leads to the condensation of water vapor out of the air, which can lead to the formation of dew or frost.

To maintain a relative humidity of 100%, the air must remain saturated. So, if the temperature of the room continues to decrease for one more hour, some of the water vapor in the air will condense out to maintain saturation. The amount of water vapor that condenses out depends on the initial temperature, the final temperature, and the amount of water vapor in the air.

To calculate the amount of water vapor that condenses out, we can use the concept of dew point temperature. The dew point temperature is the temperature at which the air becomes saturated and condensation begins to occur. If the temperature of the room reaches the dew point temperature, the air will be fully saturated, and any further decrease in temperature will lead to the condensation of water vapor.

Assuming that the initial temperature of the room was above the dew point temperature, we can estimate the amount of water vapor that would condense out after one hour of cooling by calculating the difference between the initial temperature and the dew point temperature, and then using a psychrometric chart or an online calculator to determine the corresponding amount of water vapor that would have to condense out.

For example, if the initial temperature of the room was 25°C and the dew point temperature was 20°C, and the room cooled to 19°C after one hour, then we can estimate that about 9 grams of water vapor per kg of air would have to condense out to maintain saturation and a relative humidity of 100%. However, this is just an estimate, and the actual amount of water vapor that condenses out depends on many factors, including the humidity and air circulation in the room.

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The probability that the mean cost of the 9 weddings is more than the mean cost of the 6 showers is approximately 0.0207 or 2.07%.

The probability that the mean cost of the 9 weddings is more than the mean cost of the 6 showers can be found using the Z-score and the difference between the means of two normally distributed variables.

1: Calculate the difference in means and standard deviations.
Δμ = μ_weddings - μ_showers = $82.30 - $65 = $17.30
Δσ = sqrt((σ_weddings²/n_weddings) + (σ_showers²/n_showers)) = sqrt((18.20²/9) + (17.73²/6)) = $8.47

2: Calculate the Z-score.
Z = (Δμ - 0) / Δσ = (17.30 - 0) / 8.47 ≈ 2.04

3: Determine the probability using a Z-table.
P(Z > 2.04) ≈ 0.0207

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For every 3 squares, there are 5 total shapes. Therefore, the simplified ratio of squares to total shapes is 3 : 5.

If the unsimplified ratio of squares to total shapes is 6 : 16, then we can express this as 6/16, then,we can simplify this ratio by dividing both the numerator and denominator by 2, giving us 3/8.

This means that for every 3 squares, there are 5 total shapes. Therefore, the simplified ratio of squares to total shapes is 3 : 5.

Full question "Unsimplified ratio of squares to total shapes: 6 : 16. For every 3 squares, there are ____ total shapes; Therefore, the simplified ratio of squares to total shapes is __ : __.

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The 11th term of this geometric sequence 7 -28, 112, ....​ include the following: 7,340,032.

In Mathematics, the nth term of a geometric sequence can be calculated by using this mathematical equation (formula):

aₙ = a₁rⁿ⁻¹

Where:

Next, we would determine the common ratio as follows;

Common ratio, r = a₂/a₁

Common ratio, r = -28/7

Common ratio, r = -4

For the 11th term, we have:

a₁₁ = 7(-4)¹¹⁻¹

a₁₁ = 7,340,032.

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The probability of the next elk caught being unmarked is 0.96 when the total number of elks is 5625 and the number of elks marked is 225.

A probability is given by the number of desired outcomes divided by the number of total outcomes.

Total number of elks = 5625

Number of elks marked = 225

We need to find the total number of elks not marked or  unmarked we can find it by,

= 5625 - 225

= 5500

Therefore, the total number of elks unmarked is 5500.

We can determine determined the likelihood of the next elk caught being unmarked by using probability. The probability is given by:

P = 5500/5625

= 44 / 45

= 0.96

Therefore, The total number of elks unmarked is 0.96.

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The complete question is,

Describe the likelihood of the next elk caught being unmarked.

If you are earning $12.50/hr in a part-time job, then your gross "base-pay" after deductions is $253.48.

