Which are correct representations of the inequality –3(2x – 5) < 5(2 – x)? Select two options. x < 5 –6x – 5 < 10 – x –6x + 15 < 10 – 5x A number line from negative 3 to 3 in increments of 1. An open circle is at 5 and a bold line starts at 5 and is pointing to the right. A number line from negative 3 to 3 in increments of 1. An open circle is at negative 5 and a bold line starts at negative 5 and is pointing to the left.

A parabolic function is represented by h = -16T² + 32T + 151.

A function with the equation f(x) = ax²+ bx + c is said to be parabolic. It is a second-degree quadratic expression in x². A parabola-like form can be seen in the graph of a parabolic function. The range value for the parabolic function is the same for two different domain values. Parabolic motion refers to a variety of motions, such as the motion of a stone thrown under the influence of gravity.

. The function's greatest or minimum value is where the parabola's vertex is located. In this instance, the formula -b/2a,

where a = -16 and b = 321, can be used to get the vertex of the parabola. As a result, this parabola's vertex is located at t = 1 second and h = 31 feet.

The parabola's vertex is located at t = 1 second and h = 31 feet. These are the main characteristics of this graph.

The parabola's axis of symmetry is a vertical line that passes through its vertex.

Because the coefficient of t² is negative, the parabola begins to slope downward.

- The maximum height that Miyako reaches is 31 feet.

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Answer:

x = -5

Y = -22/3

Step-by-step explanation:

Y= 2/3x - 4                     Y = -x + 1

We put -x + 1 in for y to solve for x

-x + 1 = 2/3x - 4

-5/3x + 1 = -4

-5/3x = -5

x = -5

Now put -5 in for x and solve for y

Y= 2/3(-5) - 4

Y = -10/3 - 4

Y = -22/3

So, there are only one solution x = -5 and y = -22/3

To create identical donation bags, you will need 55 hats and 40 scarves. You produce the most donation bags without any extra apparel. Each donation package contains 11 hats and 8 scarves.

To make the greatest amount of donation bags with no clothing left over, we need to find the greatest common factor (GCF) of 40 and 55. We can do this by finding the prime factors of both numbers:

[tex]40 = 2 * 2 * 2 * 5[/tex]

55 = 5 x 11

The GCF is the product of the common prime factors, which is 5. This means we can make 5 identical donation bags.

To find out how many scarves and hats are in each donation bag, we can divide the total number of scarves and hats by the number of donation bags:

Number of scarves per bag = 40 / 5 = 8 scarves

Number of hats per bag = 55 / 5 = 11 hats

Therefore, each donation bag will contain 8 scarves and 11 hats.

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The required polynomial of exact degree 5 which interpolates the four given points is equals to  p(x) = x^3/2 -4x^2 + 63x/6 -6.

Polynomial of degree exactly 5 that interpolates the four points (1, 1), (2, 3), (3, 3), (4, 4).

Apply Lagrange interpolation.

Standard form of the Lagrange interpolation polynomial of degree n passing through n+1 distinct points (x0, y0), (x1, y1), ..., (xn, yn) is,

L(x) = Σ [yi × Π (x - xj) / (xi - xj)], for i = 0, 1, ..., n

where Π is the product operator and j takes all values different from i.

For the given four points, we have n = 3, so the polynomial has degree n+1 = 4.

Using the formula, we have,

L(x) = (1× (x - 2)(x - 3)(x - 4) / ((1 - 2)(1 - 3)(1 - 4)))

+ (3× (x - 1)(x - 3)(x - 4) / ((2 - 1)(2 - 3)(2 - 4)))

+ (3× (x - 1)(x - 2)(x - 4) / ((3 - 1)(3 - 2)(3 - 4)))

+ (4× (x - 1)(x - 2)(x - 3) / ((4 - 1)(4 - 2)(4 - 3)))

Simplifying this expression, we get,

⇒L(x) = (-x^3 + 9x^2 - 26x + 24)/6 + (3x^3/2 - 12x^2 + 57x/2 - 18) +(-3x^3/2 + 21x^2 /2- 21x + 12) + (2x^3/3 - 4x^2 + 22x/3 - 4)

⇒L(x) = x^3/2 -4x^2 + 63x/6 -6

Therefore, the polynomial of degree exactly 5 that interpolates the four points (1, 1), (2, 3), (3, 3), (4, 4) is  p(x) = x^3/2 -4x^2 + 63x/6 -6.

