Evaluate the expression Evaluate the expression \dfrac{4^x}{2^x}
2
x

4
x

​
start fraction, 4, start superscript, x, end superscript, divided by, 2, start superscript, x, end superscript, end fraction for x=3x=3x, equals, 3.

Answer:

x=3

Step-by-step explanation:

Binomial distributions and hypergeometric distributions are similar in that they both have two possible outcomes (success or failure) and the probability of success is constant in each trial. They also both assume that each trial is independent of each other.

Both binomial distributions and hypergeometric distributions are similar in the following ways: they are both discrete distributions, they both have two possible outcomes (success or failure), the probability of success is constant in each trial, and they both assume that each trial is independent of each other.

A binomial distribution is a probability distribution that models the number of successes in a given number of independent trials with the same probability of success for each trial. In a binomial distribution, the random variable X is the number of successes in the given number of trials.

The probability of a success for each trial remains constant. The mean and variance of the binomial distribution are given by the formula: μ = np and σ2 = npq, respectively.

A hypergeometric distribution is a probability distribution that models the probability of selecting a certain number of successes in a given number of draws from a finite population without replacement. In a hypergeometric distribution, the random variable X is the number of successes in the given number of draws.

The probability of a success for each draw remains constant.

The mean and variance of the hypergeometric distribution are given by the formula: μ = nx/N and σ2 = nx(N-n)(N-x)/N2(N-1), respectively.

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The measure of angle D is 98°

We know that a quadrilateral is cyclic if and only if its opposite angles are supplementary.

Here, quadrilateral ABCD is inscribed in a circle.

From the attached figure, we can observe that angles A and C are opposite angles, so they are supplementary angles

∠A +  ∠C = 180°

(9x - 1) + 64 = 180

9x + 63 = 180

x = 13

So, the angle B would be:

m∠B = 6x+4

        = 6(13)+4

        = 82°

We can observe that angles B and D are opposite angles, so they are supplementary.

⇒ m∠B + m ∠D = 180°

⇒ 82° + m ∠D = 180°

⇒ m∠D = 98°

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The complete question is:

Quadrilateral ABCD is inscribed in a circle. m∠A is 64°, m∠B is (6x + 4)°, and m∠C is (9x - 1)° What is m∠D?

This table is an example of exponential growth.

The equation that represents this exponential growth is y = 2ˣ.

Exponential growth is characterized by an equation of the form y = bˣ, where b is a positive constant.

This table is an example of exponential growth.

This is because the y-values are increasing exponentially as the x-values increase.

The equation that represents this exponential growth is y = 2ˣ.

This can be seen by the fact that the y-values are equal to 2ˣ, where x is the corresponding x-value.

When x = 0, y = 2⁰ = 2.

When x = 1, y = 2¹ = 6.

When x = 2, y = 2² = 18.

When x = 3, y = 2³ = 54.

And finally, when x = 4, y = 2⁴ = 162.

Here, b = 2.

Exponential growth is different from linear and quadratic growth in that the rate of increase is not constant; instead, it increases as the x-values increase.

This is because the exponent in the equation increases as the x-values increase. This is why the y-values increase exponentially as the x-values increase.

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Standard deviation of the following data is 0.655.

The standard deviation is equal to the variance's positive square root. It is a fundamental statistical analysis method. The amount by which data values deviate from the mean is referred to as "standard deviation," or "SD."

Whereas a big standard deviation implies that the values are much outside the mean, a low standard deviation shows that the values are frequently only a few standard deviations from the mean.

Here in the question,

Total kids = 20.

Mean = 9+9.5+9.5+10+10+10+10+10.5+10.5+10.5+10.5+11+11+11+11+11+11+11.5+11.5+11.5/20

= 10.52

Now square of distance between mean and ages.

