use the properties of geometric series to find the sum of the series. for what values of the variable does the series converge to this sum?
7−14z+28z2−56z3+⋯
sum =
domain =
(Give your domain as an interval or comma separated list of intervals; for example, to enter the region x<−1 and 2

250 is the total surface area of the solid.

How do you determine surface area?

The whole surface of a three-dimensional form is referred to as its surface area. The surface area of a cuboid with six rectangular faces may be calculated by adding the areas of each face.

                         Instead, you may write out the cuboid's length, width, and height and apply the formula surface area (SA)=2lw+2lh+2hw.

Each side of a cube with side length = 5  has an area of 25; the overall area is 6 x 25 = 150

A cube with sides of length 5 has an area of 25 on each side, making its overall area 6 x 25 or 150.

Both have a combined area of 150 + 150 = 300

300 - 25 - 25 = 250 is the result from each of the two cubes.

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The probability that 10 or more of 15 marketing personnel are extroverts is 0.719.

Since 80% of all marketing personnel are extroverts, the probability of any single marketing personnel being an extrovert is 0.8. The probability that 10 or more marketing personnel at the party of 15 are extroverts can be calculated using the Binomial Distribution formula:

P(X>=10) = 1 - [P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9)]

P(X>=10) = 1 - [15C0*0.80*0.215 + 15C1*0.81*0.214 + 15C2*0.82*0.213 + 15C3*0.83*0.212 + 15C4*0.84*0.211 + 15C5*0.85*0.210 + 15C6*0.86*0.29 + 15C7*0.87*0.28 + 15C8*0.88*0.27 + 15C9*0.89*0.26]

P(X>=10) = 0.719

Therefore, 0.79 is the probability that 10 or more of the 15 marketing personnel at the party are extroverts.

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To implement the distributing principle to the phrase 7(2x+5) - 4(2x+5), we must first divide the 7 and 4 within the parenthesis to their respective terms:

7(2x+5) - 4(2x+5) = (72x + 75) - (42x + 45)

Within the parenthesis, each term is simplified:

= (14x + 35) - (8x + 20)

We may now reduce the phrase by grouping similar terms:

= 14x - 8x + 35 - 20

= 6x + 15

As a consequence, using the distributive property on 7(2x+5) - 4(2x+5) yields 6x + 15.

By expansion or multiplying, the distribution principle is frequently employed to compress statements and solve problems. It enables us to translate complicated statements into shorter language and conversely. The distributive principle is commonly utilized in mathematics, mathematics, as well as other mathematical subjects.

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The equations without absolute value for the given conditions are: 1. y = -3x + 6 if x > 5; 2. y = -x - 6 if x < -2; 3. y = x - 6 if 3 ≤ x ≤ 5, and y = -x - 6 if x < 3.

1. When x > 5, the expression (x - 3) is positive, (x + 2) is positive, and (x - 5) is positive. Thus, to get absolute value we can rewrite the equation as:

y = (x - 3) + (x + 2) - (x - 5)

Simplifying this, we get:

y = 2x - 4

2. When x < -2, the expression (x - 3) is negative, (x + 2) is negative, and (x - 5) is negative. Thus, we can rewrite the equation as:

y = -(x - 3) - (x + 2) + (x - 5)

Simplifying this, we get:

y = -2x + 6

3. When -2 ≤ x ≤ 3, the expression (x - 3) is negative, (x + 2) is positive, and (x - 5) is negative. Thus, we can rewrite the equation as:

y = -(x - 3) + (x + 2) - (x - 5)

Simplifying this, we get:

y = 10 - x

When x > 3, the expression (x - 3) is negative, (x + 2) is positive, and (x - 5) is positive. Thus, we can rewrite the equation as:

y = -(x - 3) + (x + 2) + (x - 5)

Simplifying this, we get:

y = -2x + 6

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To make 2 gallons of a 20% salt solution, combine 0.86 gallons of the 12% salt solution and 1.14 gallons of the 26% salt solution.

Let x be the amount of the 12% salt solution needed in gallons, and y be the amount of the 26% salt solution needed in gallons to make 2 gallons of a 20% salt solution.

