can someone help me do this (will give brainliest if possible

after her 2nd paycheck, Sally had $375 in her bank account, after her 3rd paycheck she had $525, after her 4th paycheck she had $675, after her 5th paycheck she had $825, after her 6th paycheck she had $975, and after her 7th paycheck she had $1125.

How to solve the problem?

Let's call the amount that Sally gets paid every week "x". We know that after her 1st paycheck, she had $225, so we can write:

1st paycheck: $225

2nd paycheck: $225 + x

3rd paycheck: $225 + 2x

4th paycheck: $225 + 3x

5th paycheck: $225 + 4x

6th paycheck: $225 + 5x

7th paycheck: $225 + 6x

8th paycheck: $225 + 7x = $1275

To solve for "x", we can start by subtracting $225 from both sides of the equation for the 8th paycheck:

$225 + 7x = $1275

7x = $1050

x = $150

So Sally gets paid $150 every week.

To find the amount of money she had in her account after each paycheck, we can substitute "x" into the equations above:

1st paycheck: $225

2nd paycheck: $225 + $150 = $375

3rd paycheck: $225 + 2($150) = $525

4th paycheck: $225 + 3($150) = $675

5th paycheck: $225 + 4($150) = $825

6th paycheck: $225 + 5($150) = $975

7th paycheck: $225 + 6($150) = $1125

8th paycheck: $225 + 7($150) = $1275

Therefore, after her 2nd paycheck, Sally had $375 in her bank account, after her 3rd paycheck she had $525, after her 4th paycheck she had $675, after her 5th paycheck she had $825, after her 6th paycheck she had $975, and after her 7th paycheck she had $1125.

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Answer:

the regular polygon has 15 angles

Answer:

1. I ............. she will to the party, I really want to see her.

a) expect

c) want

d) believe

Step-by-step explanation:

You're welcome.

Based on the sample data, 7th grade boys, would be more likely to hold a handstand for about 19 seconds or more.

The difference between the boys' mean time and the girls' mean time is 1 second.

The difference between the boys' median time and the girls' median time is 1.5 seconds.

A subset of data collected from a larger population. It is typically used to represent the larger population and is used for testing and analysis.

In this data set, the mean time of the 7th grade boys= 19.375 seconds, while the mean time of the 7th grade girls = 17.875 seconds.

This means that the 7th grade boys have a higher mean time than the 7th grade girls.

The difference is 1.5 seconds.

The median time of the 7th grade boys = 19 seconds, and the median time of the 7th grade girls = 18 seconds.

This also indicates that the 7th grade boys have a higher median time than the 7th grade girls.

The difference is 1 second.

Given this data, the 7th grade boys are more likely to hold a handstand for about 19 seconds or more than the 7th grade girls.

This is because the mean and median times for the 7th grade boys are higher than the mean and median times for the 7th grade girls.

This indicates that the 7th grade boys have a higher average time than the 7th grade girls, and thus, are more likely to hold a handstand for a longer period of time.

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Based on the information provided, I think that Ben is correct.

Let's analyze and compare the given expressions:

Adzo says the graph is of the function y = 2-¹x - √25.

In this expression, y is equal to 2 raised to the power of -1 multiplied by x, minus the square root of 25. This can also be written as y = (1/2)x - 5.

Ben says it is y - 5 = 0.5x.

In this expression, y - 5 is equal to 0.5 times x.

Comparing the two expressions, we can see that Ben's expression, y - 5 = 0.5x, is a linear equation in slope-intercept form, where the slope is 0.5 and the y-intercept is -5, representing a straight line on a graph.

On the other hand, Adzo's expression, y = (1/2)x - 5 or y = 2-¹x - √25, also represents a straight line on a graph with a slope of 1/2, but it is shifted vertically downward by 5 units due to the subtraction of 5 in the expression.

Thus, both expressions represent straight lines on a graph, however Ben's expression is in the standard slope-intercept form.