Your base pay is $12.50/hr and you worked for 18 hours, so the base pay would be:

⇒ $12.50/hr × 18 hrs = $225;

The tips are $4.60/hr on average, so the "total-tips" earned would be:

⇒ $4.60/hr × 18 hrs = $82.80;

To calculate the "total-deductions", we first find the total percentage deducted, which is the sum of FICA and Federal Tax Withholding:

⇒ 7.65% + 10% = 17.65%,

To find the total amount deducted, we multiply the total percentage deducted by the "base-pay" + "tips";

⇒ 17.65% x ($225 + $82.80) = $54.32,

So the pay, before any deductions, is the sum of "base-pay" and "tips";

⇒ $225 + $82.80 = $307.80;

To find the "net-pay", we subtract "total-deductions" from pay;

We have,

⇒ $307.80 - $54.82 = $253.48,

Therefore, your gross base pay is $253.48.

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Answer:

add the tips and pay together , this equals 17.10

then multiply by the 18 hours and you’ll get 307.80

Step-by-step explanation:

Answer:

Step-by-step explanation:

16 - (y^2/sqrt(49)) +12

at y = -5

16 - ( (-5)^2 / sqrt(49)) +12    

16 - ( 25 / 7) + 12

28 - (25/7)

(28*7 - 5) / 7    (LCM)

(196 - 5) / 7

(191/7)

27.2857 = Ans

Answer:

y=4x+4

Step-by-step explanation:

The slope formula is:
[tex]y=mx+b[/tex]

with m being the slope and b being the y-intercept

Given: slope=4, y-intercept=4

We can substitute the slope and the y-intercept into the question:

y=4x+4

Hope this helps! :)

The equation of the line in slope-intercept form of a line with slope 4 and y-intercept 4 is [tex]\text{y} = 4\text{x} + 4[/tex].

The ratio that y increase as x increases is the slope of a line. The slope of a line reflects how steep it is, but how much y increases as x increases. Anywhere on the line, the slope stays unchanged (the same).

[tex]\text{m}=\dfrac{\text{y}_2-\text{y}_1}{\text{x}_2-\text{x}_1}[/tex]

[tex]\text{m}=\dfrac{(\text{y}\bar{\text{a}}-\text{y}\bar{\text{a}})}{(\text{x}\bar{\text{a}}-\text{x}\bar{\text{a}})}[/tex]

It is given that:

A line with slope 4 and y-intercept 4.

The linear equation in one variable can be made:

As we know,

The standard equation of the line is:

[tex]\text{y} = \text{mx} + \text{c}[/tex]

Here m is the slope and c is the y-intercept.

[tex]\text{m} = 4[/tex]

[tex]\text{c} = 4[/tex]

[tex]\boxed{\bold{y = 4x + 4}}[/tex]

Thus, the equation of the line in slope-intercept form of a line with slope 4 and y-intercept 4 is [tex]\text{y} = 4\text{x} + 4[/tex].

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Answer:

To find the area of a triangle given its vertices, we can use the formula:

Area = 1/2 * |(x1*(y2-y3) + x2*(y3-y1) + x3*(y1-y2))|

where (x1,y1), (x2,y2), and (x3,y3) are the coordinates of the three vertices.

Using this formula, we can calculate the area of the triangle with vertices at (5,0), (-3,1), and (2,6) as follows:

Area = 1/2 * |(5*(1-6) + (-3)(6-0) + 2(0-1))|

= 1/2 * |-25 - 18 - 2|

= 1/2 * |-45|

= 22.5

Therefore, the area of the triangle is 22.5 units squared. So the answer is option C: 22.5 units squared.

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Answer:

Step-by-step explanation:

The mean of the thickness of paper is 39 inches, and the standard deviation is 0.1 inches.

To find the mean and standard deviation of the thickness of paper after dividing by 0.0001, you should follow these steps:

1. Divide the mean by 0.0001:
Mean after dividing = 0.0039 inches / 0.0001
Mean after dividing = 39 inches

2. Divide the standard deviation by 0.0001:
Standard deviation after dividing = 0.00001 inches / 0.0001
Standard deviation after dividing = 0.1 inches

The mean of the thickness of paper after dividing by 0.0001 is 39 inches, and the standard deviation is 0.1 inches.

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To determine a second pair of corresponding angles from KJN and LJM that are congruent, we can start by identifying the first pair of corresponding angles.

angle JKN is  harmonious to angle LJM. thus, we need to find another brace of corresponding angles that involve these same two angles.   One possibility is to look at the  perpendicular angles formed by the  crossroad of KJ and JM. Angle KJM is  perpendicular to angle NJL. therefore, angle KJM in KJN corresponds to angle NJL in LJM. thus, these two angles are  harmonious.  

