The number c that satisfies the conclusion of the Mean Value Theorem for the function f(x) = x + 3/x on the interval [1, 14] is approximately c = 3.75.
To find the number c that satisfies the conclusion of the Mean Value Theorem for the function f(x) = x + 3/x on the interval [1, 14], follow these steps:
1. Verify the conditions of the Mean Value Theorem:
The function f(x) = x + 3/x is continuous on the closed interval [1, 14] and differentiable on the open interval (1, 14).
2. Calculate the average rate of change of the function on the interval:
f(14) = 14 + 3/14 ≈ 14.214
f(1) = 1 + 3/1 = 4
Average rate of change = (f(14) - f(1)) / (14 - 1) ≈ (14.214 - 4) / 13 ≈ 0.786
3. Differentiate the function to find the instantaneous rate of change:
f'(x) = 1 - 3/x²
4. Set f'(x) equal to the average rate of change and solve for x:
1 - 3/x² = 0.786
x² = 3 / (1 - 0.786)
x² ≈ 14.056
x ≈ √14.056 ≈ 3.75
The number c that satisfies the conclusion of the Mean Value Theorem for the function f(x) = x + 3/x on the interval [1, 14] is approximately c = 3.75.
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In the figure, the vertices of ∆ ABC are A ( 0, 6 ), B ( 8, 0 )
and C ( 5, 8 ). If CD ⊥ AB, then find the length of altitude CD.
The length of altitude CD is 133/25 units.
What is the length if altitude?
To find the length of altitude CD, we need to find the length of the line segment CD, which is perpendicular to AB and passes through the point C.
First, let's find the equation of the line AB. We can use the two-point form:
y - y1 = m(x - x1)
where m is the slope of the line and (x1, y1) is one of the points on the line. Using points A and B, we get:
m = (0 - 6)/(8 - 0) = -3/4
y - 6 = (-3/4)(x - 0)
y = (-3/4)x + 6
Next, let's find the slope of the line CD, which is the negative reciprocal of the slope of AB. This is because perpendicular lines have slopes that are negative reciprocals of each other. Therefore:
m_CD = 4/3
Now we can use the point-slope form to find the equation of line CD. We know that the line passes through point C (5, 8), so:
y - 8 = (4/3)(x - 5)
Simplifying, we get:
y = (4/3)x - 4/3 + 8
y = (4/3)x + 20/3
Now we need to find the intersection of line CD with line AB. We can solve the system of equations:
y = (-3/4)x + 6
y = (4/3)x + 20/3
Setting the two expressions for y equal to each other, we get:
(-3/4)x + 6 = (4/3)x + 20/3
Multiplying both sides by 12, we get:
-9x + 72 = 16x + 80
25x = -8
x = -8/25
Substituting this value of x into either equation, we get:
y = (-3/4)(-8/25) + 6 = 57/25
Therefore, the point of intersection is (-8/25, 57/25).
Finally, we can use the distance formula to find the length of CD, which is the distance between C and the point of intersection:
d = √[(5 - (-8/25))² + (8 - 57/25)²]
d = √[(125/25 + 8/25)² + (200/25 - 57/25)²]
d = √[(133/25)² + (143/25)²]
d = √(17689/625)
d = 133/25
Therefore, the length of altitude CD is 133/25 units.
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in a relation r over a set of people, we have that a person x is related to person y if x's birth date is prior to y's birth date. no two people in the group have the same birth date. indicate the choice that correctly describes the relation r.
The relation R is a partial order for the given set of data regarding over the set of people. (Option 1)
A partial order is a binary relation that is reflexive, antisymmetric, and transitive. In this case, the relation R is reflexive since a person is always born before themselves. It is also transitive since if person A is born before person B, and person B is born before person C, then person A must be born before person C.
However, the relation R is not symmetric, since if person A is born before person B, then person B cannot be born before person A. Therefore, the relation R is not a total order.