(9-10.52) ² = 2.31

(9.5-10.52) ² = 1.04

(10-10.52) ² = 0.27

(10.5-10.52) ² = 0.0004

(11-10.52) ² = 0.23

(11.5-10.52) ² = 0.96

Now sum of all the differences = 2.31 + 1.04 + 1.04 + 4×0.27 + 4× 0.0004 + 6× 0.23+ 3×0.96

= 8.73

Now standard deviation = √8.73/20

= √0.43

= 0.655

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Answer:

6.6km

Step-by-step explanation:

Area of trapezium = 0.5(a+b)h

201 = 0.5(17+b)17

201 = 8.5(17+b)

201 = 144.5 + 8.5b

201 - 144.5 = 8.5b

56.5 = 8.5b

b = 6.65km

Answer:

C=15.7cm

A=78.54cm^2

Step-by-step explanation:

C=πr = π(5cm) = 15.7cm

A=πr^2 =π (5cm)^2 =78.54cm^2

The area of the final image is greater than 121 square centimeters, so the final image will not fit into a rectangular space that has an area of 121 square centimeters.

What is the scale factor?

A scale factor is defined as the ratio between the scale of a given original object and a new object, which is its representation but of a different size (bigger or smaller).

The original image has a length of 6 cm and a width of 9 cm, and when it's enlarged by a scale factor of 3, its new length becomes 6 cm x 3 = 18 cm, and its new width becomes 9 cm x 3 = 27 cm.

Then, the designer reduces the enlarged image by a scale factor of 0.5, which means the new length becomes 18 cm x 0.5 = 9 cm, and the new width becomes 27 cm x 0.5 = 13.5 cm.

So, the final image has a length of 9 cm and a width of 13.5 cm.

To check if the final image will fit into a rectangular space that has an area of 121 square centimeters, we need to calculate its area:

Area of the final image = length x width

Area of the final image = 9 cm x 13.5 cm

Area of the final image = 121.5 square centimeters

Therefore, the area of the final image is greater than 121 square centimeters, so the final image will not fit into a rectangular space that has an area of 121 square centimeters.

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The surface area of the triangular prism is 2160 square centimeters.

To calculate the surface area of a triangular prism, you need to find the area of each face and add them up.

In this case, the triangular prism has two identical triangular faces and three rectangular faces.

The area of one triangular face is:

0.5 x base x height = 0.5 x 30 cm x 24 cm = 360 cm²

Since there are two triangular faces, the total area of the triangular faces is:

2 x 360 cm² = 720 cm²

The area of each rectangular face can be calculated as:

length x height

Two of the rectangular faces have a length of 30 cm (the same as the base of the triangular face) and a height of 24 cm.

The area of each of these faces is:

30 cm x 24 cm = 720 cm²

The third rectangular face has a length of 30 cm (the same as the base of the triangular face) and a height of the height of the prism, which is also 24 cm.

The area of this face is:

30 cm x 24 cm = 720 cm²

Therefore, the total surface area of the triangular prism is:

720 cm² + 720 cm² + 720 cm² = 2160 cm²

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Answer:

2,5,7,9,0.1,8

Step-by-step explanation:

Hence, the present age of the father is 30 years, and the present age of the son is 4 years.

To find someone's age, you need to know their date of birth and the current date. Then, you can calculate the difference between the two dates to determine their age. Here are the steps you can follow:

Ask the person for their date of birth.

Use a calendar or an online tool to determine the current date.

Calculate the difference between the two dates in years.

If the person has not had their birthday yet this year, subtract one year from the result.

Let's assume the present age of the son is "x" years.

According to the problem statement, the present age of the father is 10 years more than 5 times the age of the son, which can be mathematically represented as:

[tex]Father's present age = 5x + 10[/tex]

Also, it is given that the father was 28 years old two years ago. So, his present age would be 28 + 2 = 30 years.

Now, equating the two expressions for the father's age, we get:

[tex]5x + 10 = 30[/tex]

Solving for x, we get:

[tex]5x = 20\\x = 4[/tex]

Therefore, the present age of the son is 4 years.

And the present age of the father is:

[tex]5x + 10 = 5(4) + 10 = 30 years.[/tex]

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Gareth had £4,199,999.94 in his account at the start of 2015 at rate 5%.

Compound interest is computed as a percentage of the principle plus the accrued interest, whereas simple interest is determined as a fixed percentage of the original amount. In other words, whereas compound interest is calculated on the principle amount plus any interest that has previously been earned, simple interest is computed on the initial principal amount. So, over time, compound interest often produces larger returns than simple interest, particularly for long-term investments.

