Point A and B lie on a circle with a radius of 1, and arc AB has a length pi/3. What fraction of the circumference of the circle is the length of arc AB?
a. 1/6
b. 1/8
c. 1/12

The steps that she could use to solve the quadratic equation is determined as 2(x² + 6x + 9) = 18 + 3.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides. LHS = RHS is a common mathematical formula.

Coefficients, variables, operators, constants, terms, expressions, and the equal to sign are some of the components of an equation. The "=" sign and terms on both sides must always be present when writing an equation.

Given:

2x² + 12x - 3 = 0

2x² + 12x = 3

divide through by 2

x² + 6x = ³/₂

Take half of coefficient of x, square it and add it both sides of the equation

(x + 3)² = ³/₂ + 3²

x² + 6x + 9 = 9 + ³/₂

2(x² + 6x + 9) = 18 + 3

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the probability that the sample mean cost for the 36 schools is between $25,575 and $26,500 is 54.14%.

Percentage is a way of expressing a number as a fraction of 100. The symbol for percentage is "%". For example, if you have 75%, it means 75 out of 100 or 0.75 as a decimal.

Given by the question.

We can solve these problems by using the central limit theorem and the formula for the standard error of the mean.

The standard error of the mean is:

SE = σ/√n

where σ is the population standard deviation, n is the sample size, and √ is the square root.

Using the given values, we have:

SE = 3204/√36 = 534

To find the probability that the sample mean cost for the 36 schools is less than $25,000, we need to calculate the z-score and then use a z-table to find the probability.

z = (25000 - 26489) / 534 = -2.79

Using a standard normal distribution table or calculator, we find that the probability of a z-score less than -2.79 is 0.0026 or 0.26%.

Therefore, the probability that the sample mean cost for the 36 schools is less than $25,000 is 0.26%.

To find the probability that the sample mean cost for the 36 schools is more than $26,000, we need to calculate the z-score and then use a z-table to find the probability.

z = (26000 - 26489) / 534 = -0.91

Using a standard normal distribution table or calculator, we find that the probability of a z-score more than -0.91 is 0.8186 or 81.86%.

Therefore, the probability that the sample mean cost for the 36 schools is more than $26,000 is 81.86%.

To find the probability that the sample mean cost for the 36 schools is between $25,575 and $26,500, we need to calculate the z-scores for both values and then use a z-table to find the probabilities.

z1 = (25575 - 26489) / 534 = -1.73

z2 = (26500 - 26489) / 534 = 0.21

Using a standard normal distribution table or calculator, we find that the probability of a z-score less than -1.73 is 0.0418 and the probability of a z-score less than 0.21 is 0.5832.

Therefore, the probability that the sample mean cost for the 36 schools is between $25,575 and $26,500 is the difference between these probabilities:

0.5832 - 0.0418 = 0.5414

Multiplying by 100, we get a probability of 54.14%.

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The minimum wage increased by approximately 13.43% from 1987 to 1988.

The minimum wage increased by approximately 3.88% from 2004 to 2005.

a) Assuming minimum wage increased from $3.35 per hour in 1987 to $3.80 per hour in 1988.

To calculate the percent increase, we can use the formula:

percent increase = (new value - old value) / old value * 100%

percent increase = (3.80 - 3.35) / 3.35 * 100% = 13.43%

b) Assuming minimum wage increased from $5.15 per hour in 2004 to $5.35 per hour in 2005.

percent increase = (5.35 - 5.15) / 5.15 * 100% = 3.88%

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8. M(x,y) and N(x,y) are functions of x and y, not just y, so the given differential equation is not linear in y.

9. Both M(u,v) and N(u,v) are functions of u, so the given differential equation is linear in u.

8. To match the given differential equation with the first differential equation given in (7), we need to write the given equation in the form of M(x,y)dx + N(x,y)dy = 0. If the resulting M(x,y) and N(x,y) are both functions of y, then the differential equation is linear in y.

