edmentum question:
-post test: similarity and proof
in the diagram, the ratios __ and ___ are equal

Answer:

  150 m²

Step-by-step explanation:

You want the total area of a figure consisting of a 12 m wide rectangle 5 m high with a triangle above that bringing the total height to 20 m.

The area of the rectangle is the product of its length and width:

  A = LW

  A = (12 m)(5 m) = 60 m²

The area of the triangle is half the product of its base and height.

  A = 1/2bh

  A = 12/(12 m)(20 -5 m) = 90 m²

The area of the whole figure is the sum of the areas of its parts:

  total area = rectangle area + triangle area

  total area = 60 m² +90 m² = 150 m²

The area of the figure is 150 m².

__

Additional comment

Since you know a triangle's area is equivalent to the area of a rectangle that has half the height, the 15 m high triangle can be considered as a rectangle 7.5 m high. Adding that to the 5 m area of the rectangle already there gives the equivalent area as that of a rectangle 12 m wide and 5+7.5 = 12.5 m high: (12 m)(12.5 m) = 150 m².

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To convert from degrees to radians, we use the conversion factor that 180 degrees is equal to π radians (or π/180 radians per degree).

Given that the angle is 45 degrees, we can calculate the radian measure as follows:

Radian measure = (45 degrees) * (π/180 radians per degree)

Radian measure = 45π/180

Simplifying further:

Radian measure = π/4

Therefore, the radian measure of a 45 degree angle is π/4.

The function g(x) in the context of this problem is given as follows:

A. [tex]g(x) = \sqrt[3]{x + 4} + 3[/tex]

A translation happens when either a figure or a function is moved horizontally or vertically on the coordinate plane.

The four translation rules for functions are defined as follows:

The parent function in this problem is given as follows:

[tex]f(x) = \sqrt[3]{x}[/tex]

The translated function was moved 4 units left and 3 units up, hence the function is defined as follows:

[tex]g(x) = \sqrt[3]{x + 4} + 3[/tex]

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To find the other two numbers in the set, we need to determine the missing numbers that satisfy the given conditions.

The mode of a set of numbers is the value(s) that appear(s) most frequently. In this case, the mode is given as 19, which means that 19 appears more times than any other number in the set.

The mean of a set of numbers is calculated by summing all the numbers and dividing by the total count. In this case, the mean is given as 20.

Given that the seven numbers provided are 16, 17, 19, 19, 21, 21, and 25, we can calculate the sum of these numbers:

16 + 17 + 19 + 19 + 21 + 21 + 25 = 138

To find the sum of all nine numbers in the set, we can use the mean formula:

Mean = (Sum of all numbers) / (Total count)

20 = (138 + x + y) / 9

Simplifying the equation, we have:

180 = 138 + x + y

x + y = 42

Now, since the mode is 19, we know that either x or y (or both) must be 19. Since the mode appears twice, we can assign 19 to both x and y:

x = 19

y = 19

Therefore, the other two numbers in the set are both 19.

The correct option is: 5x² - 8x + 10 - 25/(x + 2)

Given that we need to determine, what is the quotient of 5x³+2x2-6x-5 divided by x+2,

Long division can be used to determine the polynomial 5x³+2x2-6x-5 divided by x + 2.

Here is how to accomplish it:

         5x² - 8x + 10

   ____________________

x + 2 | 5x³ + 2x² - 6x - 5

       - (5x³ + 10x²)

       _________________

                  -8x² - 6x

             - (-8x² - 16x)

             _______________

                       10x - 5

                  - (10x + 20)

                  ___________

                          -25

The quotient is 5x² - 8x + 10, and the remainder is -25/(x + 2).

Therefore, the correct option is: 5x² - 8x + 10 - 25/(x + 2)

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The unit vector for n = (4, -3) is V = (4/5, -3/5)

An unit vector will be a vector that has the same direction than the given one, but a magnitude of 1 unit.