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Answer: 13.51 (not in inches format)

Step-by-step explanation:

tan(x) = 4.384/12

x = tan^-1 (4.384/12)

x = 20.068°

tan(20.068) = y/(12+25)

tan(20.068) = y/37

20.068 = tan^-1 (y/37)

y = 13.51

Answer:

Step-by-step explanation:

To solve this with similar triangles - you need to set up a proportion.

I changed it all to inches.

12 ft = 144 in

4.384 ft = 51. 84 inches

12 + 25 (you need to add the base of the triangle) = 37 ft = 444 inches

set up smaller triangle in proportion to bigger triangle

51.84"/144" = x/444

cross mulitply then divide

51.84 times 444 = 23016.96

23016.96 ÷ 144 = 159.84 inches

Change back to feet by dividing by 12

159.84 ÷ 12 = 13.32 ft

The coordinate of the segment in the ratio is (-2,-2).

What is ratio?

When two numbers are compared, the ratio between them shows how often the first number contains the second. As an illustration, the ratio of oranges to lemons in a dish of fruit is 8:6 if there are 8 oranges and 6 lemons present.

Partitions the line segment in the ratio A to B... then,

=> x-coordinate is : x1 +[tex]\frac{ A(x2-x1)}{(A+B)}[/tex]

=> y-coordinate is: y1 + [tex]\frac{A(y2-y1)}{(A+B)}[/tex]

Here A = 2 , B = 1 and [tex](x_1,y_1)=(-10,10)[/tex] , [tex](x_2,y_2)=(2,-8)[/tex] .

x-coordinate = -10+[tex]\frac{2(2+10)}{(2+1)}[/tex] =  -10 + [tex]\frac{2(12)}{3}[/tex] = -10 + 2(4) = -10+8 = -2

y-coordinate = 10+[tex]\frac{2(-8-10)}{(2+1)}=10+\frac{2(-18)}{3}[/tex] = 10+2(-6) = 10-12 = -2.

Hence the coordinate of the segment in the ratio is (-2,-2).

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According to the question the word "Liberty" on the quarter is about 0.872 inches long.

In geometry, a circle's diameter is any straight segment whose endpoint is on the circle and which passes through its centre. Another name for it is the circle's longest chord. Either idea can be used to determine a sphere's diameter. The diameter is indeed the length of both the line perpendicular to the two points at each end of the circle. If you think of length as the distance between two points, then diameter is length. The diameter of a circle is the separation between its two farthest points.

Since the quarter has a diameter of about 1 inch, we can use its diameter as a reference to estimate the length of the word "Liberty" on the quarter.

If the word "Liberty" makes about a 100° angle, that means it takes up about 100/360 or 5/18 of the circumference of the quarter. The circumference of a circle is given by:

C = 2πr

where C is indeed the circumference, pi (roughly 3.14), is a mathematical constant, and r is the circle's radius. The quarter's approximate 1 inch diameter means that the radius equals 1/2 inch, approximately 0.5 inches.

The circumference of the quarter is therefore:

C = 2π(0.5 inches) = π inches

The length of the word "Liberty" on the quarter is approximately 5/18 of this circumference, which is:

(5/18) × π inches ≈ 0.872 inches

So the word "Liberty" on the quarter is about 0.872 inches long.

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Percentage of students who were undecided is  59% of the students were undecided about their future careers.

what is Percentage?

A percentage is a fraction or ratio expressed as a fraction of 100. It is a way of expressing a part of a whole as a proportion of the whole. It is denoted by the symbol '%'. For example, if there are 100 students in a class and 20 of them are girls, then the percentage of girls in the class is 20%.

In the given question,

To solve the problem, we need to first find out the percentage of students who were undecided. We know that 25% of the students wanted to become doctors and 16% wanted to become teachers. The percentage of students who were undecided can be found by subtracting the sum of the percentages of those who wanted to become doctors and teachers from 100%:

Percentage of students who were undecided = 100% - (25% + 16%) = 59%

Therefore, 59% of the students were undecided about their future careers.

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The answer using probability are shown:

(A) 43/60 projection is used.

(B) He shouldn't alter the estimate in light of this knowledge.

Simply put, the probability is the likelihood that something will occur.

When we don't know how an occurrence will turn out, we can discuss the likelihood or likelihood of various outcomes.

Statistics is the study of occurrences that follow a probability distribution.

Mathematics' study of random events is known as probability, and there are four major kinds of probability: axiomatic, classical, empirical, and subjective.

So, he would calculate the amount by dividing the overall number of bricks by the proportion of green bricks. The likelihood of choosing a green block out of all 60 bricks would be that.