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Complete Question:
in a relation r over a set of people, we have that a person x is related to person y if x's birth date is prior to y's birth date. no two people in the group have the same birth date. indicate the choice that correctly describes the relation r.
partial ordersymmetricasymmetrictotal orderI have a pool in shape of a rectangle with a length of 28 feet and a width of 14 feet. I also have the stairs that form a trapezoid with the measurements of a=4 feet and b=6 feet and the height is 2 feet. what is the total area of both shapes combined for the pool? *(rectangle area = A=b x h) *Trapezoid Area =A = 1/2 (a+b)xh
Answer:
Step-by-step explanation:
28x14x4x6x2+18816
13xy^2 - 12x^2y + 5x^2y simplify
Select the best answer for the question.
4. A large facility that manufacturers send large quantities of product to then be distributed to stores is called:
O A. Store
B. Distribution Center
C. Carton
D. Garage
The statement that describe the term in the question is (b) the distribution center
How to determine the statement that describe the statementA distribution center is a large facility used by manufacturers to store and send out large quantities of products to be distributed to retail stores or directly to customers.
The products are often received in bulk and then sorted, packaged, and shipped to their final destination. The distribution center may also handle other tasks such as inventory management, quality control, and returns processing.
The purpose of the distribution center is to streamline the distribution process, reduce shipping costs, and ensure that products are delivered to their final destination in a timely and efficient manner.
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there is a population of 15,500 bacteria in a colony. if the number of bacteria doubles every 234 hours, what will the population be 936 hours from now?
If the number of bacteria doubles every 234 hours, we can use the exponential growth formula to find the population at a future time t:
N(t) = N0 * 2^(t/T)
where N(t) is the population at time t, N0 is the initial population, T is the doubling time (234 hours in this case), and t is the time elapsed.
We are given that the initial population is 15,500 bacteria. We want to find the population 936 hours from now. Therefore, we can substitute the values into the formula:
N(936) = 15,500 * 2^(936/234)
N(936) = 15,500 * 2^4
N(936) = 15,500 * 16
N(936) = 248,000
Therefore, the population of bacteria in the colony will be 248,000 after 936 hours.
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Nicholas and three other members from the Culinary Team had 100 fliers to post
around the school. Last week, they posted 1/4 of them. This week, they posted 2/4
of the remaining fliers. How many fliers have they still not posted?
The number of fliers that Nicholas and three other Culinary Team members have not still posted is 37 fliers.
How is the number determined?The remaining number of fliers not yet posted is a function of mathematical operations.
Basically, mathematical operations involve addition, subtraction, division, and multiplication.
The total number of fliers, Nicholas and three other Culinary Team members are posting around the school = 100
The fraction posted last week = 1/4
= 25 fliers (100 x 1/4)
The remaining fliers after posting 25 fliers = 75 (100 - 25)
The fraction posted this week = 2/4 = 1/2
1/2 of 75 = 37.5 or 38 fliers
The remaining fliers they still have not posted = 37 (75 - 38)
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16. Movie tickets for an adult and three children cost $29. An adult's
ticket is $3 more than a child's ticket. Find the cost of an adult's ticket.
In a set of three lines, a pair of parallel lines is intersected by a transversal. The intersection forms eight special angles. Two of the special angles, ∠A and ∠B, are corresponding angles.
For the set of parallel lines intersected by a transversal,A = 4x andB = 2(x+14).
Use an equation to represent the corresponding angle relationship betweenA and B.
Use the equation to find the measures ofA andB.
We can solve this equation by using corresponding angle relationship between A and B , to get A= 56° and B=56°
What is corresponding angle ?
In geometry, when two lines are intersected by a transversal, the angles formed at each intersection point are called the corresponding angles. Corresponding angles are angles that occupy the same relative position at each intersection point where the transversal intersects two parallel lines.
What is equation?
In mathematics, an equation is a statement that shows the equality between two mathematical expressions using an equals sign (=). An equation can contain variables (represented by letters) and constants (fixed values).
In the given question,
The corresponding angle relationship between angle A and angle B can be represented by the equation:
angle A = angle B
Substituting the given values of angle A and B we get
4x = 2(x+14)
4x = 2x + 28
2x = 28
x = 14
Therefore angle A = 4x = 4(14) = 56°
Therefore angle B = 2(x+14) = 2(14+14) = 56°
So both angle A and B have a measure of 56°.