Then we can define the vector V = k*n

Where k > 0 is a real number, then the unit vector is:

V = (4k, -3k)

But notice that this must have a magnitude of 1, then:

1 = √( (4k)² + (-3k)²)

1 = √25k²

1 = 5k

1/5 = k

Then the unit vector is:

V = (4/5, -3/5)

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The domain for n in the given Geometric sequence is the set of positive integers: {1, 2, 3, 4, 5, ...}.

In a geometric sequence, each term is obtained by multiplying the previous term by a constant value called the common ratio (r).

In the given geometric sequence, the first term (a1) is -9 and the common ratio (r) is 5. Therefore, the sequence can be written as:

-9, -9 * 5, -9 * 5^2, -9 * 5^3, ...

To determine the domain for n, we need to determine the values of n for which the sequence is defined. In a geometric sequence, the term number n represents the position of a term in the sequence.

In this case, the first term corresponds to n = 1, the second term corresponds to n = 2, and so on. The domain for n consists of all the positive integers starting from 1, as the sequence extends indefinitely.

Therefore, the domain for n in the given geometric sequence is the set of positive integers: {1, 2, 3, 4, 5, ...}.

the domain does not include zero or negative values, as these positions in the sequence do not exist. The sequence only starts from the first term (n = 1) and continues indefinitely in the positive integer domain.

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The total amplitude of the tip of King Arthur's sword is s +  [tex]\sqrt{3}[/tex]t/2.

Given that the length of the line segment connecting the centre of the regular hexagon to one of its vertices as  the length of the line segment connecting centre of the regular pentagon to one of its vertices.

To find the amplitude by adding these two lengths.

For the regular hexagon, the distance from the centre to a vertex is equal to the radius of the circumscribed circle. The radius  of the circumscribed circle is equal to length of a side.

Therefore, the amplitude of the hexagon is s.

For a regular pentagon, divide it into three triangles. One of these triangles is an isosceles triangle where the base is the side of the pentagon and the other two sides are radii of the circumscribed circle.

The amplitude of pentagon is the height of the isosceles triangle.

In an isosceles triangle, the height(h) is calculated from formula

h =  [tex]\sqrt{3}[/tex]t/2.

Therefore, the amplitude of the pentagon is   [tex]\sqrt{3}[/tex]t/2.

Hence, the total amplitude of the tip of King Arthur's sword is s +  [tex]\sqrt{3}[/tex]t/2.

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Free variables are variables used to represent parameters

Free variables are placeholders or symbols in mathematics and logic that are not constrained by any quantifiers or other specified conditions within an expression, formula, or equation.

When the expression is evaluated, they stand for values that can change or be given different values. In equations or functions, free variables are frequently employed to represent unknowns or parameters.

Free variables enable flexibility and generality in mathematical reasoning since they can be given various values to investigate various situations or solutions. In contrast, bound variables have a specific scope within a certain context or statement and are constrained by quantifiers.

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In this dataset, the mean is approximately 65.77, the median is 65, and the mode is 5 and 75.

Part A: Finding the mean of the data:

To find the mean, we need to calculate the average of all the data points.

Step 1: Identify the stems and their corresponding leaves:

3 | 5

4 | 4, 5

5 | 3, 6

6 | 2, 5

7 | 5, 5, 6

8 | 2, 5

9 | 2

Step 2: Assign numerical values to each stem-leaf combination:

3 | 5 = 35

4 | 4 = 44, 5 = 45

5 | 3 = 53, 6 = 56

6 | 2 = 62, 5 = 65

7 | 5 = 75, 5 = 75, 6 = 76

8 | 2 = 82, 5 = 85

9 | 2 = 92

Step 3: Calculate the sum of all the numerical values:

35 + 44 + 45 + 53 + 56 + 62 + 65 + 75 + 75 + 76 + 82 + 85 + 92 = 855

Step 4: Determine the count of all the data points:

The count is the total number of data points, which can be determined by adding up the frequencies of each stem-leaf combination:

1 (stem 3) + 2 (stem 4) + 2 (stem 5) + 2 (stem 6) + 3 (stem 7) + 2 (stem 8) + 1 (stem 9) = 13

Step 5: Calculate the mean by dividing the sum of all values by the count:

Mean = Sum of all values / Count = 855 / 13 = 65.77 (rounded to two decimal places)

The mean of the data is approximately 65.77.