Rewriting (y² - 1)dx + xdy = 0 as xdy = (1 - y²)dx and dividing by x(1 - y²), we get (1/x)dy - [(y² - 1)/(x(1 - y²))]dx = 0.

Therefore, we have M(x,y) = (y² - 1)/(x(1 - y²)) and N(x,y) = 1/x. Both M(x,y) and N(x,y) are functions of x and y, not just y, so the given differential equation is not linear in y.

9. To match the given differential equation with the first differential equation given in (7), we need to write the given equation in the form of M(u,v)du + N(u,v)dv = 0. If the resulting M(u,v) and N(u,v) are both functions of u, then the differential equation is linear in u.

We can rewrite the given equation as u dv + (v + uv) du - [tex]ue^u[/tex] du = 0.

Therefore, we have M(u,v) = v + uv - [tex]ue^u[/tex] and N(u,v) = u. Both M(u,v) and N(u,v) are functions of u, so the given differential equation is linear in u.

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Based on the given age ratio, Mina is 10 years old.

From the case, we know that the age ratio between these 3 people are:

L : M = 1 : 2

M : N = 3 : 4

L = 2M

N = 4/3 M

30 < L + M + N < 50

We can try to find Mina's age range as:

30 < 2M + M + 4/3 M < 50

30 < 13/3 M < 50

120 < 13M < 150

9 < M < 11

Please note that we round down the age range of Mina.

We take M = 10; then:

L = 2M = 2(10) = 20

N = 4/3 M = 4/3 (10) = 13

L + M + N = 20 + 10 + 13

L + M + N = 43 --> PROVEN!

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The inside length of the longer NBA caskets in inches is, 85.6745

A statistical measure of the degree of variation or dispersion in a set of values is the standard deviation. It offers a way to measure how far apart the individual values in a data set are from the set's mean (average). The square root of the variance is used to calculate the standard deviation, which is denoted by the symbol (sigma).

We are given the distribution here as:

From standard normal tables, we have here:

P(Z < 2.0537) = 0.98

Therefore, P(Z > 2.0537) = 1 - 0.98 = 0.02

Therefore 2.0537 is the z score that would be used here. ( As we are given here that for tallest 2% of the male basketball players, the special caskets for the NBA would be too tall )

Therefore the longest length here is computed as:

= Mean + 2.0537xSD

= 79 + 2.0537x3.25

= 85.6745 inches is the required length here.

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If the remaining milk is sold for $3.94 per half gallon, then the store initially bought 26 half gallons of milk for the profit of $46.76

What is meant by profit?

Profit is the term used to describe the financial gain experienced when the revenue from a company activity outpaces the costs, costs, and taxes incurred to support the activity in question.

Profit is the amount of money you have after paying for business expenses. Gross profit, operating profit, and net profit are the three basic categories of profit.

If the store bought x half gallons of milk at $2.90 per half gallon, the total cost of the milk would be:

                           total cost = 2.90x

Since the store stored the milk in two refrigerators, there were initially x/2 half gallons in each refrigerator.

Unfortunately, 14 half gallons were ruined, so the number of half gallons of milk remaining for sale is:

                            x - 14

The store then sells the remaining milk for $3.94 per half gallon, so the total revenue from the sale is:

                                total revenue = 3.94(x - 14)

The profit the store made is the difference between the total revenue and the total cost:

                         profit = total revenue - total cost

We know that the store made a profit of $46.76, so we can set up an equation:

46.76 = 3.94(x - 14) - 2.90x

Simplifying and solving for x:

46.76 = 3.94x - 55.16

102.92 = 3.94x

x = 26

So the store initially bought 26 half gallons of milk.

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Answer: 50 above zero

Step-by-step explanation:

Opposite numbers 5 and -5 are opposite because they are an equal distance from zero.

Absolute value, like |-5| = 5, means that the number is at five units of distance from zero.

The opposite of a number is obtained exchanging just the signal of the number.

The opposite of a number x is then given as follows:

-x.

