The series [tex]\sum_{n=3}^{\infty} \frac{6}{n^2 - 1}[/tex] is a convergent series
How to determine whether the series is convergent or divergentFrom the question, we have the following parameters that can be used in our computation:
[tex]\sum_{n=3}^{\infty} \frac{6}{n^2 - 1}[/tex]
Factorize the denominator
[tex]\sum_{n=3}^{\infty} \frac{6}{n^2 - 1} = \sum_{n=3}^{\infty} \frac{6}{(n - 1)(n + 1)}[/tex]
Decompose into partial fraction
[tex]\sum_{n=3}^{\infty} \frac{6}{n^2 - 1} = \sum_{n=3}^{\infty} \frac{A}{(n - 1)} + \frac{B}{(n + 1)}[/tex]
So, we have
An + A + Bn - B = 6
This means that
A + B = 0
A - B = 6
When evaluated, we have
A = 3 and B = -3
So, we have
[tex]\sum_{n=3}^{\infty} \frac{6}{n^2 - 1} = \sum_{n=3}^{\infty} \frac{3}{(n - 1)} - \frac{3}{(n + 1)}[/tex]
Expand the series
[tex]\sum_{n=3}^{\infty} \frac{6}{n^2 - 1} = [\frac{3}{2} - \frac{3}{4}] +[\frac{3}{3} - \frac{3}{5}] +[\frac{3}{4} - \frac{3}{6}] + ........ + [\frac{3}{N} - \frac{3}{N + 2}][/tex]
When simplified, we have
[tex]\sum_{n=3}^{\infty} \frac{6}{n^2 - 1} = \frac{3}{2} - [\frac{3}{N} - \frac{3}{N + 2}][/tex]
The above implies that the series converges
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Classify each variable according to the set of numbers that best describes its values.
the circumference C of a circle found by using the formula C=2πr
The variable C (circumference) in the formula C=2πr can be classified as a positive real number.
The variable in the formula C=2πr is the circumference (C) of a circle. To classify this variable according to the set of numbers that best describes its values, we need to consider the values that the radius (r) can take.
The radius of a circle can be any positive real number or zero. Since the circumference is calculated using the formula C=2πr, the values of the circumference will depend on the values of the radius.
The set of numbers that best describes the values of the circumference (C) in this case is the set of positive real numbers. This is because the circumference can be any positive real number greater than zero, depending on the radius of the circle.
For example, if the radius of a circle is 1 unit, then the circumference will be 2π units. If the radius is 2 units, then the circumference will be 4π units. The circumference can take on infinitely many values depending on the radius.
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2 Enter the correct answer in the box. ming Exponential Functions: Mastery Test The function f(x) = 7x + 1 is transformed to function g through a horizontal compression by a factor of. What is the equation of function g? Substitute a numerical value for k into the function equation. (0) 0 0 Vo 4 g(x) = X = (7) kx +1 < AI TT a A P E P Reset sin cos tan sin-¹ costan-¹ csc sec cot log log, In I Next 11 1 2 A O U
Answer:
Substitute a numerical value for k into the function equation. g(x)=(7)^kx fill in the given formula
Step-by-step explanation:
This is giving you a step by step answer so follow it thoroughly and you will be good
what is the common ratio between successive terms in the sequence? 2, –4, 8, –16, 32, –64, ... –2 –6 6 2
The common ratio between successive terms in the given sequence is -2.
To find the common ratio in a geometric sequence, we divide any term by its previous term. Let's examine the sequence provided:
2, -4, 8, -16, 32, -64, ...
If we divide each term by its previous term, we get:
-4/2 = -2
8/-4 = -2
-16/8 = -2
32/-16 = -2
-64/32 = -2
As we can see, each term divided by its previous term yields the common ratio of -2. This indicates that the sequence is a geometric sequence with a common ratio of -2.
In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor known as the common ratio. In this case, multiplying each term by -2 will give us the next term in the sequence. The negative sign indicates that each subsequent term has the opposite sign of its previous term, and the absolute value of the terms doubles with each step.
Hence, the common ratio between successive terms in the sequence is -2.
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A total of 323 was collected from 40 people to cover the exact cost of their dinners. Some ordered steak at 8.50 per person, others ordered chicken at 7.50 per person. How many people ordered chicken?