The compound interest is given by:

[tex]A = P(1 + r/n)^{(n*t)}[/tex]

Substituting the given values, we get:

[tex]P = 4,862,025 / (1 + 0.05/1)^{(1*3)}\\P = 4,862,025 / 1.157625\\P = 4,199,999.94[/tex]

Gareth had £4,199,999.94 in his account at the start of 2015.

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The maximum revenue can be obtained when x = 100 dollars. Thus, the store should offer a rebate of 100 to maximize its revenue.

Given information:

A store has been selling 200 DVD burners a week at 350 each. A market survey indicates that for each 10 rebate offered to buyers, the number of units sold will increase by 20 a week.

Let's assume x is the number of 10 rebate and y is the number of units sold. It is given that a store has been selling 200 DVD burners a week at 350 each. So the initial price is 350 for 200 DVD burners.

Therefore, the demand function y(x) can be given as:

y(x) = 200 + 20x...(i)It is also known that for each 10 rebate offered to buyers, the number of units sold will increase by 20 a week. So the price of DVD burners will be (350 - 10x) for the additional units sold.

Therefore, the revenue function R(x) can be given as:

R(x) = (350)(200 + 20x) + [(350 - 10x)(20x)]...(ii)

The above equation (ii) is the revenue function.

Now we will find the value of x that maximizes R(x).

For the above revenue function, we need to find the value of x that maximizes R(x).

So differentiate R(x) w.r.t x to get the value of x that maximizes R(x).

dR(x)/dx = 350(20) - 10(200 + 20x) + 20(350 - 10x) = 0

Simplifying the above expression:

7x - 700 = 0x = 100

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[tex]a = 7[/tex]

[tex]d = 12 - 7 = 5[/tex]

[tex]L = 102[/tex]

[tex]\implies a + (n-1)d = 102[/tex]

[tex]\implies 7 + (n-1)(5) = 102[/tex]

[tex]\implies (n-1)(5) = 95[/tex]

[tex]\implies n - 1 = 19[/tex]

[tex]\implies n = 20[/tex]

[tex]\text{Required sum = n/2} \ (a + L)[/tex]

[tex]= 20 \div2 \ ( 7 + 102)[/tex]

[tex]= 10 \times 109[/tex]

[tex]\bold{= 1090}[/tex]

Answer:

The sum is 1090

Step-by-step explanation:

S=7+12+17+22+...+102 ---> reason=5

We define the values, before starting:

Ratio = r , Term n = Tn , Term number = n , Term one = T1

To do this we first find the term number of 102, with the following formula:

Tn = t1 + (n - 1) r

102 = 7 + (n - 1) 5

102 = 7 + 5n - 5

102 = 5n + 2

100 = 5n

20 = n

Then we realize that 102 is the term N°20, now what we have to do is find the sum of the arithmetic series, with the following values and the following formula:

t1 = 7   ,   tn = 102   ,   n = 20

S = ( [tex]\frac{t1+tn}{2}[/tex]) . n

S = ( [tex]\frac{7+102}{2}[/tex]) . 20

S = 109 . 10

S = 1090

Observation would be the most appropriate data collection method for studying how kids engage in play at a Montessori elementary school.

Therefore the answer is A) Observation.

This method involves observing and recording the behaviors and interactions of the children during playtime, which can provide valuable insights into their play patterns, preferences, and social dynamics.

Observation is particularly well-suited for studying young children because it allows researchers to capture their behavior in a naturalistic setting, without relying on self-report or other potentially unreliable sources of data. By watching how children interact with their peers and the materials available to them, researchers can gain a deeper understanding of their social, cognitive, and emotional development.

Observation can be conducted in a variety of ways, such as through structured or unstructured observation, video or audio recording, or field notes. Researchers may also choose to use multiple methods of observation to gain a more comprehensive understanding of children's play behavior.

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--The question is incomplete, answering to the question below--

"Answer in a very short answer of 1-3 sentence and an explanation of more than 100 words

For a study exploring how kids engage in play at Montessori elementary school, which would be the most appropriate data collection method?

A) observation

B) interviews

C) documents

D) audiovisual

E) audio materials"

Answer:

Step-by-step explanation:

i think i is -1

The list that gives the sample space for pulling two slips of paper out of the bucket with replacement is List 4.