Then, the probability would be:

= 43/60

Andre shouldn't alter his assessment, in other words.

The explanation is that we do not know whether the bricks he saw were green in color; we are only told that she looked in the bag and found 6 bricks.

The only time Andre needs to adjust his guess is if it turns out that the six bricks Mai saw in the bag were, in fact, green.

The estimate should not be altered until this proof is received.

Therefore, the answer using probability are shown:

(A) 43/60 projection is used.

(B) He shouldn't alter the estimate in light of this knowledge.

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Complete question:
Andre picks a block out of a bag 60 times and notes that 43 of them were green. What should Andre estimate for the probability of picking out a green block from this bag? Mai looks in the bag and sees that there are 6 blocks in the bag. Should Andre change his estimate based on this information? If so, what should the new estimate be? If not, explain your reasoning.

The surface areas are 756.87 km², 2567.6 in², 853.91 yd², 749.72 cm² and the area to paint is 3268.60 in²

Figure 9

This is calculated as

Area = Base area + 1/2 * Perimeter * Slant height

So, we have

Surface area = (1/4 * 14² * √3) + 1/2 * (14 + 14 + 14) * 32

Surface area = 756.87 km²

Figure 10

This is calculated as

Area = Sum of side areas

So, we have

Surface area = 2 * 1/2 * (35 + 10) * 20.3 + 32 * 17 + 10 * 17 + 20.3 * 17 + 35 * 17

Surface area = 2567.6 in²

Figure 11

Start by calculating the radius using

(2r)² = 19.4² - 14.4²

2r = 13

So, we have

r = 6.5

This area is then calculated as

Area = 2πr(h + r)

So, we have

Area = 2 * 22/7 * 6.5(14.4 + 6.5)

Surface area = 853.91 yd²

Figure 12

This area is calculated as

Area = lw + l√[(w/2)² + h²] + w√[(l/2)² + h²]

So, we have

Area = 15 * 18 + 18√[(15/2)² + 12²] + 15√[(18/2)² + 12²]

Surface area = 749.72 cm²

Without painting the bottom, the area is

Area = 2 * 1/2 * 34 + 42.8 + 36 * √(34² + 42.8²) + 34 * 36

Evaluate

Area = 3268.60 in²

Unclear dimensions

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The probability that a randomly selected newborn baby boy's weight is between 3682 and 4552 grams is approximately 0.1979, or 19.79% when rounded to four decimal places.

To find the probability that a newborn baby boy's weight  Convert the weights to z-scores:

z = (X - μ) / σ, where X is the weight, μ is the mean, and σ is the standard deviation.

First, calculate the standard deviation:

σ = √variance = √385,641 ≈ 621

Now, calculate the z-scores for the given weights:

z1 = (3682 - 3186) / 621 ≈ 0.7986

z2 = (4552 - 3186) / 621 ≈ 2.1965

Use a standard normal table or a calculator with a standard normal probability function to find the probabilities corresponding to these z-scores:

P(Z ≤ 0.7986) ≈ 0.7879

P(Z ≤ 2.1965) ≈ 0.9858

Find the probability that the weight is between the two z-scores:

P(0.7986 ≤ Z ≤ 2.1965) = P(Z ≤ 2.1965) - P(Z ≤ 0.7986) ≈ 0.9858 - 0.7879 ≈ 0.1979

So, the probability that a randomly selected newborn baby boy's weight is between 3682 and 4552 grams is approximately 0.1979, or 19.79% when rounded to four decimal places.

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The transformation of the graph of f(x) to g(x) is defined as: 2 units shift on the left side and 5 units shift upward.

Think about the function f (x). We measure the movement using the x- and y-axes on a coordinate grid. The following are guidelines for function transformations that could be used with function graphs. All algebraic and graph of transformations are possible.

Transformations can be divided into four categories: translation, rotation, reflection, plus dilation.

Given :

Transformation of the graph of f(x) --->  g(x)

f(x) = x² g(x) = (x - 2)² + 5

There is a shift of 2 units on the left. Subtract 2 from x to get the value (x - 2)

There is a shift of 5 units upward. Add 5 on the function f(x)  to get the function g(x).

Thus, the transformation of the graph of f(x) to g(x) is defined as: 2 units shift on the left side and 5 units shift upward.

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For the given values, the standard deviation for this data set is approximately 3.67 minutes.

What is standard deviation?

In statistics, standard deviation is a measure used to quantify the amount of spread or dispersion of a set of numerical data around the mean. This measure indicates how much the data values vary from the average value of the dataset.