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The length of the ladder is approximately 30.6 feet. The height of the building is 90 degrees.

Let "θ" be the angle between the ladder and the ground. Then, we have:

sin(θ) = 30 ÷ x

Solving for "x", we get:

x = 30 ÷ sin(θ)

Since the ladder is the hypotenuse, we know that the angle opposite the height of the building is 90 degrees. Therefore, the angle between the ladder and the ground is the complement of this angle, which is:

θ = 90 - arcsin(30 ÷ x)

Substituting this expression for "θ" into the equation for "x", we get:

x = 30 ÷ sin(90 - arcsin(30 ÷ x))

Using the identity sin(90 - θ) = cos(θ), we can simplify this to:

x = 30 / cos(arcsin(30 ÷ x))

Using the identity cos(arcsin(x)) = [tex]\sqrt[] 1-x^{2}[/tex] , we can simplify this further to:

x = [tex]\frac{30}{\sqrt{(1-(30/x^{2}))\\} }[/tex]

Squaring both sides and simplifying, we get:

x² = 30² + 6²

x² = 900 + 36

x² = 936

x = [tex]\sqrt{936}[/tex]

x = 30.6 feet is height of ladder

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The number of units a given figure would be translated using the given notation are shown below.

In Mathematics, the translation of a graph to the left is a type of transformation that simply means subtracting a digit from the value on the x-coordinate of the pre-image while the translation of a graph to the right is a type of transformation that simply means adding a digit to the value on the x-coordinate of the pre-image.

Based on each of the transformation rule, the number of units a given figure would be translated using the given notation include the following:

(x, y)   →   (x + 7, y - 6)

7 units right.

6 units down.

(x, y)   →   (x - 9, y + 2)

9 units left.

2 units up.

(x, y)   →   (x - 11, y)

11 units left.

0 units up/down.

(x, y)   →   (x - 8, y - 5)

8 units left.

5 units down.

(x, y)   →   (x + 13, y + 3)

7 units right.

3 units up.

(x, y)   →   (x, y + 9)

0 units right/left.

9 units up.

(x, y)   →   (x - 14, y + 12)

14 units left.

12 units up.

(x, y)   →   (x, y - 4)

0 units right/left.

4 units down.

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Answer:

The population is the total number of people who go to the movie theater, and the sample is the first 150 people at the theater.

Step-by-step explanation:

The population is the potential people that could be sampled, and the sample is the people who are being asked the question.

Answer:ccccccccccccccccccccccccccc

Step-by-step explanation:

Answer:

x=2

Step-by-step explanation:

4 x1,2 =2

The correct statement which compares the graph of function g with the graph of function f is,

⇒ The graph of function g is a vertical compression of the graph of function f.

A relation between a set of inputs having one output each is called a function. and an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

Given that;

Functions are,

f (x) = eˣ - 4

g (x) = 1/2eˣ - 4

Now, We have to find that;

The graph of function g is a vertical compression of the graph of function f.

Hence, The correct statement which compares the graph of function g with the graph of function f is,

⇒ The graph of function g is a vertical compression of the graph of function f.

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Answer:

We know that <APC + < APD = 180. We also know that <APD + <DPB = 180

Thus <APC =  <DPB

Answer:

angle APC is congruent to angle DPB according the theorem -Veritical Angle Theorem.

Step-by-step explanation:

Theorem 2.8 (Vertical Angles Theorem)

If two angles are vertical angles, then they are congruent.

Erica can buy a maximum of 4 sweatshirts by rounding the cost to the nearest ten.

A number is approximated to a certain degree of accuracy by rounding it off. The purpose of doing this is usually to make the number easier to deal with or to portray it in a clearer, more accessible manner.

You must take the following actions in order to round off a number:

The digit in the number that you want to round off is known as the rounding digit. The rounding digit is the third digit following the decimal point, or 1, in the example above, if you wish to round off 3.1416 to two decimal places.