We can prove that these two angles are  harmonious using the  perpendicular angles theorem, which states that  perpendicular angles are always  harmonious. Since KJ and JM  cross at point J, angles KJM and NJL are  perpendicular angles and must be  harmonious.   thus, we've shown that the alternate brace of corresponding angles from KJN and LJM that are  harmonious are angle KJM and angle NJL.

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To find the estimated probability that it will take at least five patients to find one patient on whom the medicine would not be effective, we can use the geometric distribution. The probability of not curing the disease is 1 - 0.70 = 0.30.

Let X be the number of patients needed to find one patient on whom the medicine would not be effective. Then X follows a geometric distribution with p = 0.30. The probability that it will take at least five patients to find one patient on whom the medicine would not be effective is:

P(X ≥ 5) = (1 - p)^(5-1) = 0.7^4 ≈ 0.2401

Therefore, the estimated probability that it will take at least five patients to find one patient on whom the medicine would not be effective is approximately 0.2401.

To find the estimated probability that the medicine will be effective on exactly three out of four randomly selected patients, we can use the binomial distribution. Let X be the number of patients out of four who are cured by the medicine. Then X follows a binomial distribution with n = 4 and p = 0.70. The probability that the medicine will be effective on exactly three out of four randomly selected patients is:

P(X = 3) = (4 choose 3) * (0.70)^3 * (1 - 0.70)^(4-3) = 4 * 0.343 * 0.3^1 ≈ 0.4116

Therefore, the estimated probability that the medicine will be effective on exactly three out of four randomly selected patients is approximately 0.4116.

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The given circle has a radius of 11.2,

Given radius of the circle (r) = 11.2

The diameter of the given circle = 2*r = 2*11.2 = 22.4

The circumference of the given circle = 2πr = 2*3.14*11.2 = 70.336

The area of the given circle =  πr² = 3.14*(11.2)² = 393.88

From the above analysis, we can conclude that the diameter of the circle is 22.4 and the circumference of the circle is 70.336 and the area of the circle is 383.88.

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The solution to the given equation is x = -3/4 or x = 1.

In order to find the solution, we start by squaring both sides of the equation to eliminate the square roots:

12x² - 4 = 4x + 8

Next, we simplify the equation by moving all the terms to one side:

12x² - 4x - 12 = 0

Now we can factor the quadratic equation:

4x² - x - 3 = 0

By factoring or using the quadratic formula, we find that the equation can be written as:

(4x + 3)(x - 1) = 0

Setting each factor equal to zero gives us the solutions:

4x + 3 = 0 or x - 1 = 0

Solving for x in each equation yields:

x = -3/4 or x = 1

Therefore, the solution to the given equation is x = -3/4 or x = 1.

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The correct confidence interval for the population mean of people in the world who own their homes is CI ≈ (45.3%, 51.3%).

To construct a confidence interval for the population mean of people in the world who own their homes, we can use the sample data and calculate the margin of error. The confidence interval will provide an estimated range within which the true population mean is likely to fall.

Given the sample size of 760 people and 367 individuals who own their homes, we can calculate the sample proportion of individuals who own their homes as follows:

Sample proportion (p-hat) = Number of individuals who own their homes / Sample size

p-hat = 367 / 760 ≈ 0.483

To construct the confidence interval, we can use the formula:

CI = p-hat ± Z * sqrt((p-hat * (1 - p-hat)) / n)

Where:

CI = Confidence Interval

p-hat = Sample proportion

Z = Z-score corresponding to the desired confidence level (95% confidence level corresponds to a Z-score of approximately 1.96)

n = Sample size

Plugging in the values, we get:

CI ≈ 0.483 ± 1.96 * sqrt((0.483 * (1 - 0.483)) / 760)

Calculating the expression inside the square root:

sqrt((0.483 * (1 - 0.483)) / 760) ≈ 0.0153

Substituting back into the confidence interval formula:

CI ≈ 0.483 ± 1.96 * 0.0153

CI ≈ (0.483 - 0.0300, 0.483 + 0.0300)

CI ≈ (0.453, 0.513)

Therefore, the correct confidence interval for the population mean of people in the world who own their homes is CI ≈ (45.3%, 51.3%). None of the provided answer choices match the correct confidence interval.

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It's "D"

It's self explanatory but y gets 1/2 of whatever x is and adds 1. So if x = 2, then it'll get half of two (which is one) and add one to it, getting two.

