The Montreal Biosphere is a geodesic dome that surrounds an environmental
museum in Montreal, Canada. The dome has a volume of 6,132,812.5 cubic feet.
The structure is 75% of a full sphere. What is the length of its diameter?

We can use the coordinates of point P (-5, 5) to find the values of the trigonometric functions for the angle formed between the positive x-axis and the line passing through the origin and point P.

First, we can find the distance from the origin to point P using the Pythagorean theorem:

r = sqrt((-5)^2 + 5^2) = sqrt(50)

Next, we can use the fact that the sine of an angle is equal to the y-coordinate of a point on the unit circle and the cosine of an angle is equal to the x-coordinate of a point on the unit circle. Since the radius of the unit circle is 1, we need to divide the x and y coordinates of point P by sqrt(50) to get the values for sine and cosine:

sin(θ) = y/r = 5/sqrt(50) = (5/10)sqrt(2) = (1/2)sqrt(2)

cos(θ) = x/r = -5/sqrt(50) = -(5/10)sqrt(2) = -(1/2)sqrt(2)

Next, we can use the fact that the tangent of an angle is equal to the sine of the angle divided by the cosine of the angle:

tan(θ) = sin(θ)/cos(θ) = (1/2)sqrt(2)/(-(1/2)sqrt(2)) = -1

Similarly, we can use the reciprocal identities to find the values of the other three trigonometric functions:

csc(θ) = 1/sin(θ) = sqrt(2)

sec(θ) = 1/cos(θ) = -sqrt(2)

cot(θ) = 1/tan(θ) = -1

Therefore, the exact values of the six trigonometric functions for the angle formed by the line passing through the origin and point P (-5, 5) are:

sin(θ) = (1/2)sqrt(2)

cos(θ) = -(1/2)sqrt(2)

tan(θ) = -1

csc(θ) = sqrt(2)

sec(θ) = -sqrt(2)

cot(θ) = -1

IG:whis.sama_ent

The standard deviation of the number of green M&M's in a sample of 5 is 0.91

Standard deviation is a measure of the amount of variation or dispersion of a set of data values. It is calculated by finding the square root of the variance, which is the average of the squared differences from the mean.

To solve this problem, we can use the binomial probability formula:

[tex]P(X=k) = C(n,k) * p^k * (1-p)^{(n-k)[/tex]

where:

X is the number of green M&M's in the sample

k is the number of green M&M's we are interested in (3 or 4 or at most 3 or at least 3)

n is the sample size (5)

p is the probability of getting a green M&M in one trial

We can use the given percentages to calculate the probability of getting a green M&M:

p = 0.15

Now we can calculate the probabilities for the different scenarios:

Probability that exactly three of the five M&M's are green:

[tex]P(X=3) = C(5,3) * 0.15^3 * 0.85^2 = 0.0883[/tex]

Probability that three or four of the five M&M's are green:

[tex]P(X=3\ or\ X=4) = P(X=3) + P(X=4) = C(5,3) * 0.15^3 * 0.85^2 + C(5,4) * 0.15^4 * 0.85^1 = 0.1527[/tex]

Probability that at most three of the five M&M's are green:

P(X≤3) = [tex]P(X=0) + P(X=1) + P(X=2) + P(X=3) = C(5,0) * 0.15^0 * 0.85^5 + C(5,1) * 0.15^1 * 0.85^4 + C(5,2) * 0.15^2 * 0.85^3 + C(5,3) * 0.15^3 * 0.85^2 = 0.9053[/tex]

Probability that at least three of the five M&M's are green:

P(X≥3) [tex]= P(X=3) + P(X=4) + P(X=5) = C(5,3) * 0.15^3 * 0.85^2 + C(5,4) * 0.15^4 * 0.85^1 + C(5,5) * 0.15^5 * 0.85^0 = 0.2413[/tex]

To find the expected number of green M&M's in a sample of size 5, we can use the formula:

E(X) = n * p

where E(X) is the expected value of X.