Based on the provided data, we can construct the following system of two equations:

X + y = 2 (total volume of the mixture is 2 gallons)

0.12x + 0.26y = 0.2(2) (total salt content of the mixture is 20% of 2 gallons)

Simplifying the second equation, we get:

0.12x + 0.26y = 0.4

Multiplying the first equation by 0.12 and subtracting it from the second equation, we get:

0.14y = 0.16

Y = 1.14

Substituting y = 1.14 into the first equation, we get:

X + 1.14 = 2

X = 0.86

In order to create 2 gallons of a 20% salt solution, 0.86 gallons of the 12% salt solution and 1.14 gallons of the 26% salt solution must be combined.

The complete question is:-

How much of a 12% salt solution must combined with a 26% salt solution to make 2 gallons of a 20% salt solution?

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As the length of the side immediately across from the angle, choice (c) 6 units is the correct answer.

Three straight lines that cross at three different locations create the two-dimensional geometric outline of a triangle. A triangle's vertices, which are the three places at which those three lines intersect, are referred to as the triangle's sides. The dimensions of a triangle's edges and angles can be used to classify it. For instance, an isosceles triangle has two equal sides and two equal angles while an equilateral triangle has three equal sides and three equal angles of 60 degrees. An angle or side of a scalene triangle cannot be equivalent.

given

The right-angled triangle XYZ in the provided illustration has a side length of 6 units and an angle opposite to it that is labelled as 30°. The extent of the side YZ, denoted as x, must be determined.

To find x, we can use the trigonometric sine relation. The length of the side directly across from the angle divided by the length of the hypotenuse is known as the sine of an angle. The hypotenuse in this instance is designated as 2x.

As a result, we have:

sin 30° = (6/2x)

Adding two times to both sides:

2x * sin 30° = 6

Using sin 30°, which has a value of 0.5:

x = (6/(2 * 0.5)) = 6/1 = 6

Consequently, the side YZ is 6 units long.

As the length of the side immediately across from the angle, choice (c) 6 units is the correct answer.

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One advantage of best-guess experiments is that they are often faster and more cost-effective than factorial or design experiments.

Best-guess (trial and error) experiments involve making a hypothesis and testing it through a series of trials until a satisfactory result is achieved. On the other hand, factorial or design experiments involve manipulating multiple variables simultaneously to determine their individual and interactive effects on a response variable.

Both approaches have their advantages and disadvantages depending on the specific research question and goals. They may also be useful in situations where there is limited knowledge about the variables of interest or when the system is too complex to be modeled accurately.

However, best-guess experiments may suffer from issues such as biased or subjective interpretation of results, a lack of control over extraneous variables, and a potential for false positives or negatives.

In contrast, factorial or design experiments provide a more systematic approach to testing hypotheses and offer greater control over variables, leading to more reliable and generalizable results. They may, however, be more time-consuming and expensive to conduct.

Ultimately, the choice between best-guess and factorial or design experiments depends on the research question, available resources, and desired level of precision and control.

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If the population growth rate is 3.5 percent every year then the population of the town after 7 years would be 14940.

Given that population grows 3.5 percent every year.

So, the increase in population after one year

= 3.5% of 12000

= (3.5/100) × 12000

= 420

Thus the increase in population after 7 year would be,

= population increase in one year × 7

= 420×7 = 2940

Hence population of the town after 7 years = (present population + increase in population)

= 12000 + 2940

= 14940

So the population of the town after 7 years with 3.5 % growth every year would be 14490.

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The conditional probability that Eric is a high-risk driver, given that he has had no traffic accidents in the past three years, is very low. Generally, insurance companies use the number of traffic violations and/or the number of claims a driver has had within a certain time period as indicators of their riskiness.

As Eric has had no accidents or traffic violations, the probability that he is a high-risk driver is very low. However, this does not mean that the probability is zero. There are many other factors which can contribute to a driver's risk, such as age, gender, experience, and location.

If Eric is an experienced driver, who has been driving for many years with no traffic accidents, then the probability of him being a high-risk driver will be lower than the average driver. On the other hand, if Eric is a new driver, or is located in an area with a high rate of traffic accidents, then the probability of him being a high-risk driver may be higher than the average driver.

Overall, the conditional probability that Eric is a high-risk driver, given that he has had no traffic accidents in the past three years, is very low. However, this probability can change depending on other factors, such as his age, experience, and location.

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The answer is: 1 triangle is possible  given the following side maesurment: 3 feet , 5 feet, 4 feet.

To determine how many triangles are possible with these side measurements, we can use the triangle inequality theorem, which states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

What is inequality theorem?