Laura is paying each month:

extra payment = $550 - monthly payment

To calculate how much extra Laura is paying each month, we first need to calculate the total cost of her car financing. Here are the steps:

Calculate the trade-in value of Laura's old car. We don't have the exact value, but we know that she received 85% of the trade-in value listed below, so we can set up an equation:

0.85 * trade-in value = value Laura received

Solving for the trade-in value, we get:

trade-in value = value Laura received / 0.85

Add the trade-in value to the list price of the new car to get the total cost before taxes and fees:

total cost before taxes and fees = $19,858 + trade-in value

Add the sales tax, registration fee, and documentation fee to get the total cost of the car financing:

total cost = (1 + 0.0927) * total cost before taxes and fees + $988 + $77

Calculate the monthly payment using the formula for a loan with monthly compounding:

monthly payment = (principal * monthly interest rate) / (1 - (1 + monthly interest rate)^(-number of months))

We know that Laura is financing the rest of the cost after her trade-in value, so:

principal = total cost - value Laura received

monthly interest rate = 0.095 / 12

number of months = 48 (since it's a four-year financing)

Substituting these values into the formula, we get:

monthly payment = ($19,858 + trade-in value - value Laura received) * 0.007916 / [tex](1 - (1 + 0.007916)^{(-48)})[/tex]

Now that we have the total monthly payment, we can calculate how much extra Laura is paying each month:

extra payment = $550 - monthly payment

Note that this assumes that Laura doesn't have any other fees or interest charges on her car financing, such as late payment fees or penalties for paying off the loan early. If there are any additional fees or charges, the calculation may be different.

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Therefore, the solutions of the equation in the interval [0, 2) are: x = π/6, -π/6 (in radians) or x = 30°, -30° (in degrees). Note that the solutions are listed in ascending order.

The equation is:

sec(x) = 2√3/3

First, we can find the values of x for which sec(x) = 2√3/3. Recall that sec(x) = 1/cos(x), so we have:

1/cos(x) = 2√3/3

Multiplying both sides by cos(x), we get:

1 = 2√3/3 cos(x)

cos(x) = 3/(2√3) = √3/2

Now, we can use the unit circle to find the solutions of the equation in the interval [0, 2).

cos(x) = √3/2 when x is π/6 or 11π/6 (in radians), or 30° or 330° (in degrees), since these are the angles in the unit circle where the x-coordinate is √3/2.

However, we need to make sure that these solutions are in the interval [0, 2). Since the period of sec(x) is 2π, we can add or subtract 2π to any solution to get another solution. Therefore, we need to find the solutions in the interval [0, 2π) that correspond to the solutions we found above.

π/6 is already in the interval [0, 2π), so it is a solution in the interval [0, 2). To find the other solution in the interval [0, 2), we can add 2π to 11π/6:

11π/6 + 2π = 23π/6

23π/6 is not in the interval [0, 2), so we need to subtract 2π instead:

11π/6 - 2π = -π/6

-π/6 is in the interval [0, 2), so it is also a solution in the interval [0, 2).

Therefore, the solutions of the equation in the interval [0, 2) are:

x = π/6, -π/6 (in radians) or x = 30°, -30° (in degrees)

Note that the solutions are listed in ascending order.

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Answer:

3/y

Step-by-step explanation:

15x÷5xy

15x/5xy....you will cancel x by x and simplify by 5(GCF)

3/y ...... it is the simplify form of the given equetion.

The value of b = 4

The function becomes undefined when the denominator goes to 0

3x² - 48 = 0

3x² = 48

x² = 16

x = +/ 4

From the given choices, it's x = 4

Rational expressions that utilize ratios of polynomial expressions are referred to as rational expressions. These can be written in the format p(x)/q(x), where both p(x) and q(x) are polynomials with the constraint that q(x) cannot equal zero.

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The degree of the monomial expression -3xy²z(-2x²yz) is derived to be equal to 8.

A monomial is a type of algebraic expression that consists of only one term. It is an expression in which the variables and their exponents are multiplied together, with no addition or subtraction involved. The degree of a monomial is the sum of the exponents of its variables.

Given the monomial:

-3xy²z(-2x²yz)

We can simplify it by multiplying the coefficients and adding the exponents of the variables:

-3(-2)x^(2+1) y^(2+1) z^(1+1)

= 6x³y³ z²

Therefore, the degree of the given monomial expression is: 3 + 3 + 2 = 8.