Look at the digit that is directly above it: You should round up if this digit is 5 or higher. You should round it down if it is less than 5.

Replace the digits to the right of the rounding digit with zeros: If you are rounding to a certain number of decimal places, you will need to add zeros to the right of the rounding digit to maintain the correct level of precision. For example, if you are rounding 3.1416 to two decimal places, you will need to replace the digits to the right of the rounding digit with zeros, giving you 3.14.Adjust the rounding digit if necessary: If you round up, you will need to

To estimate the maximum number of sweatshirts Erica can buy, we need to divide her total amount of money by the cost of each sweatshirt rounded to the nearest ten.

Rounding $28.99 to the nearest ten gives us $30.

Dividing $120 by $30 gives us:

$120 ÷ $30 = 4

Therefore, Erica can buy a maximum of 4 sweatshirts by rounding the cost to the nearest ten.

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The domain of the relation is the set {-5, -2, 0, 1, 3, 5}.

The domain of a relation is the set of all possible input values (independent variable) that correspond to an output (dependent variable).

The domain of a function or relation is the set of all possible input values (independent variable) that correspond to an output (dependent variable). It represents the values for which the function or relation is defined.

Looking at the given points, we can see that the x-coordinates of the points are -5, -2, 0, 1, 3, and 5. Therefore, the possible input values for this relation (i.e., the domain) are:

Domain: {-5, -2, 0, 1, 3, 5}

So the domain of the relation is the set {-5, -2, 0, 1, 3, 5}.

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Actual time is much longer than the 50-year timeframe suggested by the speaker.

How to verify the statement?

This claim is based on the assumption that the rate of pollution reduction from each factory remains constant at 80%, and the number of factories increases by 3% per annum.

To verify this claim, we can use the following formula to calculate the expected total pollution after n years:

Total pollution after n years = [tex]Present \: pollution \: level x (1 + 0.03)^n \times (1 - 0.8)^n[/tex]

Here, the first factor represents the increase in pollution due to the growth in the number of factories, and the second factor represents the reduction in pollution due to the 80% reduction in pollution from each factory.

We can set the expected total pollution after n years equal to the present pollution level and solve for n:

[tex]Present \: pollution \: level = Present \: pollution \: level \times (1 + 0.03)^n \times (1 - 0.8)^n[/tex]

Simplifying this equation, we get:

[tex]1 = 1.03^n \times 0.2^n[/tex]

Taking the logarithm of both sides of the equation, we get,

[tex]n \times log(1.03) + n \times log(0.2) = 0 \\ n \times (log(1.03) + log(0.2)) = 0 \\ n = \frac{0} { (log(1.03) + log(0.2))} \\ n = 271.56[/tex]

Therefore, according to this calculation, it would take approximately 272 years for the total annual pollution to return to its present level if the number of factories in the country increases by 3% per annum, and they all immediately reduce the amount of pollution they produce by 80%. This is much longer than the 50-year timeframe suggested by the speaker.

It is important to note that this calculation assumes that the pollution reduction from each factory remains constant at 80% and does not take into account any other factors that may affect pollution levels, such as changes in technology or regulations. Therefore, this should be considered as a rough estimate and not as an exact prediction.

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Correct question is "A speaker claimed that if the number of factories in the country increases by 3% per annum, then even if they all immediately reduced the amount of pollution they produce by 80%, the total annual pollution will be back to its present level in about 50 years. Verify the statement. (use logarithm)"

ABCD is a square, rhombus, and rectangle.

Checking whether ABCD with vertices A(6, 2), B(3, 5), C(0, 2), and D(3, 1) fits the definition of each shape can help us determine whether it is a rectangle, rhombus, or square.

A parallelogram with four right angles is a rectangle. We can determine the slopes of AB and CD and BC and AD separately to see if ABCD is a rectangle. It is a rectangle if the opposite sides are perpendicular and they are negative reciprocals of one another.