In this case, we have three side measurements: 3 feet, 5 feet, and 4 feet. Let's call these sides a, b, and c, respectively. Using the triangle inequality theorem, we can see that:

a + b > c

a + c > b

b + c > a

Substituting in the values of a, b, and c, we get:

3 + 5 > 4

3 + 4 > 5

4 + 5 > 3

All three of these inequalities are true, so it is possible to form a triangle with these side measurements.

To determine how many distinct triangles are possible, we can use the fact that any two triangles are distinct if and only if they have at least one side with a different length. In this case, all three sides have different lengths, so there is only one distinct triangle that can be formed with these side measurements.

Therefore, the answer is: 1 triangle.

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Complete question is: 1 triangle is possible given the following side maesurment: 3 feet , 5 feet, 4 feet.

The angle of elevation of the sun is 38.24 degrees (rounded to two decimal places).

The ratio of the height of the tree and the length of the shadow can be found by using the formula for tan which is given by:

tan(x) = (opposite/hypotenuse)

In the given problem statement, the height of the tree is opposite to the angle we are trying to find and the length of the shadow is the hypotenuse of the right-angled triangle. Therefore, we can say that:

tan(x) = opposite/hypotenuse = 86/110

We can use the inverse tangent function to find the value of x. This is because tan is the ratio of the opposite side (height of the tree) to the adjacent side (length of the shadow), and the inverse tangent function gives the angle that has that ratio.

So, tan x = 86/110 => x = tan⁻¹ (86/110) = 38.24 degrees (rounded to two decimal places). Therefore, the angle of elevation of the sun is 38.24 degrees.

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If we repeated a hypothesis test 1000 times, the number of times we would expect to commit a Type I error, assuming the null hypothesis were true, would depend on the significance level (α) of the test.

A Type I error occurs when we reject the null hypothesis when it is actually true. The significance level of a test (α) is the probability of making a Type I error when the null hypothesis is true. In other words, if we set a significance level of α = 0.05, we are saying that we are willing to tolerate a 5% chance of making a Type I error.

Assuming a significance level of α = 0.05, if we repeated the test 1000 times, we would expect to make a Type I error in approximately 50 tests (0.05 x 1000 = 50). This means that in 50 out of the 1000 tests, we would reject the null hypothesis even though it is actually true.

However, it is important to note that the actual number of Type I errors we make in practice may differ from our expectation, as it depends on the specific characteristics of the population being tested and the sample sizes used in each test.

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If we see the scatter plot we can conclude that the possible relation between x and y is linear and with a positive correlation since when the values of x increase the values for y increases as well.

[tex]Cov(x,y) =\frac{\sum_1^n(x_i-X')(y_i-Y')}{n-1}[/tex]

 [tex]\sum_1^5(6-16)(6-10)+(11-16)(9-10)....(27-16)(12-10)=106\\\\and\\Cov(x,y)=\frac{106}{4}=26.5\\\\r=0.693[/tex]

For this part we use excel in order to create the scatterplot and we got the result on the figure attached

If we see the scatter plot we can conclude that the possible relation between x and y is linear and with a positive correlation since when the values of x increase the values of y increase as well

The correlation coefficient is a "statistical measure that calculates the strength of the relationship between the relative movements of two variables". It's denoted by r and its always between -1 and 1.

And in order to calculate the correlation coefficient we can use this

[tex]Cov(x,y) =\frac{\sum_1^n(x_i-X')(y_i-Y')}{n-1}[/tex]

:  

[tex]Cov(x,y) =\frac{\sum_1^n(x_i-X')(y_i-Y')}{n-1}[/tex]

 [tex]\sum_1^5(6-16)(6-10)+(11-16)(9-10)....(27-16)(12-10)=106\\\\and\\Cov(x,y)=\frac{106}{4}=26.5\\\\r=0.693[/tex]

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A minimum score of 88.95 is required to receive an "A" grade on the exam.

To determine the minimum score required to receive an "A" grade on an exam, we must first understand the meaning of standard deviation and mean. The mean is the average of a set of values, whereas the standard deviation is a measure of how far apart the values are from the mean. The minimum score required to receive an "A" grade is determined by calculating the z-score that corresponds to the top 12.3 percent of exam scores.