17 people ordered chicken.
Let's solve the system of equations to find the number of people who ordered chicken. We have:
Equation 1: x + y = 40 (total number of people)
Equation 2: 8.50x + 7.50y = 323 (total cost)
We can multiply Equation 1 by 7.50 to eliminate y:
7.50(x + y) = 7.50(40)
7.50x + 7.50y = 300
Now we have a system of two equations:
8.50x + 7.50y = 323
7.50x + 7.50y = 300
Subtracting the second equation from the first, we get:
8.50x - 7.50x = 323 - 300
1x = 23
x = 23
Substituting x = 23 into Equation 1, we find:
23 + y = 40
y = 40 - 23
y = 17
Therefore, 17 people ordered chicken.
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Which expression is equivalent to (sinθ)(\secθ) ?
A. cos θ
B. tan θ
C. sin θ
D. csc θ
The expression (sinθ)(\secθ) can be simplified to cosθ, which is option A.
The expression (sinθ)(\secθ) involves the product of the sine of an angle θ and the secant of the same angle. To simplify this expression, we can use the trigonometric identity: secθ = 1/cosθ.
Substituting the value of secθ in the given expression, we get:
(sinθ)(\secθ) = (sinθ)(1/cosθ)
Next, we can simplify further by multiplying the terms:
(sinθ)(1/cosθ) = sinθ/cosθ
Since sinθ/cosθ is equivalent to tanθ (the ratio of sine to cosine), we can conclude that the expression (sinθ)(\secθ) is equivalent to tanθ. Therefore, option B (tanθ) is the correct answer.
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Write a polynomial function with rational coefficients so that P(x)=0 has the given roots.
4,16, and 1+19 i .
The polynomial function with roots 4,16, and 1+19i is P(x) = a(x - 4)(x - 16)(x² - 2x + 362)
Given data:
To construct a polynomial function with rational coefficients using the given roots, we can use the concept of complex conjugate pairs. Since the given roots are 4, 16, and 1+19i, we know that the complex conjugate of 1+19i is 1-19i. Therefore, the roots of the polynomial function are 4, 16, 1+19i, and 1-19i.
To find the polynomial function, we can use the factored form of a polynomial:
P(x) = a(x - r₁)(x - r₂)(x - r₃)(x - r₄)
where r₁, r₂, r₃, and r₄ represent the roots of the polynomial.
Plugging in the given roots, we have:
P(x) = a(x - 4)(x - 16)(x - (1+19i))(x - (1-19i))
To simplify the expression, we can multiply the complex conjugate terms:
P(x) = a(x - 4)(x - 16)[(x - 1 - 19i)(x - 1 + 19i)]
Multiplying the complex conjugates, we get:
P(x) = a(x - 4)(x - 16)[(x - 1)² - (19i)²]
Simplifying further:
P(x) = a(x - 4)(x - 16)[(x - 1)² + 361]
Expanding the square term:
P(x) = a(x - 4)(x - 16)(x² - 2x + 1 + 361)
Combining like terms:
P(x) = a(x - 4)(x - 16)(x² - 2x + 362)
Hence, the polynomial function is P(x) = a(x - 4)(x - 16)(x² - 2x + 362)
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ab is rotated 120 degrees clockwise about b. then ab is rotated 45 degrees counterclockwise about a. what is the image of a as a composition of transformations?
The image of a as a composition of transformations is expressed as:
(r(45°, A) ⚬ r(–120°, B))(A)
What are the coordinates of the transformation?There are different types of transformation such as:
Translation
Rotation
Reflection
Dilation
Now, when we rotate 120° clockwise rotation, it is also equal to saying we rotate counterclockwise by 240°
Now, due to the fact that AB is rotated 120° clockwise and clockwise is negative and the other is rotated by an angle of 45° counterclockwise and anything counterclockwise is positive.
Then we can conclude that the image as a composition of transformations is (r(45°, A) ⚬ r(–120°, B))(A)
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Cody and Monette are playing a board game in which you roll two dice per turn.
a. In one turn, how many outcomes result in a sum of 8?