As the number and it's opposite is the same distance from zero, they have the same absolute value.

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The height of the tree to the nearest hundredth of a meter will be 6.40 meters.

Trigonometry is the branch of mathematics which set up a relationship between the sides and angles of right-angle triangles.

Let h be the height of the tree in meters. Then, based on the similar triangles formed by Ethan, his shadow, the tree, and its shadow, we have:

h / x = Ethan's height/distance to Ethan's feet from the tree

and

h / (x + 28.45) = Ethan's height/distance to Ethan's feet from the tip of the shadows

where x is the length of Ethan's shadow in meters. We can solve for x using the first equation as follows:

x = (Ethan's height/distance to Ethan's feet from the tree) * h

Substituting the given values, we get:

x = (1.85 / 24.3) x h = 0.076 x h

We can now substitute this value of x into the second equation and solve for h:

h / (0.076 x h + 28.45) = 1.85 / 24.3

Multiplying both sides by 0.076h + 28.45, we get:

h = (1.85 / 24.3) x (0.076h + 28.45)

Expanding the right side and simplifying, we get:

h = 0.070 x h + 5.95

Subtracting 0.070h from both sides, we get:

0.930h = 5.95

Dividing both sides by 0.930, we get:

h = 6.40 meters (rounded to the nearest hundredth)

Therefore, the height of the tree is approximately 6.40 meters.

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The dividends per share for the preferred stock are $2.50, $0.80, and $3.00 for 20Y1, 20Y2, and 20Y3, respectively.

The dividends per share for the common stock are $0.75, $0.24, and $0.90 for 20Y1, 20Y2, and 20Y3, respectively.

To determine the dividends per share for preferred and common stock, we need to divide the total dividends by the number of shares outstanding for each type of stock.

For the preferred stock:

20Y1: $37,500 ÷ 15,000 shares = $2.50 per share

20Y2: $12,000 ÷ 15,000 shares = $0.80 per share

20Y3: $45,000 ÷ 15,000 shares = $3.00 per share

For the common stock:

20Y1: $37,500 ÷ 50,000 shares = $0.75 per share

20Y2: $12,000 ÷ 50,000 shares = $0.24 per share

20Y3: $45,000 ÷ 50,000 shares = $0.90 per share

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The given statement follows the commutative property.

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

The given equation is x+y=y+x

The commutative property states that the change in the order of two numbers in an addition or multiplication operation does not change the sum or the product.

Therefore, the given statement follows the commutative property.

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Convert the percent to a decimal: according to the local weather report, the probability of rain in Boston on February 9 is 78% is 0.78%.

Probability is the measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Probability theory is used to study random phenomena, such as the outcome of a coin toss or the results of an election. It is also used to make predictions about the future, such as whether it will rain tomorrow.

The probability of choosing a student selected for the AP math class is 13/40 (6 girls and 7 boys selected for the AP math class). Therefore, the probability of choosing a boy or a student selected for the AP math class is 32/40 (19 boys plus 13 students selected for the AP math class out of 40 students).

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The complete question is: According to the local weather report, the probability of rain in Boston on February 9 is 78%. What is 78% converted to a decimal?

It takes Mrs. Brown 6 hours and 15 minutes to catch up to Mr. Brown.

What does average speed refer to?

The entire distance travelled by the object in a specific amount of time is its average speed.

What is a mile?

A common unit of measurement is the mile. It is typically used to describe the length of rivers, highways, and distances between cities. The unit mile is represented by the letter "mi."

Mr. Brown departs at 50 mph five hours early, giving him a 250mile head start.

So:

50n+250=90n

40n=250

n=6.25 hours after Mrs. Brown leaves before she overtakes Mr. Brown.

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The difference between the highest and the lowest guess, on the whole number, is 120.

Subtraction is a mathematical operation. Which is used to remove terms or objects in the expression.

Given:

A table that shows the amount of butter.

Amount of butter

120

130

150

240

From the table;

the highest guess is 240.

And the lowest guess is 120.