E(X) = 5 * 0.15 = 0.75

Therefore, on average we expect 0.75 green M&M's in a sample of 5.

To find the standard deviation, we can use the formula:

SD(X) = sqrt(n * p * (1-p))

SD(X) = sqrt(5 * 0.15 * 0.85) = 0.9138 (rounded to two decimal places)

Therefore, the standard deviation of the number of green M&M's in a sample of 5 is 0.91

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Answer: The probability of drawing a blue marble and then a yellow marble is 1/12 or 0.0833.

Step-by-step explanation:

There are 16 marbles in the bag. To calculate the probability of drawing a blue marble and then a yellow marble, we can use the formula:

P(blue and yellow) = P(blue) x P(yellow|blue)

The probability of drawing a blue marble on the first draw is 5/16. After one blue marble has been drawn, there are 15 marbles left in the bag, so the probability of drawing a yellow marble on the second draw, given that the first marble was blue, is 4/15.

P(blue and yellow) = (5/16) x (4/15) = 1/12 or 0.0833

The number that does not belong to the set of numbers is given as follows:

B. Three

Numbers are classified as even or odd depending on whether they are divisible by 2 or not, as follows:

For this problem, we have that the number five does not belong to the set of even numbers, as the numbers two, four and six do.

The options are:

A.Two

B.Three

C.Four

D.Six​

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The value of the inequality is g≤2.5

In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size.

The given inequality is 4g≤10

To solve the inequality, divided bo sides of the inequality by the coefficient of g which is 4

This gives the value

4g/g≤10/4

⇒ that g = 2.5

Therefore the value of the inequality is g≤2.5

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The table is completed as below

                             Interested in            NOT Interested in      Total

                             Athletics                   Athletics

Interested in

Academic Clubs        26                              58                          84

Not Interested in

Academic Clubs        118                              38                          156

Total                         144                               96                         240

To fill in the table, we start with the total number of students, which is 240.

We know that 35% of students are interested in athletics, so we can find the number of students interested in athletics as:

0.35 x 240 = 84

Similarly, we know that 3/5 of students are interested in academic clubs, so we can find the number of students interested in academic clubs as:

(3/5) x 240 = 144

We also know that 26 students are interested in both athletics and academic clubs, so we can fill in that cell as 26.

To fill in the remaining cells, we can use the fact that the row and column totals must add up correctly. For example, in the "Interested in athletics" row, we know that there are a total of 84 students interested in athletics. We also know that 26 of these students are interested in both athletics and academic clubs. Therefore, the number of students interested in athletics but not academic clubs is:

84 - 26 = 58

We can use similar reasoning to fill in the remaining cells. For example, in the "Not interested in athletics" column, we know that there are a total of 214 students who are not interested in athletics. We also know that 144 students are interested in academic clubs. Therefore, the number of students who are not interested in athletics but are interested in academic clubs is:

144 - 26 = 118

Let x represent "not interested in academics club" and "not interested in athletic". Using the total in the horizontal (bottom row) we have:

144 + 58 + x = 240

x = 240 - 202

x = 38

Hence 58 + 38 = 96 and 118 + 38 = 156

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Answer:

9/5x+7

Step-by-step explanation:

You can keep the slope the same, because it is parallel, but the y-intercept must change to plus seven in order to go through (-5, -2)

Answer:

171.76 Pounds

Step-By-Step Explanation:

So the explanation to how I got the answer is in the link below.

Answer:

m∠A = 50°

Step-by-step explanation:

The ∑ of all interior angles of any given triangle is 180°. Set all angle measurements equal to 180:

[tex](x + 30) + (2x - 10) + 70 = 180[/tex]

First, simplify. Combine like terms. Like terms are terms that share the same amount of the same variables:

[tex](x + 2x) + (30 - 10 + 70) = 180\\(3x) + (90) = 180[/tex]

Next, isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.