Part B: Finding the median of the data:

To determine the median, we need to arrange the data in ascending order and find the middle value.

Arranging the data in ascending order: 35, 44, 45, 53, 56, 62, 65, 75, 75, 76, 82, 85, 92

There are 13 data points, the median will be the value in the middle. In this case, the middle value is the 7th value, which is 65.

The median of the data is 65.

Part C: Finding the mode of the data:

The mode represents the value(s) that occur with the highest frequency.

From the stem-leaf plot, we can see that the leaves with the highest frequency are 5 and 75. Both of these frequencies occur twice.

The mode of the data is 5 and 75.

Part D: Comparing the mean, median, and mode:

In this dataset, the mean is approximately 65.77, the median is 65, and the mode is 5 and 75.

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The correct statement that explains the association between being male and favoring the color blue is:

B. There is a negative association because the number of males who chose blue is less than the number of females who chose blue.

In the given table, it can be observed that out of the total 90 students surveyed, 9 students (4 males and 5 females) decided the color blue as their favorite primary color. Since the number of males (4) who selected blue is less than the number of females (5) who chose blue, there is a negative association between being male and favoring the color blue.

Final speed after slowing down by 15 miles per hour and reducing the speed by one third is 25 miles per hour.

Let's break down the steps to determine the final speed:

Step 1: Convert the speed from miles per minute to miles per hour.

Since you're driving one and a half miles per minute, we need to convert it to miles per hour. There are 60 minutes in an hour, so we multiply 1.5 by 60 to get 90 miles per hour.

Step 2: Slow down by 15 miles per hour.

Subtract 15 from the initial speed of 90 miles per hour, resulting in 75 miles per hour.

Step 3: Reduce the speed by one third.

To find one third of 75 miles per hour, we divide it by 3, which gives us 25 miles per hour.

Therefore, the final speed after slowing down by 15 miles per hour and reducing the speed by one third is 25 miles per hour.

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The length of the segment shown in the graph is 10 units

The endpoints on the line segment are : (2, - 1) and (-4,7)

The Length of a line segment can obtained using the relation :

Length = √(x2 - x1)² + (y2 - y1)²

Here,

x2 = - 4, x1 = 2, y2 = 7, y1 = - 1

Length of segment = √((-4) - 2)² + (7 - (-1))²

Length of segment = √(-6)² + (8)²

Length of segment = √36 + 64

Length of segment = √100 = 10 units

Therefore, the length of the segment on the graph shown is 10 units.

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The transformed vertices are;

A''  => (-7, -1)

B''  => (-7, 4)

C''=> (-9,4)

D''=> (-9, -1)

Here, we have,

given that,

we have to translate by (x,y) => (x-5, y+4)

so, the rectangle will be transformed as;

the transformed vertices are;

A' => ( 4-5, 3+4) => (-1, 7)

B' => (9-5, 3+4) => (4,7)

C'=>(9 -5, 5 +4 ) => (4, 9)

D'=> (4 -5, 5+4 ) => (-1, 9)

now, For clockwise rotation of a triangle by 90 degree, then x coordinate is similar to the y coordinate of original point and y coordinate is negative times of x coordinate of original point.

So (x,y) changes to (-y,x).

the transformed vertices are;

A'' => ( 4-5, 3+4) => (-1, 7) => (-7, -1)

B'' => (9-5, 3+4) => (4,7) => (-7, 4)

C''=>(9 -5, 5 +4 ) => (4, 9) => (-9,4)

D''=> (4 -5, 5+4 ) => (-1, 9)=> (-9, -1)

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The area of a triangle is 24 square feet and the area of a trapezium is 50 square feet.