There are 5 outcomes that result in a sum of 8
How many outcomes result in a sum of 8?From the question, we have the following parameters that can be used in our computation:
Rolling of two dice
The sample space of each die is
S = {1, 2, 3, 4, 5, 6}
When the two outcomes are added, we have
(2, 6), (3, 5), (4, 4), (5, 3) and (6, 2)
Hence, the outcomes are 5
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solve the equation below
|6w+4|=2w-10
Answer:
The equation [tex]\(|6w+4|=2w-10\)[/tex] has no solutions.
You can verify this by considering the two cases separately:
1. When [tex]\(6w+4 = 2w-10\)[/tex], which simplifies to [tex]\(4w = -14\)[/tex], or [tex]\(w = -\frac{7}{2}\)[/tex]. However, if you substitute this value back into the original equation, you'll find that the absolute value of a negative number is not equal to a negative number.
2. When [tex]\(6w+4 = -(2w-10)\)[/tex], which simplifies to [tex]\(8w = 6\)[/tex], or [tex]\(w = \frac{3}{4}\)[/tex]. Again, if you substitute this value back into the original equation, you'll find that the absolute value of a positive number is not equal to a negative number.
Therefore, there are no solutions to this equation.
Perform the indicated operations.
-5d(13 d²+7 d+8)
The result of the indicated operation is -65d³ - 35d² - 40d.
To perform the indicated operations, we need to distribute -5d to the terms inside the parentheses.
-5d(13d² + 7d + 8) = -5d * 13d² - 5d * 7d - 5d * 8
Simplifying each term, we have:
-5d * 13d² = -65d³
-5d * 7d = -35d²
-5d * 8 = -40d
Putting it all together, the expression becomes:
-65d³ - 35d² - 40d
Therefore, the result of the indicated operation is -65d³ - 35d² - 40d.
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Tom is running for president of the chess club, and he received 60 votes. There are 80 members in the club. What percentage of the club members voted for Tom?
Answer:
75%
Step-by-step explanation:
[tex]\frac{60}{80}[/tex] = .75
To change a decimal to a percentage move the decimal 2 places to the right.
75%
Helping in the name of Jesus.
determine the space of all of the possible outcomes of choosing a card nimbered 1,2,3, or 4 and a blue, green, or yellow marble. how many out comes involve choosing a blue marble?
There are 4 outcomes that involve choosing a blue marble.
To determine the space of all possible outcomes, we first need to list all the possible combinations of card numbers and marble colors:
Card Numbers: 1, 2, 3, 4
Marble Colors: Blue, Green, Yellow
The possible outcomes are as follows:
Card 1, Blue Marble
Card 1, Green Marble
Card 1, Yellow Marble
Card 2, Blue Marble
Card 2, Green Marble
Card 2, Yellow Marble
Card 3, Blue Marble
Card 3, Green Marble
Card 3, Yellow Marble
Card 4, Blue Marble
Card 4, Green Marble
Card 4, Yellow Marble
There are a total of 12 possible outcomes.
Now, let's determine how many outcomes involve choosing a blue marble. From the list above, we can see that there are 4 outcomes involving choosing a blue marble:
Card 1, Blue Marble
Card 2, Blue Marble
Card 3, Blue Marble
Card 4, Blue Marble
Therefore, there are 4 outcomes that involve choosing a blue marble.
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The printer at the Central Library can print out one A1 page every 12 seconds. In 2. 8 hours time, how many pages can be printed out?
The printer at the Central Library can print out approximately 840 A1 pages in 2.8 hours. This is calculated by converting the given time into seconds and then dividing the total time by the time it takes to print one page.
The number of pages that can be printed out, we need to convert 2.8 hours into seconds.
There are 60 minutes in an hour and 60 seconds in a minute, so 2.8 hours is equal to 2.8 * 60 * 60 = 10,080 seconds.
Next, we divide the total time in seconds by the time it takes to print one page (12 seconds) to find the number of pages that can be printed:
10,080 seconds / 12 seconds per page = 840 pages
Therefore, the printer at the Central Library can print out approximately 840 A1 pages in 2.8 hours.