The sample space for pulling two slips of paper out of the bucket with replacement means that after drawing the first slip of paper, it is returned to the bucket before drawing the second slip.

This means that each draw is independent of the previous draw, and the possible outcomes of each draw remain the same for all subsequent draws.

The possible outcomes for each draw are {Red, Blue, Yellow}, and there are three possible outcomes for each draw.

Since we are drawing with replacement, each draw has the same possible outcomes and the same number of outcomes. So, the total number of possible outcomes for two draws is simply the number of outcomes for a single draw raised to the power of 2:

3 x 3 = 9

Therefore, the list that gives the sample space for pulling two slips of paper out of the bucket with replacement is List 4.

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The slope of the graph would give us the speed of the motion and that is  40 m/min.

We can use this principle to show that the slopes of two lines are equal if they are parallel and intersected by a transversal line. Here's how:

Draw two parallel lines, labeled as line A and line B.

Draw a transversal line that intersects both lines A and B at two points.

Draw a perpendicular line segment from one of the intersection points on line A to the transversal line, and label it as segment 1.

Draw a perpendicular line segment from the corresponding intersection point on line B to the transversal line, and label it as segment 2.

The triangles formed by segment 1, the transversal line, and a segment on line A, and by segment 2, the transversal line, and a segment on line B, are similar triangles.

Since the corresponding sides of similar triangles are proportional, we can set up the following proportion: segment 1 / segment on line A = segment 2 / segment on line B.

Simplifying the proportion, we get segment 1 / segment 2 = segment on line A / segment on line B.

Since segment 1 and segment 2 are both perpendicular to the transversal line, they represent the rise of the lines A and B, respectively. The segments on lines A and B represent the run of the lines A and B, respectively. Therefore, we can rewrite the proportion as rise of line A / run of line A = rise of line B / run of line B.

This shows that the slopes of the two parallel lines, line A and line B, are equal.

The slope of the graph is;

m = 140 - 60/2 - 0

m = 40 m/min

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Rounded to the nearest hundredth, the volume of empty space is approximately 6651.72 cm³

Diameter is a straight line passing through the center of a circle or a sphere, and connecting two points on the circumference or surface respectively.

The diameter of each ball is 14 centimeters, so the height of the cylindrical container must be at least 42 centimeters in order to stack three balls inside. Let's assume that the height of the container is exactly 42 centimeters.

The three tennis balls weigh a combined amount of:

3 × (4/3)πr³ = 3 × (4/3)π(7 cm)³ ≈ 1436.76 cm³

The volume of the cylindrical container is:

πr²h = π(7 cm)²(42 cm) ≈ 8088.48 cm³

So the volume of the empty space left inside the container is:

8088.48 cm³ - 1436.76 cm³ ≈ 6651.72 cm³

Rounded to the nearest hundredth, the volume of empty space is approximately 6651.72 cm³.

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The length of AD is 23.5 cm.

The perimeter of a rectangle is the sum of the lengths of all four of its sides. So,

Perimeter = AD + AB + BC + CD

38 cm = AD + (x + 4) + (3x - 12) + (x + 4)

38 cm = 4x + 4

34 cm = 4x

x = 8.5 cm

Therefore, AD = 3x - 12 = 3(8.5) - 12 = 23.5 cm

The formula used to calculate the perimeter of a rectangle is P = 2(l + w) where P is the perimeter, l is the length and w is the width. The length of AD can be calculated by solving the equation 38 cm = 2(l + w). We know that AB = x + 4 and BC = 3x - 12, so l + w = x + 4 + 3x - 12 = 4x - 8. This means that 38 cm = 2(4x - 8), which can be rearranged to 34 cm = 4x. Therefore, x = 8.5 cm, so l = 8.5 cm + 4 cm = 12.5 cm and AD = 12.5 cm - 8.5 cm = 23.5 cm.

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The equation is a linear combination of three angles that add up to 180°, he might consider using systems of linear equations or other algebraic methods to solve for the variables.

given that za + 2d + g = 180°, some observations that could potentially be useful to Jason include:

Identifying any relationships or constraints between the variables za, d, and g that might help him simplify or solve the equation. For example, if he knows that za and g are complementary angles (i.e., they add up to 90°), he could substitute 90 - za for g in the equation to get a new expression in terms of za and d.