So, the difference between the highest and the lowest  guess is,

= 240 - 120

= 120.

And here, 120 is a whole number.

Therefore, the required value is 120.

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Third-degree Taylor polynomial for f(x) = e centered at a = 1 using a graphing calculator estimate the value of eˣ near x = 1.

A polynomial is an expression consisting of variables and coefficients, which involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Now, let's focus on the Taylor polynomial. The Taylor polynomial of degree n for a function f(x) centered at a is given by the formula:

[tex]Tn(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^{2/2!} + f'''(a)(x - a)^{3/3!} + ... + f^{n}(a)(x - a)^{n/n!}[/tex]

In this formula, f'(a), f''(a), f'''(a), and f^(n)(a) are the first, second, third, and nth derivatives of f(x), respectively, evaluated at a. The symbol n! denotes the factorial of n, which is the product of all positive integers from 1 to n.

In our case, we're given the function f(x) = e and a = 1. The first derivative of e^x is e^x, the second derivative is e^x, and the third derivative is also eˣ. Evaluating these derivatives at a = 1, we get:

f(1) = e¹ = e

f'(1) = e¹ = e

f''(1) = e¹ = e

f'''(1) = e¹ = e

Substituting these values into the formula for the Taylor polynomial, we get:

[tex]T3(x) = e + e(x - 1) + e(x - 1)^{2/2!} + e(x - 1)^{3/3!}[/tex]

Simplifying this expression, we get:

[tex]T3(x) = e + e(x - 1) + e(x - 1)^{2/2} + e(x - 1)^{3/6}[/tex]

This is the third-degree Taylor polynomial for f(x) = e centered at a = 1. It can be used to approximate the value of eˣ near x = 1.

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Complete Question:

A graphing calculator is recommended. Find the Taylor polynomial T3(x) for the function f centered at the number a. f(x) = e, a = 1.

Answer:

(- 3, 1.5)

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

Given system:

The first expression is ready to be substituted as no further operation is required to simplify it.

Find x:

The density of the liquid is 2g/cm³

The density of a substance is the relationship between the mass of the substance and how much space it takes up (volume). The mass of atoms, their size, and how they are arranged determine the density of a substance. Density equals the mass of the substance divided by its volume; D = m/v.

Where D is the density

m is the mass and

v is the volume

mass = 100kg = 100000 g( 1kg = 1000g)

50l = 50 × 1000 = 50000cm³

density = 100000/50000

= 2g/cm³

therefore the density of the liquid is 2 g/cm³

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Madeline's school building model is 5.33 meter.

Two triangles are similar if the angles are the same size or the corresponding sides are in the same ratio. Either of these conditions will prove two triangles are similar.

Given that, Madeline walks a distance of 9.45 meters from the building, then places a mirror flat on the ground, marked with an X at the center.

Use corresponding parts of similar triangles.

Madeline's eyes height/Madeline's distance from x = Buildings height/Buildings distance from X

1.55/2.75 = x/9.45

2.75x=1.55×9.45

2.75x=14.6475

x=14.6475/2.75

x=5.33

Therefore, Madeline's school building model is 5.33 meter.

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We would expect John to weigh approximately 199.8 pounds at the end of week 8 if he continues to burn 6100 calories per week and maintains a proper diet.

Finding the best linear relationship between the independent and dependent variables is the aim of linear regression. To put it another way, it seeks out the line (or plane, or hyperplane in higher dimensions) that best fits the data. The dependent variable's future values for new values of the independent variable can then be predicted using this line.

We may build a linear regression model using the provided data, with activity (in calories) serving as the independent variable and weight (in pounds) serving as the dependent variable. Using this model, we can then forecast John's weight at the conclusion of week eight depending on the number of calories he burned in that week.

Using a spreadsheet or statistical software, we can calculate the regression equation:

Weight = -0.0049 x Exercise + 229.85

We can plug in the exercise value for week 8 (6100 calories) and solve for weight:

Weight = -0.0049 x 6100 + 229.85

Weight ≈ 199.8 pounds

Therefore, we would expect John to weigh approximately 199.8 pounds at the end of week 8 if he continues to burn 6100 calories per week and maintains a proper diet.