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The following graph represents a function.
-8-6-4
39
8
Domain: [Select]
6
P
P
-2-
-4
-6
-8
2019 StrongMind
What are the domain and range of the function?
The domain of the function is: Domain = {-8, -6, -4, -2, 1, 2} and range of the function is :- Range = {-8, -6, -4, -2, 0}.
Define function?A function is a rule that maps each element in one set, called the domain, to a unique element in another set, called the range.
It is a relation between two sets where each input in the domain corresponds to a unique output in the range.
Functions are often represented by equations, graphs, or tables, and they are used to model real-world phenomena and solve mathematical problems in various fields such as science, engineering, economics, and more.
The domain of the function is the set of all possible input values, which in this case are the x-coordinates of the given points on the graph. So, the domain is:
Domain = {-8, -6, -4, -2, 1, 2}
The range of the function is the set of all possible output values, which in this case are the y-coordinates of the given points on the graph. So, the range is:
Range = {-8, -6, -4, -2, 0}
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I need the answer for 4x+8<24 it is for my homework
Answer:
Step-by-step explanation:
x = 8
higgins iced cream shop sells ice cream in cone containers. The diameter of the cone containers in 2.5 inches, and the height is 6 inches. One tub of Higgins Ice cream is 384 ounces, and 1 cubic inch is equivalent to .55. What is the maximum number of cone containers higgins ice cream shop can fill completely with one tub of ice cream?
There isn't a way we get a fractional number or cones, we roll down to the closest full amount: 28 cones. The great majority of cone boxes that Higgins Shop can fill to the brim by one ice cream tub is 28.
What does 3 + 4 represent as a fraction?or as one number followed by the other number inside a line of text, as in: 3/4. 3 out of 4 components are represented by the fraction 3/4, or three quarters. The term "numerator" refers to the higher number, 3, whereas the term "denominator" refers to the lower number, 4.
Its radius is 2.5/2 ≈ 1.25 inches because the cone container's diameter is specified as 2.5 inches. It states that the height is 6 inches.
Thus, one cone container has a volume of: V = (1/3)(1.25)2(6)V 7.35 cubic inches.
We must now determine which of these cone-shaped containers one tub of cream can fit into.
First, we must convert 1 tub of ice cream's volume from inches to cubic centimeters.
211.2 cubic inches from 384 ounces divided by 0.55 cubic inches per ounce.
Secondly, we divide a volume of a cone container by volume of an ice cream tub:
28.77 cones out of 28.12 cubic inches at 7.35 cubic inches each
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in a large population, 64% of the people have been vaccinated. if 3 people are randomly selected, what is the probability that at least one of them has been vaccinated?
The probability that at least one of three randomly selected people has been vaccinated in a large population where 64% of people have been vaccinated is 0.784.
The probability that at least one of three randomly selected people has been vaccinated in a large population where 64% of people have been vaccinated can be calculated using the binomial probability formula.
The binomial probability formula is used to calculate the probability of a given number of successes from a given number of trials when the probability of success is constant. In this case, the number of trials is 3 (i.e. the three randomly selected people) and the probability of success is 64% (i.e. the probability that each of the randomly selected people has been vaccinated).
To calculate the probability that at least one of the three randomly selected people has been vaccinated, we first calculate the probability of none of them having been vaccinated:
P(none) = (1 - 0.64)^3 = 0.216
Then, the probability of at least one of them having been vaccinated is 1 - P(none) = 1 - 0.216 = 0.784.
In conclusion, the probability that at least one of three randomly selected people has been vaccinated in a large population where 64% of people have been vaccinated is 0.784.
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about 10% percent of workers (ages 16 years and older) in the united states commute to their jobs by carpooling. you randomly select eight workers. what is the probability that exactly four of them carpool to work?
Probability that exactly four workers carpool to work is 0.214990847, or approximately 21.5%.