[tex]Slope \ of \ AB = \frac{(5 - 2)} { (-3 + 6)} = 1 \ Slope\ of\ CD = \frac{(-1 - 2)} { (-3 - 0)} = -1 \ Slope \ of\ BC =\frac{ (2 - 5)}{ (0 + 3)} = -1 \ Slope\ of\ AD = \frac{(2 - (-1))}{ (-6 + 3)} = 1[/tex]

The opposite sides are perpendicular to one another since AB and CD have negative reciprocal slopes and BC and AD do too. In light of this, ABCD is a rectangle.

A parallelogram with four congruent sides is called a rhombus. We can determine the separation between each pair of vertices in ABCD to determine if it is a rhombus. It is a rhombus if the lengths of the four sides are equal.

[tex]Distance\ between\ A \ and\ B = \sqrt{{(5 - 2)}^2 + {(-3 + 6)}^2} = \sqrt(10) \ Distance\ between\ B \ and\ C = \sqrt{({(2 - 5)}^2 + {(0 + 3)}^2)} = \sqrt(10) \ Distance\ between\ C\ and\ D = \sqrt{((-1 - 2)^2 + (-3 - 0)^2) }= \sqrt(10) \ Distance \ between\ D \ and \ A = \sqrt{((2 - (-1))^2 + (-6 + 3)^2)} = \sqrt(10)[/tex]

ABCD is also a rhombus since the lengths of all four sides are equal.

A parallelogram having four identical sides and four right angles is called a square. We can combine the tests for rectangles and rhombuses to determine whether ABCD is a square.

Since ABCD satisfies the requirements for both rectangles and rhombuses, it is also required to be a square.

Hence, ABCD is a square, rhombus, and rectangle.

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a finite group with an even number of elements contains an even number of elements such that. x inverse = x.

a similar statement for a finite group with an odd number of elements

For a finite group with an even number of elements, we can prove that it contains an even number of elements such that their order is 2.

Let G be a finite group of even order.

We can define t(G) as the set { g ∈ G | g ≠ g − 1 }.

if g ∈ t(G), then g − 1 ∈ t(G), and by definition g − 1 ≠ g.

Now let A = { { g, g − 1 } | g ∈ t (G) }.

Thus we have | t (G) | = ∑ α ∈ A | α | = 2 k for some positive integer k;

in particular, t (G) contains an even number of elements 1.

Hence [tex]x^{-1}=x[/tex]

For a finite group with an odd number of elements, we can prove that it contains an odd number of elements such that their order is 2. For any element in G-{e}, there must be an inverse of that element in G-{e}. Take any element in G-{e}, say b, If bb=e, then the proof is done. If the inverse of b is not itself, then there must be one element in G-{e}, say c, such that bc=e. Because there are an odd number of elements, I keep doing this until I have left with one

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The median price charged is $78.00

The median of a set of data is the middle value when the data is arranged in numerical order.

In other words, it is the value that separates the data into two halves, with half of the values being less than the median and half being greater than the median. If there is an even number of data points, the median is the average of the two middle values.

Here we have

The prices of 9 dog training programs were:

$ 78.00, $ 78.00, $80.00, $77.00, $78.00, $77.00, $78.00, $77.00, $79.00

To find the median price, we need to arrange the prices in order from least to greatest:

$77.00,  $77.00, $77.00,  $78.00, $78.00, $78.00, $78.00, $79.00, $80.00

There are nine prices, which is an odd number,

So the median is the middle number.

In this case, the middle number is the fifth number, which is $78.00

Therefore,

The median price charged is $78.00

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Answer: 9+3z<-12

Step-by-step explanation: because first  -3(2-z)= -6+3z

then 15-6=9

so then it is 9+3z for the left side and we didn't touch the right side so it would be the same

Answer:

The first step is to calculate the average daily balance for the billing cycle. This is done by adding up the daily balances for each day of the billing cycle and then dividing by the number of days in the billing cycle. In this case, the billing cycle is from July 7 to August 6, so there are 31 days in the billing cycle.