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The correct description that best explains the domain of (g o f) (x) is: The elements in the domain of f(x) for which g(f(x)) is defined.

From the question, we have the following parameters that can be used in our computation:

(g o f) (x)

The expression (g o f)(x) is a composite function

The domain of (g o f)(x) is the set of all possible inputs x for which the composition (g o f)(x) is defined.

For the composition (g o f)(x) to be defined, it is necessary that the input x belongs to the domain of f(x), and the output of f(x) belongs to the domain of g(x).

Hence, the domain of (g o f)(x) consists of all the elements in the domain of f(x) for which g(f(x)) is defined

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Answer:

Step-by-step explanation:

He can walk 125km in 2 and a half hours

Answer:

125km

Step-by-step explanation:

If he walks 50km per hour and he walks 2 hours it's 2 x 50km = 100km

if walks for another half hour the half of 50km is 25km so 100km + 25km = 125km

The answer is (C) the absolute value of the z-score for the midterm exam grade is greater than for the final grade, so the midterm grade is more unusual.

To calculate the z-scores for the midterm and final exam grades, we first need to calculate the mean and standard deviation for each data set.

For the midterm exam grades:

mean = (80 + 78 + 85 + 82 + 79 + 79 + 78 + 86 + 80 + 84) / 10 = 81.1

standard deviation = sqrt([ (80-81.1)^2 + (78-81.1)^2 + ... + (84-81.1)^2 ] / 9) = 2.336

For the final exam grades:

mean = (81 + 88 + 68 + 69 + 69 + 81 + 82 + 86 + 76 + 71) / 10 = 77.1

standard deviation = sqrt([ (81-77.1)^2 + (88-77.1)^2 + ... + (71-77.1)^2 ] / 9) = 7.042

Now, we can calculate the z-scores for Jaymes's grade of 86 on each exam:

z-score for midterm grade = (86 - 81.1) / 2.336 = 2.094

z-score for final exam grade = (86 - 77.1) / 7.042 = 1.259

To determine which grade is more unusual, we need to compare the absolute values of the z-scores. Since the absolute value of the z-score for the midterm grade is greater than for the final exam grade (|2.094| > |1.259|), we can conclude that the midterm grade is more unusual for Jaymes. Therefore, the answer is (C).

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_____The given question is incomplete, the complete question is given below:

The midterm and final exam grades for a statistics course are provided in the data set below. Jaymes, a student in the class, scored 86 on both exams. Treat the given data sets as samples. Jaymes's wants to know which grade is more unusual, the midterm grade or the final exam grade. Use Use a TI-83, TI-83 Plus, or TI-84 calculator to calculate the z-scores corresponding to each grade. Round your answer to three decimal places.

Midterm 80, 78, 85, 82, 79, 79, 78, 86, 80, 84,

final 81, 88, 68, 69, 69, 81, 82, 86, 76, 71

The measure of the side JU of the triangle will be 24 units.

Geometry is one of the oldest branches of mathematics, along with arithmetic. It is concerned with spatial properties such as figure distance, shape, size, and relative position.

Given that the two triangles ΔLKJ and ΔTUJ are similar. The sides LK  96 and the side JK = 64. The side JU = -4 + 4x and JT = 27.

From the similarity property, the length of the segment JU will be calculated as:-

LK / TU = JK / JU

96 / 36 = 64 / ( -4 + 4x )

96( -4 + 4x ) = 36 x 64

-384 + 384x = 2304

384x = 2688

x = 7

The value of JU will be,

JU = -4 + 4x

JU = -4 + 4 x 7

JU = -4 + 28

Ju = 24

Therefore, the length of JU will be 24 units.