The formula for calculating the z-score is given as: z = (x - μ)/σ, where x is the raw score, μ is the population mean, and σ is the population standard deviation. Solving for z, we have: z = invNorm(1 - 0.123) = invNorm(0.877) ≈ 1.15. The inverse normal distribution function is used to determine the value of z that corresponds to the area to the right of the z-score. We can then use the formula for the z-score to solve for the raw score (x):
x = zσ + μ
Substituting the values we have, we get:
x = 1.15(17) + 68 ≈ 88.95
Therefore, a minimum score of 88.95 is required to receive an "A" grade in the exam.

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As a result, we should not be concerned that the sample does not accurately reflect the population.

We can learn more about average, population, and sample.

What is the population?

The entire group of people, items, or objects that we want to draw a conclusion about is known as the population. For example, if we want to learn about the average age of people in the United States, then the entire population is every individual in the United States.

What is a sample?

A smaller group of individuals, objects, or items that are selected from the population is known as a sample. A random sample is a sample in which every individual in the population has an equal chance of being selected for the sample.

What is an average?

A statistic that summarizes the central tendency of a group of numbers is known as an average.

The mean is the most commonly used average in statistics. The mean is calculated by adding up all the numbers in a group and then dividing by the number of numbers in the group. If we want to learn about the average salary of all teachers in the United States, we'd have to sample every teacher. That's not a feasible option. Instead, we take a smaller sample, which should be representative of the population, and then use the information gathered from that sample to make predictions about the population as a whole.

If we assume that the five teachers in the example are a random sample of all teachers in the United States, then we can conclude that the average salary of all teachers in the United States is around $49,000. As a result, we should not be concerned that the sample does not accurately reflect the population.

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The answer to subtracting the two polynomials is 2x2 - 10x - 7.  The process of subtracting polynomials is like that of subtracting any other type of numerical expression.

Polynomials are mathematical expressions consisting of variables, coefficients, and exponents. They are used to represent a variety of functions, such as polynomial equations, rational equations, and trigonometric functions.

In this case, our like terms are 2x2 and 4x2, and -6x and -4x, and -4 and +3. We begin by subtracting the coefficients of the like terms. We subtract the coefficient of the term with the highest power first. In this case, that is 2x2 and 4x2, we subtract 2 from 4 to get 2. Next, we subtract the coefficients of the terms with the second highest power. This is -6x and -4x, so we subtract -6 from -4 to get -2. Thus, our answer now reads 2x2 -2x.

Finally, we subtract the coefficients of the terms with the lowest power. This is -4 and +3, so we subtract -4 from 3 to get -7. Thus, our answer now reads 2x2 -2x -7.

This answer can then be placed in the proper position on the grid.

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Answer:The equation of a line that passes through a point is an algebraic equation. It can also be referred to as the Slope-Intercept Equation.The equation of the line that passes through the point (4, 4) and has a slope of 0 is written as: y = 4 The equation of the line through a point (x1, y1) can be represented by the algebraic equation:y = mx + cwhere:m = slopec = y - interceptFrom the question,(x1, y1) = (4, 4)m = slope = 0Substituting these values into the algebraic equation,4 = (0 x 4) + c Hence, y = 4The equation of the line that passes through the point (4, 4) and has a slope of zero is y = 4

Answer:

y=4

Step-by-step explanation:

If a line has a slope of 0, then it is a horizontal line, and all points on the line have the same y-coordinate. We are given that the line passes through the point (4, 4). Therefore, the equation of the line in slope-intercept form is:

y = b

where b is the y-coordinate of the point through which the line passes. In this case, b = 4, so the equation of the line is:

y = 4

Therefore, the equation of the line in slope-intercept form is y = 4.

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Answer:

Eighty biscuits.

Step-by-step explanation:

We need to find the limiting factor. We can do that by comparing ratio of mass of ingredient given to mass of ingredient needed for 20 biscuits

[tex]Butter:\\800:150\\=16:3\\=5.33\\Sugar:\\700:75=28:3\\=9.33\\Flour:\\1000:180\\=50:9\\=5.56\\Chocolate Chips:200:50\\=4:1\\=4\\[/tex]

We can clearly see that the choco. chips are the limiting factor since it has the lowest ratio, basically meaning we will run out of choco chips before anything else.

[tex]Biscuits=4*20=80[/tex]

Since we only have 4 times the choco chips needed to make 20 biscuits, we can only make 80 biscuits. Now you can see, we have other ingredients left, but choco chips have ran out which is why it was the limiting factor.