We must apply the binomial distribution formula to resolve this issue. The number of successes in a certain number of independent trials that all have the same chance of success are described by the binomial distribution, which is a type of probability distribution. In this scenario, the number of successes is the number of workers who carpool, and the probability of success is 0.1.
The binomial distribution's formula is:
[tex]P(X=k) = to (n pick k) * p*k*1-p (n-k)[/tex]
Where n is the total number of trials (in this case, 8), p is the success probability (0.1), and (n choose k) is the binomial coefficient, which is equal to [tex]n! / (k! * (n-k)![/tex] and P(X=k) is the probability of obtaining exactly k successes.
This formula can be used to determine the likelihood that four employees will carpool to work:
[tex]P(X=4) = (8 pick 4) * 0.14 * 0.94 * 0.214 = 0.214990847.[/tex]
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The list below represents the number of novels written
by each of Amir's 10 favorite authors.
1 3 3 4 5 7 8 10 12 23
What is the interquartile range of the number of novels
written by each of Amir's 10 favorite authors?
The interquartile range of the number of novels
written by each of Amir's 10 favorite authors would be = 7
How to calculate the interquartile range of the novels?The total number of the novels written by the 10 favorite authors of Amir = 1 +3 +3 +4+ 5 +7+ 8+ 10+ 12+ 23 = 76
The interquartile range of the number of novels = Q3-Q1
Where Q3 = median of second half = 10
Q1 = median of the first half = 3
The interquartile range of the number of novels = 10-3 = 7
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a local county has an unemployment rate of 4%. a random sample of 19 employable people are picked at random from the county and are asked if they are employed. round answers to 4 decimal places.
The probability that exactly 8 of the 19 people in the random sample are employed is 27.93%.
We need to calculate the probability that exactly 8 of the 19 people in the random sample are employed. The probability of a single person being employed is 4%, or 0.04.
To calculate the probability of 8 people being employed out of the 19, we can use the binomial distribution formula:
P(X=8) = nCx * (p^x) * (1-p)^(n-x) Where n = 19, x = 8, p = 0.04, and 1-p = 0.96
So, P(X=8) = 19C8 * (0.04^8) * (0.96^11) = 0.2793 or 27.93%.
Therefore, the probability that exactly 8 of the 19 people in the random sample are employed is 27.93%.
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Seven bananas weigh 1 3/4 pounds and are on sale for $3 .How much does Luke pay for 28 bananas?How much do the 28 bananas weigh?Assume each banana weighs about the same.
Answer:
The 28 bananas weigh 7 pounds
Step-by-step explanation:
We can start by figuring out the price per banana by dividing the total price by the number of bananas:
Price per banana = $3 / 7 bananas
Price per banana = $0.43 per banana (rounded to the nearest cent)
Next, we can use the price per banana to find the cost of 28 bananas:
Cost of 28 bananas = 28 bananas x $0.43 per banana
Cost of 28 bananas = $12.04
Therefore, Luke pays $12.04 for 28 bananas.
To find the weight of 28 bananas, we can use the fact that seven bananas weigh 1 3/4 pounds:
Weight of one banana = 1 3/4 pounds / 7 bananas
Weight of one banana = 1/4 pound per banana
Then we can multiply the weight of one banana by 28 to find the total weight of 28 bananas:
Weight of 28 bananas = 1/4 pound per banana x 28 bananas
Weight of 28 bananas = 7 pounds
Therefore, the 28 bananas weigh 7 pounds.
Help me on this please
Answer:
x = -4
y = 1
Step-by-step explanation:
I added a photo of my solution
What are the values of x and y? *
The values of sides x and y of the triangle are 6.75 and 11.25 units respectively.