The daily balances are as follows:

* July 7: $216.71

* July 14: $268.75

* July 17: $284.37

* July 21: $116.71

* July 24: $260.74

The average daily balance is therefore $226.81.

The next step is to calculate the finance charge. This is done by multiplying the average daily balance by the interest rate and then by the number of days in the billing cycle. In this case, the interest rate is 1.25% and the number of days in the billing cycle is 31 days.

The finance charge is therefore $7.43.

The final step is to calculate the new balance. This is done by adding the finance charge to the previous balance. In this case, the previous balance is $216.71 and the finance charge is $7.43. The new balance is therefore $224.14.

Here is the solution in mathematical form:

* Average daily balance = (216.71 + 268.75 + 284.37 + 116.71 + 260.74) / 31 = 226.81

* Finance charge = 226.81 * 0.0125 * 31 = 7.43

* New balance = 216.71 + 7.43 = 224.14

Step-by-step explanation:

Answer: x=0, x=1/8

Step-by-step explanation:

Simplifying the equation give us: 4x=32x^2

Subtract 4x from both sides: 32x^2-4x=0

Factor:4x(8x-1)=0

Using the Zero Product Property, x=0, x=1/8

Answer:

x equals 0  it wont only  let me put answer so i am typing more

Answer:

If the entire mass of the Milky Way was due to gas and stars, how would you expect the rotational speed of a star near the edge of the galaxy to compare to the rotational speed of a star near the center?

Step-by-step explanation:

The resulting values for r will be sqrt(140).

Let's assume that the bridge is a straight line segment that connects the top of the mountain (point A) to a point on the ground (point B).

Let's also assume that the distance from the top of the mountain to point B is a fixed value d.

If we attach the higher end of the bridge to the top of the mountain at a height y, the distance between point A and the attachment point can be calculated using the Pythagorean theorem as:

x = sqrt([tex]d^2[/tex] - [tex]y^2[/tex])

The length of the bridge, which is also the hypotenuse of the right triangle formed by points A, B, and the attachment point, can then be calculated as:

r = sqrt([tex]x^2[/tex] + [tex](d-y)^2[/tex])

To find the values of r for different heights y, we can simply substitute different values of y into these equations and calculate the resulting values of r.

For example, if we use the values d=10 and y=3, we get:

To calculate the values for "r" given 5 possible values for "y," we need to use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In this case, the bridge is the hypotenuse, and we know the height of the mountain (x) and the distance from the mountain to the lower end of the bridge (d). So we can write:

r^2 = [tex]x^2[/tex] + [tex](d + y)^2[/tex]

x = sqrt([tex]10^2[/tex] - [tex]3^2[/tex])

x = sqrt(91)

r = sqrt((sqrt([tex]91))^2[/tex] + [tex](10-3)^2[/tex])

r = sqrt(91 + 49)

r =  sqrt(140)

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Answer:

Step-by-step explanation:

y = 4

cause its at y

Tammy ran 2.25 miles on Sunday.

A linear equation is a mathematical expression that describes a straight line on a graph. It is of the form y = mx + b, where "m" is the slope of the line and "b" is the y-intercept.

Let's start by figuring out how far Tammy ran on Sunday. We know that she ran for 1/2 of the distance she ran on Saturday. Therefore, if we let "x" be the distance Tammy ran on Sunday, we can set up the following equation:

x = 1/2(4.5)

Here, we used the fact that Tammy ran 4.5 miles on Saturday.

To solve for "x", we simply need to simplify the right-hand side of the equation:

x = 1/2(4.5)

x = 2.25

Therefore, Tammy ran 2.25 miles on Sunday.