To find the standard deviation, you can follow these steps:

Find the mean (average) of the data set:

mean = (11 + 7 + 14 + 2 + 8 + 13 + 3 + 6 + 10 + 3 + 8 + 4 + 8 + 4 + 7) / 15 = 7.2

Subtract the mean from each data point:

(11 - 7.2), (7 - 7.2), (14 - 7.2), (2 - 7.2), (8 - 7.2), (13 - 7.2), (3 - 7.2), (6 - 7.2), (10 - 7.2), (3 - 7.2), (8 - 7.2), (4 - 7.2), (8 - 7.2), (4 - 7.2), (7 - 7.2)

Simplifying, we get:

3.8, -0.2, 6.8, -5.2, 0.8, 5.8, -4.2, -1.2, 2.8, -4.2, 0.8, -3.2, 0.8, -3.2, -0.2

Square each result:

13.44, 0.04, 46.24, 27.04, 0.64, 33.64, 17.64, 1.44, 7.84, 17.64, 0.64, 10.24, 0.64, 10.24, 0.04

Find the mean (average) of the squared results:

(13.44, 0.04, 46.24, 27.04, 0.64, 33.64, 17.64, 1.44, 7.84, 17.64, 0.64, 10.24, 0.64, 10.24, 0.04) / 15 = 12.49

Take the square root of the mean:

[tex]\sqrt{(12.49)[/tex] ≈ 3.67

Therefore, the standard deviation for this data set is approximately 3.67 minutes.

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Answer: We can find the total distance of the first route by adding the individual distances:

53 + 28 + 98 = 179 miles

Similarly, we can find the total distance of the second route:

28 + 49 + 85 = 162 miles

Therefore, the second route is shorter by (179 - 162) = 17 miles.

Step-by-step explanation:

Answer:

126 meters squared

Step-by-step explanation:

To find the surface area

For the top and bottom it will be: l x w x 2(because there's a top and bottom)

3x6x2 = 36

For the sides (left and right) it will be: l x h x 2 (There's left and right)

3x5x2 = 30

For the front and back it will be w x h  x 2 (There's the front and the back)

6x5x2 = 60

Added it all together and you get 126

The surface area is 126 meters squared



Answer:

The given equation is:

15x + 30 = 105

Here,

15x represents the total cost of the fruit baskets purchased, and

30 represents the cost of the membership fee.

To find the number of fruit baskets purchased (x), we must isolate the variable on one side of the equation.

Subtracting 30 from both sides, we get:

15x = 105 - 30

15x = 75

Dividing both sides by 15, we get:

x = 5

Therefore, Mrs. Williams purchased 5 fruit baskets for her coworkers.

The candy distributor needs to use 30 kilograms of 10% fat-content chocolate and 70 kilograms of 50% fat-content chocolate to create 100 kilograms of a 14% fat-content chocolate.

To solve this problem, we can use the method of mixture problems, which involves setting up a system of equations. Let x be the number of kilograms of the 10% fat-content chocolate and y be the number of kilograms of the 50% fat-content chocolate.

We have two equations based on the fat content and the total weight of the mixture:

0.1x + 0.5y = 0.14(100) (equation for fat content)

x + y = 100 (equation for total weight)

We can solve this system of equations using substitution or elimination. Using substitution, we can solve for x in terms of y from the second equation and substitute it into the first equation:

x = 100 - y

0.1(100 - y) + 0.5y = 0.14(100)

10 - 0.1y + 0.5y = 14

0.4y = 4

y = 10

Then we can substitute y = 10 back into the equation for x and get:

x = 100 - y = 100 - 10 = 90

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Answer:

Yes they can.

The only time they would have to be different lengths is if it was a square

Answer:

x = 8

Step-by-step explanation:

The two sides equal each other

5x - 13 = x + 19  Subtract x from both sides

5x - x - 13 = x - x + 19

4x - 13 = 19  Add 13 to both sides

4x - 13 + 13 = 19 + 13

4x = 32  Divide both sides by 4

x = 8

Helping in the name of Jesus.

The conditional probability that the coin chosen is fake given that it was flipped three times and all three flips resulted in heads is higher than the initial probability.