[tex]Flour:\\1000-4(180) = 280g[/tex]

After making 4 servings we still have 280g of flour left.

Answer:

-0.2

Step-by-step explanation:

[tex]\frac{y2-y1}{x2-x1}[/tex]

^This here is how I calculated the slope^

Y2=4

Y1= 2

4-2= 2

X2=-7

x1=3

-7-3=-10

2/-10

or -2/10

Answer:

16x + 3

Step-by-step explanation:

Simplify by combining like terms.  Add the terms with x, then add the integers.

8x + 8x + 4 - 1 = 16x + 3

Answer:

(5,2)

Step-by-step explanation:

Let's solve your system by substitution.

[tex]x+y=7{\text{ ; }}x=y+3[/tex]

Step 2: let Solve [tex]$x+y=7$[/tex] for[tex]$x$[/tex]

[tex]x+y=7[/tex]

[tex]x+y +(-x)=7+(-x)[/tex] (Add (-x) on both sides)

[tex]y=-x+7[/tex]

0+(x)=7-x-y+(x)   (Add (x) on both sides)

x = -y + 7

x/1 = -y+7/1  (divide through by 1)

x = -y + 7

Substitute -y+7 for x in x = y + 3, then solve for u

(-y + 7) = y + 3

-y + 7 = y + 3 (simplify)

-y+7+(-7) = y + 3 + (-7)    (Add (-7) on both sides)

-y=y-4

-y = y-4 (simplify)

-y+(-y)=y-4+(-y)  (Add (-y) on both sides)

-2y-=-4

-2y/-2 = -4/-2  (Divide through by -2)

y = 2

Substitute in 2 for y in x = -y + 7

x =  -y+7

x = -2+7

x = 5

Answer:

x = 5 and y = 2

Matthew can travel up to 150 miles for $50, assuming the cost of the limo ride remains constant at a $20 fixed cost plus $0.20 per mile. Let's say Matthew has $50 to spend on the limo ride.

We know that the cost per mile is $0.20, so we can set up an equation:

Cost = $20 + $0.20 x Distance

We can substitute $50 for Cost and solve for Distance:

$50 = $20 + $0.20 x Distance

$30 = $0.20 x Distance

Distance = $30 / $0.20

Distance = 150 miles

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Answer:

No, q=15 is not a solution to the inequality.

Step-by-step explanation:

As given, q=15. So, substituting is the best way to solve this problem.

Step 1: Substitute

[tex]\frac{15}{5}=3[/tex]

[tex]15-20=-5[/tex]

Step 2: Substitute values into inequality

[tex]3 < -5[/tex]

Equation is false since 3 is a bigger value than -5.

Hope this helps ya!

Answer:

Let's call the length of the hypotenuse SD as x.

Since the altitude from T to SD divides SD into two parts, let the length of the shorter part be y. Then the length of the longer part is x-y.

Using similar triangles, we have:

TU/TS = ST/TD

Substituting the values we have:

TU/(3√5) = √5/UD

TU = (3/5)UD

Using the Pythagorean theorem in triangle TUS, we have:

TU² + (3√5)² = TS²

(3/5 UD)² + 45 = ST²

9/25 UD² + 45 = ST²

Using the Pythagorean theorem in triangle TUD, we have:

TU² + UD² = TD²

(3/5 UD)² + UD² = x²

9/25 UD² + UD² = x²

34/25 UD² = x²

UD² = (25/34)x²

Substituting the value of UD² in the equation ST² = 9/25 UD² + 45, we get:

ST² = 9/25 (25/34)x² + 45

ST² = 45/34 x² + 45

Since y = x-y-1, we have y = (x-1)/2.