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The instantaneous velocity of the particle at 8 seconds is approximately the change in position over the change in time, or[tex]s(8+Δt) - s(8) / Δt[/tex]. If we use Δt = 0.1, then the instantaneous velocity is approximately s(8.1) - s(8) / 0.1. The approximate position of the particle after 10 seconds is then s(8) + (10-8) * (s(8.1) - s(8) / 0.1).

The instantaneous velocity of a particle is defined as the change in position over the change in time. This can be represented by the equation [tex]s(t+Δt) - s(t) / Δt[/tex]. Here, s(t) is the position of the particle at time t, and t is a small change in time. To approximate the instantaneous velocity at 8 seconds, we can use Δt = 0.1. This gives us an equation of s(8+0.1) - s(8) / 0.1. This equation gives us the velocity at 8 seconds. To approximate the position of the particle after 10 seconds, we can use the equation s(8) + (10-8) * (s(8.1) - s(8) / 0.1). This equation gives us an approximate position of the particle after 10 seconds, using the change in position over the change in time. This equation gives us a good approximation of the position of the particle after 10 seconds, given the information about the particle's position at 8 seconds.

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The solid is a cylinder with a typical disk or washer. It is obtained by rotating the region bounded by[tex]y = 6 - x^2[/tex]and y = 2 about the x-axis. The volume is 48π.

The region to be rotated is bounded by the equations [tex]y = 6 - x^2[/tex] and y = 2. The region is bounded horizontally by x = 0 and x = 2 and vertically by y = 2 and [tex]y = 6 - x^2.[/tex] To find the volume of the solid we can use the formula V = ∫2π0∫2y0rdrdθ. The integral inside is the area of a disk with radius r and the integral outside is the circumference of a circle with radius r. We substitute [tex]y = 6 - x^2[/tex] into the integral, giving us [tex]V = ∫2π0∫2(6 - x^2)0rdrdθ[/tex]. By solving the integral we get V = 48π.

y = 2

x² = 6×4 = 24

V = ∫2π0∫2y0rdrdθ

V = 48π

The solid is a cylinder with a typical disk or washer and is obtained by rotating the region bounded by [tex]y = 6 - x^2[/tex] and y = 2 about the x-axis.

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The correct answer is option b. When the homoskedasticity assumption is violated, The beta parameter estimates are biased while the rest of the OLS assumptions are correct.

When the homoskedasticity assumption is violated, the ordinary least squares (OLS) estimator is still consistent but no longer efficient. This means that the estimates of the regression coefficients (beta parameters) are still unbiased, but they have higher variances and covariances.

In other words, the OLS estimator is no longer the best linear unbiased estimator (BLUE) and it may be biased when the errors are heteroscedastic. Therefore, option b is correct.

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The volume of the solid formed when this region is rotated around the y-axis is V = ∫[1,9] π(8 - y/2)² dy cubic units

Volume is a measure of the amount of three-dimensional space that a solid object occupies.

To find the volume of the solid formed when the region is rotated around the y-axis, we need to use the disk method. This involves slicing the solid into thin disks perpendicular to the y-axis and adding up the volumes of all the disks.

Each disk will have a thickness of dy and a radius of x, which can be expressed in terms of y. We can find the radius of each disk by considering the distance between the y-axis and the edge of the region. Since the region is in quadrant i, we know that the edge of the region is at x = 0 (the y-axis) and x = 6 (the right boundary of the region). Therefore, the radius of each disk is given by x = 6 - (y - 5)/2, or equivalently, x = 8 - y/2.

The area of each disk is given by the formula for the area of a circle, A = πr². Substituting our expression for x into this formula, we get

=> A = π(8 - y/2)².

To find the volume of the solid, we need to integrate the area of each disk over the range of y values that covers the region. Since the region has centroid (3,5) and area 12 square units, we know that the region must extend from y = 1 to y = 9. Therefore, the volume of the solid is given by the integral:

V = ∫[1,9] π(8 - y/2)² dy cubic units

Evaluating this integral will give us the volume of the solid formed by rotating the region around the y-axis.

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