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Find the distance between each pair of points, to the nearest tenth. (3,-2),(3,5)
The distance between the points (3,-2) and (3,5) is 9.4, rounded to the nearest tenth. The distance between two points can be found using the distance formula, which states that the distance between the points (x1,y1) and (x2,y2) is:
√((x2 - x1)^2 + (y2 - y1)^2)
In this case, the points are (3,-2) and (3,5), so the distance formula becomes:
√((3 - 3)^2 + (5 - (-2))^2)
= √((0)^2 + (7)^2)
= √(49)
= 7.0
To the nearest tenth, the distance is 7.0.
The distance formula uses the Pythagorean theorem to calculate the distance between two points. In this case, the two points are (3,-2) and (3,5), which means that the x-coordinates are the same but the y-coordinates are different.
When we plug these values into the distance formula, we get √((0)^2 + (7)^2), which is equal to √(49) = 7.0.
To the nearest tenth, the distance is 7.0.
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What is the solution of |5 x-2|=7 x+14 ? Check for extraneous solutions.
The stated solution does not have extraneous solution as all the solutions validate the equation.
To solve for the extraneous solutions, let us solve by keeping the values in Right Hand Side positive and negative. Beginning with positive.
Rewriting the equation -
5x - 2 = 7x + 14
Rearranging the equation
7x - 5x = - 14 - 2
Performing subtraction on both sides of the equation
2x = - 16
x = -16/2
Performing division on Right Hand Side of the equation
x = -8
Case when values are negative on Right Hand Side
5x - 2 = -( 7x + 14 )
Rewriting the equation
5x - 2 = -7x - 14
Rearranging the equation
7x + 5x = 2 - 14
Performing addition and subtraction on Left and Right Hand Side
12x = -12
x = - 1
Now, validating the equation to find extraneous solution -
When x is -8
5(-8) - 2 = 7(-8) + 14
-40 -2 = -56 + 14
- 42 = 42
When x is -1
5(-1) -2 = 7(-1) + 14
-5 -2 = -7 + 14
-7 = 7
The absolute values of negative numbers will be positive and hence both the solutions are valid. Thus, there is no extraneous solution.
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Consider two goods case: movies and concerts. Draw indifference curves that represent the preferences of each of the following people. Define the axes as "concerts per month" and "movies per month." For each graph, label the direction of preference with an arrow. 1) Mike likes concerts but doesn't care whether or not he goes to movies. 2) Ruth like movies but dislikes concerts. 3) Marie likes concerts up until she goes to three per month and then starts to dislike extra ones. However, she likes movies no matter how many she sees. 5.1 Consider two goods case: movies and concerts. Draw indifference curves that represent the preferences of each of the following people. Define the axes as "concerts per month" and "movies per month." For each graph, label the direction of preference with an arrow. 1) Mike likes concerts but doesn't care whether or not he goes to movies. 2) Ruth like movies but dislikes concerts. 3) Marie likes concerts up until she goes to three per month and then starts to dislike extra ones. However, she likes movies no matter how many she sees
1) Copy code
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Concerts | Movies
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Concerts | Movies
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1) Mike likes concerts but doesn't care about movies:
Since Mike likes concerts but doesn't care about movies, his indifference curves will be upward sloping and parallel to the movies axis. This indicates that as the number of concerts per month increases, Mike's satisfaction increases, while the number of movies per month has no effect on his satisfaction. Here's a graph representing Mike's preferences:
perl
2)Ruth likes movies but dislikes concerts:
Since Ruth likes movies but dislikes concerts, her indifference curves will be downward sloping and parallel to the concerts axis. This indicates that as the number of movies per month increases, Ruth's satisfaction increases, while the number of concerts per month has no effect on her satisfaction. Here's a graph representing Ruth's preferences:
sql
3) Marie likes concerts up until she goes to three per month and then starts to dislike extra ones. However, she likes movies no matter how many she sees:
Since Marie likes concerts up to three per month but dislikes extra ones, her indifference curves will be convex and bend inward after the point where she starts disliking extra concerts. On the other hand, her indifference curves for movies will be upward sloping, indicating that she always likes movies, regardless of the number. Here's a graph representing Marie's preferences:
sql
Note: The indifference curves in the graph may not be perfectly accurate, as they are just a visual representation. The exact shape and curvature of the indifference curves may vary. The main idea is to capture the direction of preferences for each individual.