In summary, we used the equation x = 1/2(4.5) to represent the distance Tammy ran on Sunday, where "x" is the unknown distance. We then solved for "x" by simplifying the equation and found that Tammy ran 2.25 miles on Sunday.

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The nationalization of major industries in Hugo Chavez's Venezuela describes a move away from capitalism. Thus, option C is correct.

Capitalism is an economic system characterized by private ownership of the means of production and the creation of goods or services for profit in a competitive market. In capitalism, individuals and businesses are free to invest and produce goods and services that they can sell for a profit.

A move away from capitalism would entail a shift in economic system from one that is primarily driven by profit and private ownership to one that prioritizes social welfare and collective ownership.

Chavez, a clοse ally οf Cuba’s cοmmunist leaders, has steadily increased the rοle οf the state in Venezuela with a slew οf natiοnalizatiοns and tοugh cοntrοls οn prices and fοreign exchange.

Despite an ecοnοmy weakened by lοwer οil incοme, Chavez has sοlid apprοval ratings and is nοw pοwering ahead with his plan tο regulate and reduce the private sectοr in the OPEC cοuntry

Therefore, The nationalization of major industries in Hugo Chavez's Venezuela describes a move away from capitalism. Thus, option C is correct.

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Complete question:

Which of the following describes a move away from capitalism?

O A. Sweden's 1980s cutback on social programs

O B. the transformation of the former Soviet Union to today's Russia

O C. the nationalization of major industries in Hugo Chavez's Venezuela

O D. economic reforms following the Great Depression in the United States

For a circle [tex]k(O)[/tex] with diameter AB and CD [tex]\bot[/tex] AB, it is proved that [tex]AD\times CB=AC\times CD[/tex], using similarity of triangles.

Given:

circle [tex]k(O)[/tex] with diameter AB

CD [tex]\bot[/tex] AB

To Prove: [tex]AD\times CB=AC\times CD[/tex]

Proof:

In [tex]\Delta ADC[/tex] and [tex]\Delta CDB[/tex],

[tex]\angle ADC = \angle CDB = 90^\circ[/tex]

[∵Both are right angle triangles]

[tex]CB = CB[/tex] [Common side]

[tex]\implies\dfrac{AC}{CB} =\dfrac{CD}{DB}[/tex]

Thus, [tex]\Delta ACD[/tex] is similar to [tex]\Delta CDB[/tex] by RHS similarity.

Therefore, we can write,

[tex]\dfrac{AD}{CD} =\dfrac{AC}{CB}[/tex] [Since corresponding sides of similar triangles are proportional]

[tex]\implies AD\times CB = AC\times CD[/tex]

Hence proved.

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1) The time series plot is attached accordingly. The percentage of stocks seems to fluctuate randomly over time.

2)  the optimal value of the smoothing coefficient is 0.5.

3) Therefore, the forecast of the percentage of stocks in a typical portfolio for the second quarter of 2009 is 30.9%.

1) See the attached time series. The percentage of stocks seems to fluctuate randomly over time.

2)  To find the optimal value of the smoothing coefficient that results in a relatively small mean squared error (MSE), we can use a trial-and-error approach. We can try different values of the smoothing coefficient α and calculate the MSE for each value. The smoothing coefficient that results in the smallest MSE is the optimal value.

Using other software, we can apply exponential smoothing to the data and try different values of α. For example, we can start with α=0.1 and increment it by 0.1 until we find the optimal value.

The following table shows the results of applying exponential smoothing with different values of α: See attached.

From the table, we can see that the smallest MSE is obtained when α=0.5. Therefore, the optimal value of the smoothing coefficient is 0.5.

c. Using the optimal value of the smoothing coefficient α=0.5, we can forecast the percentage of stocks in a typical portfolio for the second quarter of 2009. To do this, we use the following formula for exponential smoothing:

F(t+1) = α x Y(t) + (1-α) x F(t)

where F(t+1) is the forecast for the next period, Y(t) is the actual value in the current period, and F(t) is the forecast for the current period.