From the given figure, the dimensions of trapezium are parallel sides a=7 ft, b= 13 ft and height = 5 ft.

Here, area of trapezium = 1/2 (Sum of parallel sides)×Height

= 1/2 ×(7+13)×5

= 1/2 ×20×5

= 50 square feet

Dimensions of triangle are base = 8 ft and height = 6 ft

Area of a triangle = 1/2 ×8×6

= 24 square feet

Therefore, the area of a triangle is 24 square feet and the area of a trapezium is 50 square feet.

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Tom's estimated travel time is approximately 13.6 hours.

Based on the given scenario, Tom is traveling a distance of approximately 340 nautical miles from Houston to New Orleans by boat. He will be using a medium-sized recreational boat with an average speed of 25 miles per hour.

To estimate the time it will take for his journey, we can use the formula:

Time = Distance / Speed

Substituting the values, we have:

Time = 340 miles / 25 miles per hour

Calculating the division, we find that Tom's estimated travel time is approximately 13.6 hours.

However, it is important to note that this estimate assumes a continuous journey without considering other factors that may affect the travel time. Various elements can impact the actual duration, such as weather conditions, water currents, navigational challenges, refueling stops, and any necessary rest breaks.

Additionally, it is essential for Tom to adhere to all boating regulations and safety guidelines to ensure a smooth and secure voyage.

To obtain a more accurate estimate, Tom should consult local boating authorities, marinas, or experienced sailors familiar with the route.

They can provide valuable insights into the specific conditions and offer guidance on potential challenges that may affect the travel time. Planning for additional buffer time is also recommended to account for unforeseen circumstances and to prioritize safety during the journey.

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The complete question may be like:

Tom plans to travel by boat from Houston, Texas, to New Orleans, Louisiana. He wants to estimate the time it will take for his journey. The distance between the two cities by water is approximately 340 nautical miles. Tom will be traveling on a medium-sized recreational boat that has an average speed of 25 miles per hour. Considering normal weather conditions and assuming a continuous journey, how long will it take Tom to reach New Orleans?

Please provide an answer based on this scenario, taking into account the distance, average speed, and any other relevant factors.

The other options provided, -3x - 1, -x - 12, and -x - 3, do not match the simplified form of the given expression. Only -3x - 4 corresponds to the original expression after simplification. It is important to carefully distribute and combine like terms to simplify expressions correctly.

The expression shown is 2 + 2(x – 3) – 5x. To find an equivalent expression, we need to distribute the 2 to both terms inside the parentheses, resulting in 2x - 6. Now we can simplify the expression further:

2 + 2x - 6 - 5x

Combining like terms, we have:

(2x - 5x) + (2 - 6)

This simplifies to:

-3x - 4

Hence, the expression -3x - 4 is equivalent to 2 + 2(x – 3) – 5x.

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The critical value of the chi-squared statistic with a significance level of 0.05 is 3.84. Therefore, the correct answer is option B.

The null hypothesis in this case would be that there is independence between the type of sleeping pill and the presence of any adverse health condition.

The critical value needed to test for this is the chi-squared statistic with a significance level of 0.05.

To calculate the critical value, we first need to calculate the degrees of freedom,  which is calculated by (rows-1) * (columns-1).  In this case, the degrees of freedom is (2-1)*(2-1) = 1.

The critical value of the chi-squared statistic with a significance level of 0.05 is 3.84. This means that if the calculated chi-squared statistic is greater than 3.84, then we can reject the null hypothesis and claim that there is a significant relationship between the type of sleeping pill and the presence of any adverse health condition.

Therefore, the critical value of the chi-squared statistic with a significance level of 0.05 is 3.84. Therefore, the correct answer is option B.

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Answer: i dont know what this is but i  do believe its the second one

Step-by-step explanation:

Answer:

B. 3188.5 cubic inches.

Step-by-step explanation:

The volume of a cone is calculated using the following formula:

Volume = (1/3) * π * r² * h

Where:

In this problem, we are given that r = 17 inches and S.h = 20 inches.