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racing speed bobby and rick are in a 10-lap race on a onemile oval track. bobby, averaging 90 mph, has completed two laps just as rick is getting his car onto the track. what speed does rick have to average to be even with bobby at the end of the tenth lap? hint bobby does 8 miles in the same time as rick does 10 miles.
Rick needs to average 112.5 mph to be even with Bobby at the end of the tenth lap.
We know that Bobby averages 90 mph and completes 8 miles in the same time it takes Rick to complete 10 miles. To find Rick's required speed, we can set up a proportion using the distance and speed ratios.
The distance ratio is 8 miles to 10 miles, which simplifies to 4/5. The speed ratio is 90 mph to Rick's speed, which we'll call R mph. Setting up the proportion, we get (4/5) = 90/R.
To solve for R, we can cross-multiply and then divide:
4R = 5 * 90.
Simplifying,
we find that 4R = 450.
Dividing both sides by 4,
we get R = 112.5.
Therefore, Rick needs to average 112.5 mph to be even with Bobby at the end of the tenth lap.
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Question 1
Johnny's Landscape, Inc. was extra busy during the summer and decided to hire extra workers to complete all the lawns they had on their routes.
Mike was working with Johnny's Landscape, Inc., and was also talking to homeowners while cutting their lawns as an employee and started to mention that he was thinking about starting up his own lawn cutting business. Some of the homeowners offered to let Mike cut their lawn each week instead of having Johnny's Landscape company do it. Mike started cutting about 5 homes each week on his own time after work.
Johnny's Landscape, Inc.'s manager found out and was upset at Mike for taking their customers. Mike feels that the homeowner can always decide who they want to cut their lawn and it was all on his own time after work hours, but Johnny's Landscape owners feel that Mike was wrong for messing with their customers.
1. Explain what arguments Johnny's Landscape, Inc. could make that Mike was wrong?
2. Explain what arguments Mike could make to defend his cutting of the 5 lawns?
Johnny's Landscape, Inc. may also argue that Mike's actions potentially harm the company's reputation, client base, and revenue. Mike might emphasize that he did not directly interfere with Johnny's Landscape.
Johnny's Landscape, Inc. may argue that Mike was wrong for taking their customers because it violates the principles of loyalty, trust, and confidentiality expected from an employee. They could claim that Mike's actions constitute a breach of his employment contract, as he engaged in competing with the company while still working for them. Johnny's Landscape, Inc. may also argue that Mike's actions potentially harm the company's reputation, client base, and revenue.
Mike, on the other hand, could defend his decision to cut the 5 lawns by asserting that he performed the work on his own time, after his regular work hours, and did not actively solicit customers while working for Johnny's Landscape, Inc. He may argue that homeowners have the right to choose who provides services for them and that he merely offered an alternative option. Mike might emphasize that he did not directly interfere with Johnny's Landscape, Inc.'s contractual agreements with the homeowners.
In this situation, the legal and ethical aspects surrounding employment obligations, competition, and customer rights need to be considered to evaluate the validity of the arguments presented by Johnny's Landscape, Inc. and Mike. The specific terms of Mike's employment contract, any non-compete clauses, and local labor laws may play a significant role in determining the rights and responsibilities of both parties involved.
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One morning, exactly at sunrise, a buddhist monk began to climb a tall mountain. A narrow path, no more than a foot or two wide, spiraled around the mountain to a glittering temple at the summit. The monk ascended at varying rates of speed, stopping many times along the way to rest and eat dried fruit he carried with him. He reached the temple shortly before sunset. After several days of fasting and meditation he began his journey back along the same path, starting at sunrise and again walking at variable speeds with many pauses along the way. His average speed descending was, of course, greater than his average climbing speed. Prove that there is a spot along the path that the monk will occupy on both trips at precisely the same time of day.
The existence of a spot along the path where the monk will occupy at precisely the same time of day during both the ascent and descent can be proven using the Intermediate Value Theorem.
The monk's journey involves ascending and descending along the same narrow path. Let's assume time is measured continuously. As the monk climbs the mountain, his speed varies, and he takes pauses along the way.
Similarly, during the descent, his speed also varies but is on average faster than his climbing speed. The key concept to consider is that the monk's position on the path is a continuous function of time.
Since time is continuous, and the monk's position changes continuously, the Intermediate Value Theorem guarantees that the monk's position will intersect at the same time of day during both the ascent and descent.