Using additional information about the angles or the situation to derive new equations or constraints that could help him solve for one or more variables. For example, if he knows that za and d are equal (i.e., za = d), he could substitute za for d in the equation to get a new expression in terms of za and g.

Identifying any patterns or structures in the equation that might suggest a particular approach or technique for solving it. For example, if he notices that the equation is a linear combination of three angles that add up to 180°, he might consider using systems of linear equations or other algebraic methods to solve for the variables.

Ultimately, the most effective observations for completing the proof will depend on the specific details of the problem and the approach that Jason is taking.

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The complete question:Jason knows that za + 2d + g = 180°. Which of the following observations, if true, would allow him to prove a statement or theorem related to the angles za, d, and g?

The sphere has a volume of around [tex]221.83[/tex] cubic inches.

All of the points on a circle are equally spaced apart from the center along a line, whereas all of the points on a sphere are equally spaced apart from the center along any of the axis. In the case of a ring, we can only calculate the area, however for a sphere, we can compute both the surface area and the volume.

The volume of a cuboid generated by stretching the dimensions of rectangles in place is equal to: Volumes = length [tex]*[/tex] width [tex]*[/tex] height, for instance, if the area of a rectangles is length [tex]*[/tex] breadth.

[tex]V = (4/3)[/tex]π[tex]r^{3}[/tex]

where [tex]r[/tex]  is the radius of the sphere.

In this problem, the radius of the sphere is half the diameter, which is [tex]11[/tex]inches. Therefore, the radius is:

[tex]r = d/2 = 11/2 = 5.5[/tex] inches

Substituting this value into the formula for the volume of a sphere, and using the approximation π [tex]=3[/tex], we get:

[tex]V = (4/3)[/tex]π[tex]r^{3}[/tex] [tex]= (4/3)(3)(5.5^{3} ) = (4/3)(3)(166.375) = 221.83 in^{3}[/tex]

Therefore, the volume of the sphere is approximately [tex]221.83[/tex] cubic inches.

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Answer: 5

Step-by-step explanation:

tan(32) = x/8

8 x tan(32) = x

x = 4.998 ≈ 5

Using equations, we can find that there are a total of 45,000 seats available at the baseball game.

An equation is a mathematical statement that has two expressions with equal values, each separated by the symbol "equal to."

Let x be the total number of seats available.

According to the problem, 18% of the seats are bleachers, and 42% of the seats are grandstands.

That means the remaining 40% of the seats are box and club seats combined since 100% - 18% - 42% = 40%.

The problem states that there are 12,600 box seats and 5,400 club seats available, which makes a total of

12,600 + 5,400 = 18,000 box and club seats.

Now, we can set up an equation to find the total number of seats:

0.40x = 18,000

To solve for x, divide both sides of the equation by 0.40:

x = 18,000 ÷ 0.40

x = 45,000

There are a total of 45,000 seats available at the baseball game.

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As it is known that the temperature of hot coffee will cool to room temperature following an exponential mode, the option which would linearize the data for temperature and minutes is "Minutes, ln(Temperature)". The Option B is correct.

Using logarithmic transformation, we can convert the exponential relationship between temperature and time into a linear relationship. By taking the natural logarithm of the temperature, we can rewrite the exponential model as:

ln(Temperature) = -kt + ln(T0)

In this model, ln(Temperature) is the natural logarithm of the temperature, k is a constant representing the rate of cooling, t is the time elapsed, and T0 is the initial temperature of the coffee.

This equation is in the form of a linear equation, y = mx + b, where ln(Temperature) is the y variable, -k is the slope, t is the x variable, and ln(T0) is the y-intercept. Therefore, plotting Minutes versus In(Temperature) will give a linear relationship.

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Answer:

90.52

Step-by-step explanation:

Answer:

Proved that tan45° × sec40° + tan60° × cosec40° =4 by using the trigonometric ratio formulas.

Given that,

We have to prove that tan45° × sec40° + tan60° × cosec40° =4

We know that,

What is trigonometry ratio?

Trigonometric ratios are ratios of triangle side lengths. These ratios in trigonometry demonstrate the relationship between the sides and angles of a right triangle. The three basic ratios in trigonometry are sine, cosine, and tangent.