First we need to find height h.

py using Pythagorous theorem,we get

c²=a²+b²

here c= slight height and a is radius

20²=17²+b²

20²-17²=b²

111=b²

b=√(111)

Plugging these values into the formula, we get:

Volume = ⅓*π* 17² *√(111) = 3188.5 cubic inches

Therefore, the volume of the cone is 3188.5 cubic inches.

The perimeter of shaded figure is 10 unit.

We know,

The perimeter of a figure is the total distance around its boundary. To calculate the perimeter, you need to sum the lengths of all the sides of the figure.

From the figure

length of rectangle = 4 unit

width of rectangle =  1 unit

Now, the perimeter of shaded figure

= 2 (l + w)

= 2 (4 +1 )

= 2 x 5

= 10 unit

Thus, the perimeter of figure is 10 unit.

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The equation of the parabola with the given focus (2, 7) and directrix y = -1 is [tex](x - 2)^2 = 12(y - 3)^2[/tex].Option C.

To find the equation of the parabola with a focus and directrix, we can use the standard form of the equation of a parabola:

For a vertical parabola:

[tex](x - h)^2 = 4p(y - k),[/tex]

where (h, k) is the vertex, and p is the distance from the vertex to the focus and directrix.

In this case, the focus is given as (2, 7), which means the vertex is also (2, 7) since the focus and vertex lie on the axis of symmetry. Additionally, the directrix is given as y = -1, which means the directrix is a horizontal line.

First, let's determine the distance from the vertex to the focus and directrix, which is the value of p. The distance is the absolute difference between the y-coordinate of the focus (7) and the y-coordinate of the directrix (-1):

p = |7 - (-1)| = 8.

Now we can substitute the values of the vertex (h, k) = (2, 7) and p = 8 into the standard form equation:

[tex](x - 2)^2 = 4(8)(y - 7).[/tex]

Simplifying further:

[tex](x - 2)^2 = 32(y - 7).[/tex]

Expanding the equation:

[tex]x^2 - 4x + 4 = 32y - 224.[/tex]

Rearranging the terms:

[tex]x^2 - 4x - 32y + 228 = 0.[/tex]

Therefore, the equation of the parabola with the given focus (2, 7) and directrix y = -1 is  [tex](x - 2)^2 = 12(y - 3)^2.[/tex] SO Option C is correct.

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The total area of the quadrilateral DEFG is determined as 154 cm² .

option A.

The total area of the quadrilateral DEFG is calculated by applying the following method.

Consider triangle GDE in the quadrilateral;

base of the triangle = 16 cm

height of the triangle = 8 cm

Area of the triangle GDE = ¹/₂ x base x height

Area of the triangle GDE = ¹/₂ x 16 cm x 8 cm

Area of the triangle GDE = 64 cm²

Consider triangle GEF in the quadrilateral;

base of the triangle = 18 cm

height of the triangle = 10 cm

Area of the triangle GEF = ¹/₂ x base x height

Area of the triangle GEF = ¹/₂ x 18 cm x 10 cm

Area of the triangle GEF = 90 cm²

The total area of the quadrilateral DEFG is calculated as;

Area =  64 cm² +  90 cm²

Area = 154 cm²

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Answer: The expression can be simplified by combining the fractions with a common denominator:

(x - 9)/(x - 4) + (x^2 - x + 5)/(x - 4)

To add these fractions, we need to find a common denominator, which in this case is (x - 4). Therefore, we can rewrite each fraction with the common denominator:

[(x - 9) + (x^2 - x + 5)] / (x - 4)

Simplifying the numerator:

(x - 9 + x^2 - x + 5) / (x - 4)

Combining like terms:

(x^2 - 2x - 4) / (x - 4)

Hence, the simplified expression is (x^2 - 2x - 4) / (x - 4).