Let A be the event that the coin chosen is fake, and B be the event that the three flips resulted in heads. We are given that the probability of choosing the fake coin is 0.1, i.e. P(A) = 0.1, and the probability of getting heads with the fake coin is 1, i.e. P(B|A) = 1. The probability of getting heads with the fair coin is 0.5, i.e. P(B|A') = 0.5, where A' is the event that the coin chosen is fair. Using Bayes' theorem, we can calculate the conditional probability of A given B:

P(A|B) = P(B|A) * P(A) / (P(B|A) * P(A) + P(B|A') * P(A'))

Plugging in the given values, we get:

P(A|B) = 1 * 0.1 / (1 * 0.1 + 0.5 * 0.9) = 0.18

Therefore, the conditional probability that the coin chosen is fake given that it was flipped three times and all three flips resulted in heads is 0.18, which is higher than the initial probability of 0.1. This means that the evidence of three consecutive heads increases the likelihood that the coin is fake.

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The height of the cone is 15cm.

From the question;

Height of cylinder = 28cm

Let the radius of the solid object = Rcm

The slant height of the cone = 17cm

Cost of painting = Rs. 140 per 100 sq cm

Cost of painting 1 cm² =  140/100 = RS1.4

The total cost of painting = Rs. 2851.2

Total area of the object =  total cost of painting/Cost of painting per sq cm

Total area of the object = 2851.2/1.4  = 2036.57143 cm²

Total surface area of object = CSA of Cylinder + CSA of cone + area of the circle at the bottom

Total surface area of the object = [tex]\pi rh + \pi rl + \pi r^2[/tex]

So, [tex]\frac{2851.2}{1.4}[/tex] =  [tex]\pi rh + \pi rl + \pi r^2[/tex]

[tex]\frac{2851.2}{1.4}[/tex] = [tex]2\pi r * 28 + \pi r * 17 + \pi r^2[/tex]

[tex]\frac{2851}{1.4}[/tex] = [tex]\frac{1232}{7}r =+\frac{374}{7} r + \frac{22}{7} r^2[/tex]

[tex]\frac{2851}{1.4}[/tex] = [tex]\frac{1606}{7} r + \frac{22}{7} r^2[/tex]

[tex]\frac{2851}{2}[/tex]= [tex]1606r + 22r^2[/tex]

[tex]14256 = 1606r + 22r^2[/tex]

[tex]22r^2 + 1606r - 14256 =0[/tex]

Solving the quadratic equation, the possible value of radius are:

r = 8cm

r = -81cm

r = -81cm will be neglected because radius can not be negative.

Therefore, the radius of the object is 8cm

To find the height of the cone, we should know that the relation between radius, height and slant height of the cone is:

[tex]l^2 = r^2 + h^2[/tex]

[tex]h=\sqrt{l^2 - r^2}[/tex]

[tex]h=\sqrt{17^2 - 8^2}[/tex]

[tex]h=\sqrt{225}[/tex]

[tex]h=15cm[/tex]

Therefore, height of  cone in the solid object is 15cm

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For the first question, the measure of variability that should be used for both sets of data to determine the location with the most consistent temperature is IQR, because it is robust to outliers and can provide a better representation of the spread of data for skewed distributions. The option "IQR, because Desert Landing is symmetric" is incorrect because the graph of Desert Landing is not symmetric.

For the second question, we need to compare the measures of center (mean and median) of the two sets of data to determine which shop typically sells the most amount of coffee per hour. To do this, we can calculate the mean and median of each set of data:

For Coffee Ground, the mean is (1.5+20+3.5+12+2+5+11+7+2.5+9.5+3+5)/12 = 6.25 gallons and the median is the middle value of the ordered set, which is 5 gallons.

For Wide Awake, the mean is (2.5+10+4+18+4+3+6.5+15+6+5+2.5)/11 = 6.136 gallons and the median is the middle value of the ordered set, which is 4.5 gallons.

Therefore, we can conclude that Wide Awake typically sells the most amount of coffee per hour, with a median value of 4.5 gallons. The option "Wide Awake, with a mean value of about 4.5 gallons" is incorrect because the mean value of Wide Awake is slightly lower than 4.5 gallons. The option "Coffee Ground, with a median value of 5 gallons" is incorrect because the median value of Coffee Ground is lower than the median value of Wide Awake. The option "Coffee Ground, with a mean value of about 5 gallons" is incorrect because the mean value of Coffee Ground is higher than 5 gallons.

Answer:

1)  5

Step-by-step explanation:

Trapezoid Midsegment Theorem:

EF = (DC + AB) / 2

2x + 2 = ((x + 3) +(5x - 9)) / 2

4x + 4 = 6x - 6

-2x = -10

x = 5

The probability shows that if the mean μ=1, then only 0.0027 is at least as small as the sample mean x=1.5. This also leads to the conclusion that if we believe that μ = 1, we must also believe that there is a 27 in 10,000 chance that our observed sample mean can be described as such.