PEMDAS is the order of operations, and stands for:

Parenthesis

Exponents (& Roots)

Multiplications

Divisions

Additions

Subtractions

~

First, subtract 90 from both sides of the equation:

[tex]3x + 90 = 180\\3x + 90 (-90) = 180 (-90)\\3x = 180 - 90\\3x = 90[/tex]

Next, divide 3 from both sides of the equation:

[tex]3x = 90\\\frac{(3x)}{3} = \frac{(90)}{3}\\ x = \frac{90}{3}\\ x = 30[/tex]

Plug in 30 for x in the given angle measurement of A and simplify:

[tex]m\angle{A} = 2x - 10\\m\angle{A} = 2(30) - 10\\m\angle{A} = (60) - 10\\m\angle{A} = 50[/tex]

50° is your answer for m∠A.

~

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Answer:

m∠A = 50°

Step-by-step explanation:

The ∑ of all interior angles of any given triangle is 180°. Set all angle measurements equal to 180:

[tex](x + 30) + (2x - 10) + 70 = 180[/tex]

First, simplify. Combine like terms. Like terms are terms that share the same amount of the same variables:

[tex](x + 2x) + (30 - 10 + 70) = 180\\(3x) + (90) = 180[/tex]

Next, isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.

PEMDAS is the order of operations, and stands for:

Parenthesis

Exponents (& Roots)

Multiplications

Divisions

Additions

Subtractions

~

First, subtract 90 from both sides of the equation:

[tex]3x + 90 = 180\\3x + 90 (-90) = 180 (-90)\\3x = 180 - 90\\3x = 90[/tex]

Next, divide 3 from both sides of the equation:

[tex]3x = 90\\\frac{(3x)}{3} = \frac{(90)}{3}\\ x = \frac{90}{3}\\ x = 30[/tex]

Plug in 30 for x in the given angle measurement of A and simplify:

[tex]m\angle{A} = 2x - 10\\m\angle{A} = 2(30) - 10\\m\angle{A} = (60) - 10\\m\angle{A} = 50[/tex]

50° is your answer for m∠A.

~

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A 95% confidence interval for the standard normal distribution is typically given by  (-1.96, 1.96) . mean that correspond to the upper and lower bounds of the interval

what is mean ?

In statistics, the standard deviation is a measure of the amount of variation or dispersion in a set of data. It is the square root of the variance and is denoted by the symbol σ (sigma). The standard deviation is expressed in the same units

In the Given question :

A 95% confidence interval for the standard normal distribution is typically given by  (-1.96, 1.96)

This means that we are 95% confident that the true value of a normally distributed variable falls within this range. The values of -1.96 and 1.96 represent the number of standard deviations from the mean that correspond to the upper and lower bounds of the interval, respectively. This interval is often used in statistical inference to estimate the population mean or proportion based on a sample.

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what 95% confidence interval for a standard normal distribution then what is the number of standard deviations from the mean that correspond to the upper and lower bounds of the interval ?

Answer:

D

Step-by-step explanation:

The domain of a function refers to the set of all possible input values (independent variable) for which the function is defined. In this case, the graph has points (6,80) and (11,40), which means that the function is defined for the values of g (the independent variable) between 6 and 11. Therefore, the domain of the function is:

D. 6≤g≤11

The domain of the function in the given graph is 6≤h≤11, as it includes all possible values of h between and including 6 and 11 on the horizontal (h) axis. So the correct option is D.

The domain of a function refers to the set of all possible input values for the function. In the given graph, which is plotted on the h-g axis and includes points (6, 80) and (11, 40), we are interested in determining the valid range for the independent variable, h.

The lowest h-value on the graph is 6, corresponding to the point (6, 80), and the highest h-value is 11, corresponding to the point (11, 40). These are the boundaries that define the domain of the function. Any value of h that falls within this range is a valid input for the function, and any value outside this range is not represented on the graph.