Take the left hand side

tan45° × sec40° + tan60° × cosec40°

We know that tan45°=1 and tan60°=√3

1×sec40°+√3×cosec40°

sec40°+√3cosec40°

We know that from formula,

sec40° = \frac{1}{cos40\textdegree}

cos40\textdegree

1

and cosec40° = \frac{1}{sin40\textdegree}

sin40\textdegree

1

\frac{1}{cos40\textdegree}

cos40\textdegree

1

+√3× \frac{1}{sin40\textdegree}

sin40\textdegree

1

1+3

4 (Approximately)

Therefore, Proved that tan45° × sec40° + tan60° × cosec40° =4 by using the trigonometric ratio formulas.

Using a calculator, we find that the bearing of A from B is approximately 165.95 degrees. Using a calculator, we find that the bearing of C from B is approximately 87.88 degrees.

Trigonometry is a branch of mathematics that deals with the study of the relationships between the angles and sides of triangles. It is based on the three primary trigonometric functions: sine, cosine, and tangent, which relate the ratios of the sides of a right triangle to the angles opposite those sides.

Here,

i.  Let x be the distance from B to A and y be the distance from B to C. Then, we have:

tan(50) = x / y

Rearranging this equation, we get:

x = y * tan(50)

Thus, the coordinates of A are (-4 sin(50), 4 cos(50) - 4 sin(130)). The bearing of A from B is then:

bearing of A from B = tan⁻¹((y * sin(50)) / (y * cos(50) - x))

Substituting x and y, we get:

bearing of A from B = tan⁻¹(sin(130) / (cos(50) - sin(50) tan(50)))

≈165.95 degrees

ii. To find the bearing of C from B, we can use the same method. Let's assume that the coordinates of B are (0, 0), and the distance from B to A is 3 units. Then, the coordinates of A are (-3 sin(50), 3 cos(50) - 3 sin(130)). The bearing of C from B is then:

bearing of C from B =  tan⁻¹((4 * sin(50)) / (4 * cos(50) - 3 * sin(50) / tan(50)))

≈87.88 degrees

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The size of the angle PRQ is approximately 26.565 degrees.

Since PQ and QR are two sides of a regular 12-sided polygon, they are congruent, and the polygon can be divided into 12 congruent isosceles triangles with base PQ and QR.

Let's call the center of the polygon O. Since the polygon is regular, the angle POQ is 360/12 = 30 degrees. Also, since PR is a diagonal of the polygon, it divides angle POQ into two congruent angles, each measuring 15 degrees.

Now, let's consider the triangle PQR. We know that angle PQR is 180 - (15 + 15) = 150 degrees (since the sum of angles in a triangle is 180 degrees).

Using the Law of Cosines, we can find the length of PR:

PR² = PQ² + QR² - 2(PQ)(QR)cos(angle PQR)

PR² = PQ² + QR² - 2(PQ)(QR)cos(150)

PR² = PQ² + QR² + (PQ)(QR)

Since PQ = QR, we can simplify this to:

PR² = 2(PQ²) + PQ²

PR² = 3(PQ²)

Therefore,

PQ² = (1/3)(PR²)

Now, let's consider the triangle PRQ. We know that PQ is a side of the polygon, so it has length 1 (we can choose any unit of length we like). Also, we know that QR is a side of a regular 12-sided polygon, so we can use the formula for the side length of a regular polygon:

QR = 2PQsin(180/12) = PQ√(6 + 2sqrt(3))

Substituting PQ = 1, we get:

QR = √(6 + 2√(3))

Finally, we can use the Law of Cosines again to find the angle PRQ:

cos(angle PRQ) = (PQ² + QR² - PR²) / (2PQQR)

cos(angle PRQ) = (1 + 6 + 2√3) - PR^2) / (2√(6 + 2√(3)))

Substituting PR² = 3(PQ²) = 3, we get:

cos(angle PRQ) = (7 + 2√(3)) / (2√(6 + 2√(3)))

Taking the inverse cosine, we get:

angle PRQ = 26.565 degrees (to 3 decimal places)

Therefore,  the size of the angle PRQ is approximately 26.565 degrees.

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