To achieve the goal of this task, use the mean μ x and the standard deviation σ x.

The task has two conditions, the first is that the standard deviation σ is 1.3 days and the second is that the standard deviation σ is 1.8 days.

If the standard deviation σ is 1.3, then the mean μ is 1.4. And if the standard deviation is σ 1.8, then the mean is μ₁. In this case, the given standard deviation is μ 1.8, so use the second condition to achieve the task goal.

we know that:

μₓ = μ and σₓ = σ/√n

Given that:

μ = 1, σ = 1.8 and n = 100

So, by the formula of standard deviation σₓ,

σₓ = σ/√n = 1.8/√100

               = 0.18

Calculate the probability with the mean μ=1, and standard deviation

σₓ =0.18.

P(x > 1.5) = P(z > 0.5/0.18)

              = P( z > 2.78)

Looking at the z table, first, find the units and decimals in the first column of the z table. In this case, search for 2.7 first. Then look in which column the 100th value is placed. In this case, 0.08 and follow the number of lines placed 2.7.

Therefore, the value of 2.78 in table z is 0.9973 because z=0.9973, and the right tail area is 0.0027 or (1−0.9973).

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Answer:

48 yards

Step-by-step explanation:

Use the Pythagorean theorem:

[tex] {a}^{2} = {52}^{2} - {20}^{2} = 2704 - 400 = 2304[/tex]

[tex]a > 0[/tex]

[tex]a = \sqrt{2304} = 48 \: yd[/tex]

Answer: 19 3/5 x 20 2/5 = 9996/25 = 399 21/25

A random sample of 38 cities was asked to report their voter turnout percentages, which can be used to estimate the average voter turnout in those cities. However, caution should be taken when making generalizations about the larger population of cities, since the sample size is relatively small.

The average voter turnout during an election can be calculated by taking the percentage of registered voters who actually voted in a sample of cities and finding the mean. In this case, a random sample of 38 cities was asked to report the percent of registered voters who voted in the most recent election. To calculate the average voter turnout, follow these steps:

1. Obtain the reported voter turnout percentages from each of the 38 cities in the random sample.
2. Add up all the percentages of voter turnout from the 38 cities.
3. Divide the sum of voter turnout percentages by the number of cities (38) to find the average voter turnout.

By following these steps, you will find the average voter turnout for the most recent election, based on the random sample of 38 cities. Keep in mind that the average voter turnout can vary depending on the election and location, as factors such as political climate, awareness, and accessibility can impact voter participation. This calculated average represents an estimation of the overall turnout and may not be completely representative of the entire population, but it provides a useful measure to understand voter engagement during elections.

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Angle B is approximately 53.15 degrees using trigonometry.

We can use trigonometry to solve for angle B in the right triangle BCD. We know that angle CBC is 37 degrees, and BD is 64.

First, we can use the Pythagorean theorem to find the length of BC, which is the hypotenuse of the right triangle:

[tex]BC^{2}[/tex] = [tex]BD^{2}+CD^{2}[/tex]

[tex]BC^{2}[/tex] = 64² + [tex]CD^{2}[/tex]

Since CD is opposite angle B, we can use trigonometry to relate CD to angle B. Specifically, we can use the tangent function:

tan(B) = CD/BD

Rearranging, we have:

CD = BDxtan(B)

Taking the square root of both sides, we have:

BC = 64[tex]\sqrt{(1+tan^{2}B)}[/tex]

Now we can use the fact that BC is the hypotenuse of the right triangle to relate it to angle CBC, which is 37 degrees. Specifically, we can use the sine function:

sin(CBC) = BD/BC

Substituting our expression for BC, we have:

sin(37) = 64/64√(1 + tan²(B))

Simplifying, we get:

sin(37) = 1/[tex]\sqrt{(1+tan^{2}B)}[/tex]

Squaring both sides, we have:

sin²(37) = 1/[tex](1+tan^{2}B)}[/tex]

Substituting the identity cos²(θ) + sin²(θ) = 1, we have:

cos²(37) = cos²(B)

Taking the square root of both sides, we have:

cos(37) = ±cos(B)

Since angle B is acute (it is less than 90 degrees because it is in a right triangle), we know that cos(B) is positive. Therefore, we can take the positive square root:

cos(B) = cos(37)

B = 53.15 degrees (rounded to two decimal places)

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