Therefore, there exists a spot along the path where the monk will be present at precisely the same time of day on both trips.
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Question 1
Find the measure of BC. Assume that the given figure is not drawn to scale.
OA) 2¹ in.
O B) 3 in.
O C) in.
OD) in.
Next Question
A
6 in.
B
C
The measure of arc BC is 6 in, which is equal to the radius of the circle.
The answer is 6 in.
The measure of an arc is equal to the measure of the central angle that intercepts it. In the diagram, the arc BC is intercepted by the central angle BOC. The measure of central angle BOC is 180 - 90 = 90 degrees. Therefore, the measure of arc BC is also 90 degrees.
The circumference of a circle is equal to 2 * pi * r. In the diagram, the radius of the circle is AB = 3 in. Therefore, the circumference of the circle is 2 * pi * 3 = 6 pi in.
The measure of arc BC is 90 degrees, which is 1/6 of the circumference of the circle. Therefore, the measure of BC is 6 pi / 6 = 6 in.
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A forest ranger in an observation tower sights a fire 39° east of north. A ranger in a tower 10 miles due east of the first tower sights the fire at 42° west of north. How far is the fire from each tower?
The fire is approximately 6.868 miles from Tower A and 7.699 miles from Tower B calculated using trigonometry.
The fire is located 39° east of north from the first observation tower and 42° west of north from the second observation tower. To find the distance to the fire from each tower, we can use trigonometry.
To solve this problem, we can use trigonometry and create a diagram to visualize the situation.
Let's label the first tower as Tower A and the second tower as Tower B. From Tower A, the fire is sighted 39° east of north.
This means that the angle between the direction of the fire and the north direction is 39°.
From Tower B, the fire is sighted 42° west of north.
This means that the angle between the direction of the fire and the north direction is 42°.
Now, let's draw a diagram to represent the situation
In the diagram, the line segment AB represents the distance between the two towers, which is 10 miles.
We need to find the distances x and y, which represent the distances from the fire to Tower A and Tower B, respectively.
Using trigonometry, we can use the tangent function to find x and y.
For Tower A: tan(39°) = x / 10 miles
For Tower B: tan(42°) = y / 10 miles
Let's calculate x and y: x = 10 miles * tan(39°) y = 10 miles * tan(42°)
Using a calculator: x ≈ 6.868 miles y ≈ 7.699 miles
Therefore, the fire is approximately 6.868 miles from Tower A and 7.699 miles from Tower B.
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State whether the sentence is true or false. If false, replace the underlined term to make a true sentence.
The \underline{\text{center}} of a regular polygon is also the center of its circumscribed circle.
The statement is true. The center of a regular polygon is indeed also the center of its circumscribed circle.
If we draw a circle inside a polygon both of their centers were coincide.
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Solve each system by substitution.
2y = y - x² + 1 y=x²- 5x - 2
The solution to the system of equations is (3, -8).
To solve the system by substitution, we substitute the expression for y from the second equation into the first equation. From the second equation, we have y = x² - 5x - 2. Substituting this into the first equation, we get 2(x² - 5x - 2) = x² - 5x - 2 - x² + 1.
Simplifying the equation, we have 2x² - 10x - 4 = x² - 5x - 1.
Rearranging terms, we get x² - 5x - 3 = 0.
Factoring the quadratic equation, we have (x - 3)(x + 1) = 0.
This gives us two possible values for x: x = 3 or x = -1.
Substituting these values back into the second equation, we can find the corresponding values of y. When x = 3, y = (3)² - 5(3) - 2 = -8. When x = -1, y = (-1)² - 5(-1) - 2 = 2.
Therefore, the solutions to the system are (3, -8) and (-1, 2).
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Determine whether the number described is a statistic or a parameter. according to a sample of college students, the average amount of sleep they get each night is 6.2 hours.
The number described, "the average amount of sleep college students get each night is 6.2 hours," is a statistic.
In the field of statistics, a statistic refers to a numerical value that is calculated from a sample of data. In this case, the average amount of sleep is based on a sample of college students. It represents a characteristic or measure of the sample.
The reason it is considered a statistic and not a parameter is because a parameter refers to a numerical value that describes a population as a whole. To obtain a parameter, data from the entire population would need to be collected and analyzed. In this case, the average amount of sleep for all college students would need to be determined, which is not feasible.