Therefore, the domain of the function is 6≤h≤11, as it includes all values of h between and including 6 and 11. This range encompasses all possible inputs for this function as depicted in the graph.

In summary, the domain represents the valid input values, and in this case, it is limited to the interval from 6 to 11 on the h-axis.

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if 1 2 3 = 2

then 3 5 4 = 5

hope it helps:)

the partial fraction decomposition of x^3-x/x^4+2x^2+1 is (x^3 - x)/(x^4 + 2x^2 + 1) = -1/4/(x^2 + 1)

To find the partial fraction decomposition of x^3-x/x^4+2x^2+1, we first factor the denominator:

x^4 + 2x^2 + 1 = (x^2 + 1)^2

So we can write:

(x^3 - x)/(x^4 + 2x^2 + 1) = A/(x^2 + 1) + B/(x^2 + 1)^2

Multiplying both sides by the denominator, we get:

x^3 - x = A(x^2 + 1) + B

Now we can solve for A and B by choosing appropriate values for x. Let's choose x = 0 first:

0 - 0 = A(0^2 + 1) + B

B = 0

Now let's choose x = 1:

1 - 1 = A(1^2 + 1) + 0

A = -1/4

So the partial fraction decomposition of x^3-x/x^4+2x^2+1 is:

(x^3 - x)/(x^4 + 2x^2 + 1) = -1/4/(x^2 + 1) + 0/(x^2 + 1)^2

or

(x^3 - x)/(x^4 + 2x^2 + 1) = -1/4/(x^2 + 1)

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Answer:

C is correct

Step-by-step explanation:

Okay, let's break this down step-by-step:

From the table, we know the x-values are 1, 3, 5, 7, and 9.

And the y-values are 4, 6, 8, ?, 12.

So at x=7, there is a missing y-value that we need to determine.

To find this, we need to look at the linear relationship between the points. Some clues:

1) There are 5 points total.

2) The x-values increase by 2 each time.

3) The y-values also seem to be increasing by 2 each time.

So the linear relationship is: y = mx + b (where m is the slope and b is the y-intercept)

In this case, the slope is 2 (because y increases by 2 for every increase of 1 in x).

And the y-intercept is 4 (because when x = 1, y = 4).

So the equation is: y = 2x + 4

Plugging in x = 7:

y = 2(7) + 4

y = 14 + 4

y = 18

Therefore, when x = 7, the second coordinate in the scatterplot is 18.

Rounded to the tenths place: 1.8

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

The angles and percentages have been calculated as shown below.

A pie chart for the information about the different make of cars is shown below.

For the total number of car makes, we have the following:

Total make of cars = 12 + 8 + 8 + 2 + 10

Total make of cars = 40.

Next, we would determine the angles and percentage as follows;

Toyota

12/40 × 360 = 108°

12/40 × 100 = 30%.

Nissan

8/40 × 360 = 72°

8/40 × 100 = 20%.

Datsun

8/40 × 360 = 72°

8/40 × 100 = 20%.

Volvo

2/40 × 360 = 18°

2/40 × 100 = 5%.

Peugeot

10/40 × 360 = 90°

10/40 × 100 = 25%.

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To calculate the expected value of playing this game, we need to multiply the probability of winning each amount by the corresponding payout and then sum them up.

Let's start by calculating the probability of selecting each type of marble:

   Probability of selecting a gold marble: 3/34

   Probability of selecting a silver marble: 8/34

   Probability of selecting a black marble: 23/34

Now, let's calculate the expected value of playing the game:

E(x) = (3/34) * $3 + (8/34) * $2 + (23/34) * (-$1)

E(x) = $0.26

So the expected value of playing this game is $0.26. This means that over many plays of the game, we would expect to win an average of $0.26 per play. However, it's important to remember that this is just an average, and in any individual play of the game, you could win more or less than this amount.