Using the Pythagorean theorem in triangle TUD, we have:

(1/4) (x-1)² + UD² = x²

(1/4) (x² - 2x + 1) + (25/34)x² = x²

(1/4)(x²) + (25/34)x² - (1/2)x + (1/4) = 0

(59/68)x² - (1/2)x + (1/4) = 0

Using the quadratic formula, we get:

x = [1/2 ± √(1/4 - 4(59/68)(1/4))]/(2(59/68))

x = [1/2 ± (3√34)/17]/(59/34)

x = 17/59 ± 6√34/59

Since x is the hypotenuse SD, we have:

UD² = (25/34) x²

UD² = (25/34) [(17/59 ± 6√34/59)²]

UD² = 136/59 ± 204√34/295

Therefore, the exact lengths of the two parts of the hypotenuse are:

SD = x = 17/59 ± 6√34/59

SU = x-y = (x-1)/2 = 8/59 ± 3√34/59

UD = y = (x-1)/2 = 8/59 ± 3√34/59

TU = (3/5) UD = (3/5) [8/59 ± 3√34/59] = 24/295 ± 9√34/295

ST² = 45/34 x² + 45 = 45/34 [(17/59 ± 6√34/59)²] + 45

ST = √[45/34 [(17/59 ± 6√34/59)²] + 45]

Serena's overall grade in the course is 75.5%. It's important to note that in a weighted grading system like this, the final exam is often a major determinant of the final grade.

To calculate Serena's overall grade, we need to first determine the weight of the final exam. We know that the assignments are worth 45% and the quizzes are worth 40%, which leaves 100% - 45% - 40% = 15% for the final exam.

Next, we can calculate Serena's grade for each component of the course. Her grade for assignments is 78% and her grade for quizzes is 65%. We can calculate her grade for the final exam by multiplying her score of 96% by the weight of the final, which is 15%:

Final grade = (0.45 * 78%) + (0.4 * 65%) + (0.15 * 96%)

Final grade = 35.1% + 26% + 14.4%

Final grade = 75.5%

Therefore, Serena's overall grade in the course is 75.5%. It's important to note that in a weighted grading system like this, the final exam is often a major determinant of the final grade. In this case, Serena's strong performance on the final exam helped to boost her overall grade, even though her scores on the assignments and quizzes were not as high. It's also worth noting that this calculation assumes that all assignments, quizzes, and the final exam were weighted equally within their respective categories (i.e., each assignment was worth the same percentage of the assignment grade).

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Answer:29160

Step-by-step explanation:

The composite function (f + g)(3) when evaluated from f(x) = −3x² + 3x and g(x) = 2x+5 is -7

Given that

f(x) = −3x² + 3x and

g(x) = 2x+5

To perform the operation (ƒ + g)(3), we need to add the functions ƒ(x) and g(x) first, and then evaluate the sum at x = 3.

ƒ(x) = −3x² + 3x

g(x) = 2x + 5

To add the functions, we simply add their corresponding terms:

(ƒ + g)(x) = ƒ(x) + g(x) = (−3x² + 3x) + (2x + 5)

When the like terms are evaluated, we have

(ƒ + g)(3) = −3x² + 5x + 5

Now, we can evaluate the sum at x = 3:

(ƒ + g)(3) = −3(3)² + 5(3) + 5

So, we have

(ƒ + g)(3) = −27 + 15 + 5

Lastly, we have

(ƒ + g)(3) = -7

Therefore, (ƒ + g)(3) = -7.

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The remainder when a 90-digit number 9999 is divided by 89 is 0, as the result of applying the divisibility rule of 89, which involves reversing the digits of the number and subtracting the smaller from the larger.

To find the remainder when a 90-digit number 9999 is divided by 89, we can use the divisibility rule of 89. The rule states that for any integer n, the number obtained by reversing the digits of n and subtracting the smaller from the larger is divisible by 89.

In this case, we reverse the digits of 9999 to get 9999 again, and subtract the smaller from the larger to get 0. Since 0 is divisible by any number, including 89, the remainder when 9999 is divided by 89 is 0.

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The third quartile is $10.6575, which means that 75% of all days the stock is below this value.

The range of the stock price is from $8.82 to $16.17, and the distribution is uniform, which means that the probability of the stock price being any value between the minimum and maximum is the same.

To find the third quartile, we need to find the value x such that 75% of the observations are less than or equal to x, and 25% of the observations are greater than or equal to x.

Since the distribution is uniform, we can find x by finding the value that separates the bottom 25% of the distribution from the top 75% of the distribution.

The bottom 25% of the distribution spans from $8.82 to some value x. Since the distribution is uniform, we can find x by setting the probability of the stock price being less than or equal to x to 0.25, which gives:

(x - 8.82) / (16.17 - 8.82) = 0.25

Solving for x, we get:

x - 8.82 = 0.25 * (16.17 - 8.82)

x - 8.82 = 1.8375

x = 10.6575

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