The Pythagorean TheoremThe square of the hypotenuse side of the right-angled triangle equals the sum of the squares of the other two sides, according to Pythagoras' Theorem. The three sides of this triangle are referred to as the perpendicular, base, and hypotenuse.
by using the Pythagorean theorem:
Equation1: y ² =x ² + 9²
Equation2: (12 + x) ² =1 5 ² + y ²
Putting values of equation1 in 2 we get
(12 + x) ² =1 5 ² +x ² + 9²
144 + x²+ 2×12x = 225 + x²+ 81
x=6.75
Putting value of x in equation1
y ² =6.75 ² + 9²
y=11.25
Hence, the values of x and y are
x = 6.75 unit
y =11.25 unit
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factor x^2=14x-18=-8x-2
Answer:
=(x+9)(x+5)
Step-by-step explanation:
Find the sales tax. Then find the total cost of the item. Sales Tax Selling Price Rate of Sales Tax Sales Tax $90.00 5%
Answer:
Sales tax: $90 × .05 = $4.50
Total cost: $90 + $4.50 = $94.50
3. What is the distance between the points (4, 5) and (-3, 5)?
7 units
8 units
1 units
10 units
Answer:
7
Step-by-step explanation:
you use your fingers and count down from 4 to -3 and you get 7.
Which of the following shows an example of two irrational numbers being multiplied to get a rational number?
Responses
3×9
0×5√
2√ ×8√
2√×3√
Step-by-step explanation:
Which of the following shows an example of two irrational numbers being multiplied to get a rational number?
Responses
option c
a) use these data to estimate the mean wrist extension for people using this new mouse design using a 90% confidence interval. (round your answers to three decimal places.) , (b) what assumptions are required in order for it to be appropriate to generalize your estimate to the population of students at this university? yes, the assumption would have to be made that the 24 students in the study formed a random sample of people in the country. no, we can generalize the estimate to the population of students at this university. yes, the assumption that the 24 students in the study formed a random sample of students at all universities. yes, the assumption would have to be made that the 24 students in the study formed a random sample of students at this university. no, we cannot generalize the estimate to any population as we have less than 30 students. to the population of all university students? no, we cannot generalize the estimate to any population as we have less than 30 students. yes, the assumption would have to be made that the 24 students in the study formed a random sample of people in the country. yes, the assumption would have to be made that the 24 students in the study formed a random sample of students from this university. no, we can generalize the estimate to the population of all university students. yes, the assumption that the 24 students in the study formed a random sample of students at all universities. (c) based on your interval from part (a), do you think there is reason to believe that the mean wrist extension for people using the new mouse design is greater than 20 degrees? explain why or why not. no, the entire confidence interval is below 20, so the results of the study would be very likely if the population mean wrist extension were greater than 20. no, the confidence interval contains 20, so the results of the study would be very likely if the population mean wrist extension were greater than 20. yes, the confidence interval contains 20, so the results of the study would be very unlikely if the population mean wrist extension were as low as 20. yes, the entire confidence interval is above 20, so the results of the study would be very unlikely if the population mean wrist extension were as low as 20.
The random sample of [tex]24[/tex] students has a mean of [tex]1.9149[/tex], and the data suggests that the mean is [tex]20^{0}[/tex].
How to Use the Sample Mean Formula to Determine the Sample Mean?x = (xi) / n is the general solution for computing the sample mean. Thus, xi refers to all X sample values, xi represents the sampling distribution, and n is the total number of specimen terms inside the data collection.
What does sample in statistics mean?The statistic known as the sampling distribution is created by arithmetically averaging the values of the variables in a group. The sample mean is an estimate of the anticipated value if the sample is taken from probabilistic with a common expected value.
(a) Sample mean [tex]= (17 + 21 + 20 + 19 + 23 + 22 + 20 + 20 + 18 + 21 + 19 + 24 + 22 + 20 + 19 + 20 + 19 + 20 + 21 + 22 + 23 + 21 + 18 + 22)/24 = 20.5[/tex]Sample standard deviation [tex]= 1.9149[/tex]
Next, we can use a t-distribution with [tex]23[/tex] degrees of freedom
Confidence interval [tex]= (20.5 - 0.8277, 20.5 + 0.8277) = (19.6723, 21.3277)[/tex]
b) The assumption required in order to generalize the estimate to the population of students at this university is that the [tex]24[/tex] students in the study formed a random sample of students at this university.
(c) There is not enough evidence to suggest that the mean wrist extension for people using the new mouse design is greater than [tex]20^{0}[/tex].