The function graphed on the axis above should be expressed as a piecewise function as follows;

f(x) = x + 1       {x < -5}

     = 5x - 10   {x > 1}

In order to determine the piecewise function, we would determine an equation that represent each of line shown on the graph. Therefore, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (-7 + 4)/(-8 + 5)

Slope (m) = -3/-3

Slope (m) = 1.

At data point (-5, -4) and a slope of 1, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y + 4 = 1(x + 5)  

y = x + 1

1, -5 2 0

For the second line, we have:

Slope (m) = (0 + 5)/(2 - 1)

Slope (m) = 5/1

Slope (m) = 5.

At data point (2, 0) and a slope of 5, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 0 = 5(x - 2)  

y = 5x - 10

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Answer:

13.56 mm²

Step-by-step explanation:

surface area = 2 circle area + the side

circle = π r² = 3.14

2 of them = 6.28

side = circumference * 1 = 2πr *1 = 6.28

total 6.28*2 = 13.56 mm²

Answer:

Your calculator should be in radian mode.

a)

[tex]f(x) = 56 \sin( \frac{2t\pi}{12} + \frac{\pi}{2} ) + 4[/tex]

b)

[tex]f(x) = 56 \cos( \frac{2t\pi}{12} - \frac{\pi}{2} ) + 4[/tex]

c)

The height of the wheel's central axle is 32 meters.

f(x) = 56sin((2(x + 4)π/12) - (π/2)) + 4

Juan traveled approximately 288.24 miles.

The direct road would be approximately 288 miles long.

Juan's trip would be approximately 0.24 miles shorter if he was able to take the direct route.

We have,

a.

Here is a diagram illustrating Juan's trip:

    Grand Canyon

         |

         |

         |

Juan's    |       x

house --------->

         y

b.

To find the total distance Juan traveled, we can use the Pythagorean theorem:

distance² = x² + y²

Juan drove 280 miles south (y direction) and 64 miles east (x direction), so we have:

distance² = 280² + 64²

distance² = 78,976 + 4,096

distance² = 83,072

Taking the square root of both sides, we get:

distance ≈ 288.24 miles

c.

To find the length of the direct road, we can use the Pythagorean theorem again:

distance² = x² + y²

The direct road forms a right triangle with legs of 280 miles and 64 miles, so we have:

distance² = 280² + 64²

distance² = 78,976 + 4,096

distance² = 83,072

Taking the square root of both sides, we get:

distance ≈ 288.24 miles

d.

To find how much shorter Juan's trip would be if he took the direct route, we can subtract the distance he traveled from the direct road distance:

288 - 288.24 ≈ -0.24

Thus,

Juan traveled approximately 288.24 miles.

The direct road would be approximately 288 miles long.

Juan's trip would be approximately 0.24 miles shorter if he was able to take the direct route.

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The empirical rule to determine the interval middle 99.7% of Caroline's finishing times in the 400 meter race is 63.5 seconds to 72.5 seconds

Given data ,

The empirical rule, also known as the 68-95-99.7 rule, states that for a normal distribution:

Approximately 68% of the data falls within one standard deviation of the mean.

Approximately 95% of the data falls within two standard deviations of the mean.

Approximately 99.7% of the data falls within three standard deviations of the mean.

Since the middle 99.7% of data falls within three standard deviations of the mean, we can calculate the upper and lower limits of this interval as follows:

Lower limit = Mean - (3 x Standard Deviation)

Upper limit = Mean + (3 x Standard Deviation)

Plugging in the values for mean and standard deviation:

Lower limit = 68 - (3 x 1.5) = 68 - 4.5 = 63.5 seconds

Upper limit = 68 + (3 x 1.5) = 68 + 4.5 = 72.5 seconds

Hence , the interval of times that represents the middle 99.7% of Caroline's finishing times in the 400 meter race is 63.5 seconds to 72.5 seconds

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The total number of different ways that the student can be assigned during a 4-day work week is 81 different ways.

A student is a person who is enrolled in an educational institution, such as a college, university, or trade school. Students are typically required to attend classes and complete assignments to gain knowledge and develop skills related to their field of study. Depending on their level of education, students may also be required to participate in research or internships in order to gain practical experience.