To apply this formula, we start with the first quarter of 2007 as the initial forecast value:

F(1) = Y(1) = 29.8

Then, we can use the formula to calculate the forecast for each subsequent quarter:


F(2) = α x Y(2) + (1-α) x F(1) = 0.5 x 31.0 + 0.5 x 29.8 = 30.4

F(3) = α x Y(3) + (1-α) x F(2) = 0.5 x 29.9 + 0.5 x 30.4 = 30.2

F(4) = α x Y(4) + (1-α) x F(3) = 0.5 x 30.1 + 0.5 x 30.2 = 30.2

F(5) = α x Y(5) + (1-α) x F(4) = 0.5 x 32.2 + 0.5 x 31.2 = 31.7

F(7) = α x Y(7) + (1-α) x F(6) = 0.5 x 32.0 + 0.5 x 31.7 = 31.9

F(8) = α x Y(8) + (1-α) x F(7) = 0.5 x 31.9 + 0.5 x 31.9 = 31.9

F(9) = α x Y(9) + (1-α) x F(8) = 0.5 x 30.0 + 0.5 x 31.9 = 30.9

Therefore, the forecast of the percentage of stocks in a typical portfolio for the second quarter of 2009 is 30.9%.

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Full Question:

Although part of your question is missing, you might be referring to this full question:

The following table (See attached image) reports the percentage of stocks in a portfolio for nine quarters from

2007 to 2009.

a. Construct a time series plot. What type of pattern exists in the data?

b. Use trial and error to find a value of the exponential smoothing coefficient that results in a relatively small MSE.

c. Using the exponential smoothing model you developed in part (b), what is the forecast of the percentage of stocks in a typical portfolio for the second quarter of 2009?

Answer:

Each side length of the cube will be a polynomial of degree 3.

Putting all these values together, we get the equation: h(t) = 4 cos(2π/3 (t - 2)) + 1 with Mike's height above the ground as a function of time t in seconds, where h(t) is measured in feet.

An equation is a mathematical statement that shows that two expressions are equal. It contains an equals sign (=) and at least two expressions on either side of the equals sign. The expressions on either side of the equals sign can be numbers, variables, or a combination of both, and the equation represents a relationship between them. Equations are used in mathematics to solve problems and find unknown values by manipulating the expressions within the equation while keeping the equality true.

Here,

Assuming that Mike's motion on the swing set follows a periodic pattern, we can model his height above the ground using a sinusoidal function. Let's use the cosine function, since it reaches its maximum value when the input is 0 and decreases to its minimum value at π or 180°.

The general form of a cosine function is:

y = A cos(Bx + C) + D

where:

A = amplitude (half the distance between the maximum and minimum values)

B = frequency (the number of cycles per unit of x)

C = phase shift (horizontal shift of the graph)

D = vertical shift (vertical shift of the graph)

We are given that Mike's highest point is at 9 feet and his lowest point is at 1 foot. Therefore, the amplitude A = (9 - 1) / 2 = 4.

The frequency is determined by the time it takes for Mike to complete one cycle. We know that it takes him 1.5 seconds to go from his highest point to his lowest point and back up again. Therefore, the period of the function is 3 seconds (2 x 1.5), and the frequency is 1/3 cycles per second. Hence, B = 2π/3.

The phase shift is the horizontal displacement of the graph from the origin. We know that at t = 2 seconds, Mike is at his highest point. Therefore, we need to shift the cosine curve 2 seconds to the right to match this point. Thus, C = -2.

Finally, the vertical shift D is the position of the center of the curve. Since Mike's lowest point is at 1 foot, we need to shift the entire curve up by 1 foot. Thus, D = 1.

Putting all these values together, we get the equation:

h(t) = 4 cos(2π/3 (t - 2)) + 1

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