The probability that a aimlessly  named person's triglyceride  position is  lower than 80 milligrams per deciliter is  roughly0.0618 or6.18.    

To calculate the probability that a aimlessly  named person's triglyceride  position is  lower than 80 milligrams per deciliter, we can use the conception of standard normal distribution.  

First, we need to  regularize the value of 80 using the z- score formula  z = ( x- μ)/ σ  Where  x =  80( the value we want to calculate the probability for)  μ =  134( mean triglyceride  position)  σ =  35( standard  divagation)  Plugging in the values, we get  z = ( 80- 134)/ 35  z = -54/ 35  z ≈-1.543

Next, we need to find the corresponding area under the standard normal distribution  wind for a z- score of-1.543. We can use a standard normal distribution table or a calculator to find this area.

Looking up the z- score in the table or using a calculator, we find that the area to the left wing of z = -1.543 is  roughly0.0618.  

thus, the probability that a aimlessly  named person's triglyceride  position is  lower than 80 milligrams per deciliter is  roughly0.0618 or6.18.

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The equation for the function shown is f(x) = (x + 1)² - 2. Option B

From the information given, we have the deductions;

The vertex of the parabola is given as (-1, -2)

Using the form;

f(x) = a(x - h)^2 + k

Such that (h, k) represents the vertex coordinates.

Substitute the vertex coordinates into formula, we get;

f(x) = a(x - (-1))² + (-2)

= a(x + 1)² - 2

Now, substitute the value for a, we get;

2 = a(-3 + 1)² - 2

find the square and collect like terms

4 = a(4)

a = 1

Put value back into the equation

f(x) = 1(x + 1)² - 2

Multiply the value, we get;

f(x) = (x + 1)² - 2

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The probability that a given student said math was their favorite subject was 3.44

The mathematical notion of probability measures the possibility of an event happening. A number between 0 and 1 is used to indicate it, with 0 denoting impossibility (the event won't happen) and 1 denoting certainty (the event will unquestionably happen). In probability theory, the probability of an event is determined based on a set of assumptions or information available. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Here, the total outcome is 299 and favorable outcome is 87.

Hence, the probability is 299/87 = 3.44

Therefore, probability that a given student said math was their favorite subject was 3.44.

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The two numbers could be 0.43× 10⁻⁵ and, 0.06× 10⁻³.

Here, we have,

given that,

the product of two number in scientific notation is: 0.0258 × 10⁻⁸

now, let the two numbers be:

0.43× 10⁻⁵ and, 0.06× 10⁻³

we get,

 0.43× 10⁻⁵ × 0.06× 10⁻³

= 0.43 × 0.06× 10⁻⁵ × 10⁻³

= 0.43 × 0.06 × 10⁻⁵⁻³

=0.43 × 0.06× 10⁻⁸

=0.0258 × 10⁻⁸

Hence, The two numbers could be 0.43× 10⁻⁵ and, 0.06× 10⁻³.

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The distance between Martin's house and the school is 5 kilometers.

To find the distance between Martin's house and the school, we can use the Pythagorean theorem.

Let's represent the distance between Martin's house and the school as x kilometers.

According to the given information:

Distance between Martin's house and Hayley's house (hypotenuse) = 13 kilometers

Distance between the school and Hayley's house (one side of the right triangle) = 12 kilometers

Using the Pythagorean theorem, we have:

x[tex]x^2 + 12^2 = 13^2[/tex]

Simplifying the equation:

[tex]x^2 + 144 = 169x^2 = 169 - 144x^2 = 25[/tex]

Taking the square root of both sides:

x = √25

x = 5

Therefore, the distance between Martin's house and the school is 5 kilometers.

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To find the measure of each interior and exterior angle of a regular 30-gon, we can use the formula:

In this case, we have a regular 30-gon, so n = 30.

Let's calculate the measures:

Therefore, each interior angle of the regular 30-gon measures 168°, and each exterior angle measures 12°.