There are 3 possible floors that the student can be assigned to, so for each day of the 4-day week, there are 3 possible assignments.
Therefore, the total number of different ways that the student can be assigned during a 4-day work week is 3 x 3 x 3 x 3
= 81 different ways.

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the three friends spent L.120.00, L.240.00, and L.360.00, respectively.

How to solve the problem?

Let's denote the amount that Alicia spends as "x". According to the problem statement, we know that Juan spends twice as much as Alicia, which means he spends 2x. Similarly, we know that Ana spends three times as much as Alicia, which means she spends 3x.

We also know that the three friends together have spent L.720.00. So, we can set up an equation to represent this:

x + 2x + 3x = 720

Simplifying this equation, we get:

6x = 720

Dividing both sides by 6, we get:

x = 120

So, Alicia spent L.120.00, Juan spent twice as much as Alicia, which is L.240.00, and Ana spent three times as much as Alicia, which is L.360.00.

Therefore, the three friends spent L.120.00, L.240.00, and L.360.00, respectively.

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The p-value associated with the test statistic of z = 2.566 is 0.0045, accurate to four decimal places.

Based on the given information, we can conduct a one-tailed right-sided test using a normal distribution calculator to find the p-value associated with the test statistic of z = 2.566.

In hypothesis testing, a one-tailed right-sided test is a statistical test where the alternative hypothesis (H1) is specified to detect a change typically towards larger values.

Using a normal distribution calculator, we can find the cumulative distribution function (CDF) of the standard normal distribution at z = 2.566.

The CDF gives us the probability that a randomly selected value from a standard normal distribution is less than or equal to a given value.

Using a normal distribution calculator:

CDF at z = 2.566 = 0.9955 (rounded to four decimal places)

Since we are conducting a one-tailed right-sided test, we are interested in the probability of getting a value greater than the test statistic. Therefore, the p-value for the given test statistic is:

p-value = 1 - CDF at z = 2.566 = 1 - 0.9955 = 0.0045 (rounded to four decimal places)

Thus, the p-value associated with the test statistic of z = 2.566 is 0.0045, accurate to four decimal places.

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The person is looking at each peak on a leg of a right triangle. The hypotenuse is the distance between the mountains.

The distance is determined by

x^2+(x+5)^2=25^2

x^2+x^2+10x+25=625

2x^2+10x-600=0

x^2+5x-300=0

(x+20)(x-15)=0

only positive root is x=15

x+5=20

The vista point is 15 miles from one mountain peak and 20 miles from the other.

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Plotting the graph of the function h(x) = [tex](x^2 - 3x - 4) / (x - 4)[/tex], we get y- intercept = 1, vertical asymptotes at x=4 and  horizontal asymptotes at y=1.

The connection between a function's input values (usually written as x) and output values (often indicated as y) is represented by the function's graph. It gives insights on the function's characteristics, including its domain, range, intercepts, asymptotes, and general form, as well as a visual representation of how the function acts.

Steps to plot the graph of the give function h(x) =[tex](x^2 - 3x - 4) / (x - 4)[/tex]:

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The range of the function is y = {11, 13, 17, 25}.

In mathematics, the range of a function refers to the set of all possible output values that the function can produce when given a set of input values. It is also known as the image of the function. The range of a function can be found by evaluating the function for all possible input values, or by analyzing the properties of the function.

To find the range of the function, we need to substitute each value of the domain into the function and record the corresponding output values.

For x = 2, f(2) = 2(2) + 7 = 11.

For x = 3, f(3) = 2(3) + 7 = 13.

For x = 5, f(5) = 2(5) + 7 = 17.

For x = 9, f(9) = 2(9) + 7 = 25.

Therefore, the range of the function is y = {11, 13, 17, 25}.

The answer is A. y = {9, 10, 12, 16} is not the correct range for this function with the given domain.

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