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Why isn't x^4+4x-21 factorable?
Answer:
The polynomial expression x^4 + 4x - 21 cannot be factored into linear factors with rational coefficients. This can be shown using the rational root theorem, which states that any rational roots (zeros) of the polynomial must be of the form p/q, where p is a factor of the constant term (in this case, 21) and q is a factor of the leading coefficient (1). The only possible rational roots of this polynomial are ±1, ±3, ±7, and ±21. However, plugging in each of these values does not give a zero of the polynomial, meaning it has no rational roots. Therefore, it cannot be factored into linear factors with rational coefficients. It may still be factorable using complex numbers, but it is not factorable using only real numbers.
In the figure, each side of square ABCD has length 1, the length of line Segment CE is 1, and the length of line segment BE is equal to the length of line segment DE. What is the area of the triangular region BCE?
A. 1/3
B. \(\frac{\sqrt{2}}{4}\)
C. 1/2
D. \(\frac{\sqrt{2}}{2}\)
E. 3/4
Thus, the area of triangular region BCE is √(1/2) / 2.
The area of triangular region BCE can be found using the formula for the area of a triangle: (1/2) * base * height. In this case, the base of triangle BCE is equal to the side length of square ABCD, which is 1. To find the height, observe that triangle BEC is a right triangle with right angle at vertex B.
Since the length of line segment BE is equal to the length of line segment DE, and CE has a length of 1, we can deduce that triangle BED is an isosceles right triangle. Therefore, the angle BEC is 45 degrees, and the angle BED is also 45 degrees.
Using the Pythagorean theorem on triangle BED, we have:
[tex]BE^2 + DE^2 = CE^2[/tex]
Since BE = DE, we can rewrite the equation as:
[tex]2(BE^2) = 1[/tex]
[tex]BE^2 = 1/2[/tex]
[tex]BE = √(1/2)[/tex]
Now that we know the length of BE, we can use it as the height of triangle BCE. Therefore, the area of triangle BCE is:
Area = (1/2) * base * height
[tex]Area = (1/2) * 1 * √(1/2)[/tex]
[tex]Area = √(1/2) / 2[/tex]
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Write the equation in standard form for the circle that has a diameter with endpoints (-4, 9) and (-4, 3)
The equation in standard form for the circle is (x + 4)² + (y - 6)² = 9.
The midpoint of the diameter is the center of the circle. Therefore, we can first find the midpoint:
Midpoint = [(x1 + x2)/2, (y1 + y2)/2] = [(-4 + (-4))/2, (9 + 3)/2] = [-4, 6]
The distance from the center to either endpoint of the diameter is the radius of the circle. In this case, the distance is:
r = |9 - 6| = 3
So we have the center (-4, 6) and the radius 3. Using the standard form equation of a circle, we can write:
(x - (-4))² + (y - 6)² = 3²
Simplifying and expanding, we get:
(x + 4)² + (y - 6)² = 9
Therefore, the equation in standard form for the circle is (x + 4)² + (y - 6)² = 9.
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A number is equal to 3 times a smaller number. Also, the sum of the smaller number and 4 is the larger number. The
situation is graphed on the coordinate plane below, where x represents the smaller number and y represents the larger
number.
Which two equations represent the situation?
A. y= 1/3x and y = x-4
B. Y = 1/3x and y= x +4
C. Y= 3x and y= x +4
D. Y= 3x and y = x- 4
Please and thank you! <3
Answer:
3x = y
x + 4 = y
Step-by-step explanation:
Letter C
What is the measure of each interior angle of a regular 20-gon?
Answer:
162°
Step-by-step explanation:
the sum of the interior angles of a polygon is
sum = 180° (n - 2) ← n is the number of sides
here n = 20 , then
sum = 180° × (20 - 2) = 180° × 18 = 3240°
The interior angles of a regular polygon are congruent
then each interior angle = 3240° ÷ 20 = 162°
midpoint of 0 and 2/7
Answer: 0, 4.5
Step-by-step explanation:
The answer in decimal form is 0 and 4.5. :)