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The value of the investment at the end of 6 years is $6,359.77 if the interest is compounded annually, $6,416.52 if it is compounded quarterly, and $6,437.70 if it is compounded monthly.

[tex]A = P(1 + r/n)^{nt}[/tex]

where A is the final amount, P is the initial investment, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

A. If the interest is compounded annually, we have:

A = [tex]5,500(1 + 0.0155/1)^{1*6}[/tex] = $6,359.77

B. If the interest is compounded quarterly, we have:

A = [tex]5,500(1 + 0.0155/4)^{4*6[/tex] = $6,416.52

C. If the interest is compounded monthly, we have:

A = [tex]5,500(1 + 0.0155/12)^{12*6[/tex] = $6,437.70

Therefore, the value of the investment at the end of 6 years is $6,359.77 if the interest is compounded annually, $6,416.52 if it is compounded quarterly, and $6,437.70 if it is compounded monthly.

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A graph of the axis of symmetry and the vertex for this function is shown below.

In Mathematics and Geometry, the graph of a quadratic function would always form a parabolic curve because it is a u-shaped. Based on the graph of this quadratic function, we can logically deduce that the graph is an upward parabola because the coefficient of x² is positive and the value of "a" is greater than zero (0).

Since the leading coefficient (value of a) in the given quadratic function h(x) = (x - 5)² - 7 is positive 1, we can logically deduce that the parabola would open upward and the solution would be on the x-intercepts. Also, the value of the quadratic function f(x) would be minimum at -7.

In conclusion, the turning point and vertex is given by the ordered pair (5, -7).

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Answer:

12

Step-by-step explanation:

multiply 4x18 then divide by 6

The best representation of the buttons in the box is the third sample.

To determine which sample is the best representation of the button in the box, we need to examine the samples and compare them to the known ratios of the colors of buttons in the box.

First sample: 6 red, 2 green, 1 blue, 1 yellow (out of 10)

Second sample: 3 red, 3 green, 3 blue, 1 yellow (out of 10)

Third sample: 2 red, 2 green, 5 blue, 1 yellow (out of 10)

To compare the samples to the known ratios, we can calculate the percentage of each color in each sample and compare it to the percentage of each color in the box.

The percentages of each color in the box are:

Red: 10/100 = 10%

Green: 30/100 = 30%

Blue: 40/100 = 40%

Yellow: 20/100 = 20%

First sample:

Red: 6/10 = 60%

Green: 2/10 = 20%

Blue: 1/10 = 10%

Yellow: 1/10 = 10%

Second sample:

Red: 3/10 = 30%

Green: 3/10 = 30%

Blue: 3/10 = 30%

Yellow: 1/10 = 10%

Third sample:

Red: 2/10 = 20%

Green: 2/10 = 20%

Blue: 5/10 = 50%

Yellow: 1/10 = 10%

Based on these percentages, we can see that the third sample is the best representation of the buttons in the box, as it is closest to the known ratios of colors in the box.

The first sample is skewed towards red buttons and away from blue and green buttons, while the second sample has equal percentages of red, green, and blue buttons, which is not reflective of the known ratios in the box.

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For two parametric equations, x = cos(t)− sin(t) ; y = cos(t) + sin(t) ; 0≤t≤2π

a) Cartesian equation for the parametric curve is represented by x² + y² = 2.

b) The sketch for this parametric curve, with arrows in the direction of curve tracing is present above figure.

A parametric curve in the x-t plane has the equations x=x(t), y=y(t). The curve associates a point of the plane (x,y) to a value of the parameter t. The rectangular form of the curve can be determined by eliminating the parameter t, i.e. determine the parameter in one equation and Substituting this value in the other equation. We have the following parametric equations,

x = cos⁡t - sin⁡ty = cos⁡(t)+ sin⁡t, 0 ≤ t ≤ 2π

(a) we have to eliminate parameter t to determine a cartesian equation for the parametric curve, use x²+ y² = (cos⁡(t) − sin⁡(t))²+ (cos⁡(t) + sin⁡(t))²

=> x² +y² = cos²t + sin²t - 2cos⁡t sin⁡t + cos²t + sin²t + 2cos⁡t sin⁡t

=> x² + y² = 2 ( sin²t + cos²t) = 2

which represents a circle curve centered at the origin and having radius √2.

(b) A sketch of this parametric curve is shown above figure and arrows are used to indicate the direction of curve trace.

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