Therefore, since the information is based on a sample, the average amount of sleep of 6.2 hours is considered a statistic. It provides insight into the sleep habits of the specific group of college students that were surveyed.
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Use a compass to draw a circle with chord -AB . Refer to this construction for the following problem.
a. Use an indirect proof to show that -CD passes through the center of the circle by assuming that the center of the circle is not on -CD .
The indirect proof shows that if the center of the circle is not on chord CD, it contradicts the property that the center is equidistant to all points on the circle, leading to the conclusion that the center must be on chord CD.
Here is an indirect proof to show that chord CD passes through the center of the circle by assuming that the center of the circle is not on the chord CD:
Assume: The center of the circle is not on chord CD.
Then: The center of the circle must be on either side of chord CD.
But: This is impossible, because the center of a circle is the only point that is equidistant to all points on the circle.
Therefore: The center of the circle must be on chord CD.
Here is a diagram that illustrates the proof:
[Diagram of a circle with chord AB and center O. Point C is on the circle, but point D is not.]
The proof follows from the following two facts:
The center of a circle is equidistant to all points on the circle.
If a point is equidistant to two points on a line, then the point is on the line.
In the diagram, point O is the center of the circle. Point C is on the circle, but point D is not. If the center of the circle were not on chord CD, then the center would have to be on either side of the chord. However, this is impossible, because the center of a circle is equidistant to all points on the circle. Therefore, the center of the circle must be on chord CD.
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Determine if the triangles below are similar. If they are, give the rule that you used to determine similarity.
Yes, the triangles above are similar based on the AA similarity theorem.
What are the properties of similar triangles?In Mathematics and Geometry, two triangles are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.
Furthermore, the lengths of three (3) pairs of corresponding sides or corresponding side lengths are proportional to the lengths of corresponding altitudes when two (2) triangles are similar.
Based on the angle, angle (AA) similarity theorem, we can logically deduce the following congruent triangles:
Triangle 1 ≅ Triangle 2
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Solve each quadratic equation by completing the square. x²+x-1 = 0 .
The solutions to the quadratic equation x² + x - 1 = 0 by completing the square are: x = -1/2 + √5/2 and x = -1/2 - √5/2.
To solve the quadratic equation x² + x - 1 = 0 by completing the square, follow these steps:
Step 1: Move the constant term to the other side of the equation:
x² + x = 1
Step 2: Take half of the coefficient of x and square it:
In this case, half of 1 is 1/2, and (1/2)² = 1/4.
Step 3: Add the squared term obtained in step 2 to both sides of the equation:
x² + x + 1/4 = 1 + 1/4
x² + x + 1/4 = 5/4
Step 4: Rewrite the left side of the equation as a perfect square:
(x + 1/2)² = 5/4
Step 5: Take the square root of both sides, considering both positive and negative square roots:
x + 1/2 = ±√(5/4)
Step 6: Solve for x:
x + 1/2 = ±√5/2
Subtract 1/2 from both sides:
x = -1/2 ±√5/2
Therefore, the solutions to the quadratic equation x² + x - 1 = 0 by completing the square are:
x = -1/2 + √5/2 and x = -1/2 - √5/2.
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Suppose that a dart lands at random on the dartboard shown at the right. Find each theoretical probability.
The dart lands in the bull's-eye,
The theoretical probability of the dart landing in the bull's-eye is 1/21.
To find the theoretical probability of the dart landing in the bull's-eye, we need to determine the ratio of the favorable outcomes (dart lands in the bull's-eye) to the total possible outcomes.
Assuming that the dartboard is divided into different regions with equal probability of landing on any particular region, we can consider the bull's-eye as a single region. Let's denote the number of regions as "n" and the number of favorable regions (bull's-eye) as "f."
In this case, since we only have one bull's-eye, f = 1.
The total number of regions on the dartboard (including the bull's-eye) is n = 1 + 20 (assuming there are 20 other regions on the dartboard).
Thus, the theoretical probability of the dart landing in the bull's-eye is:
P(Bull's-eye) = f / n
= 1 / (1 + 20)
= 1 / 21
Therefore, the theoretical probability of the dart landing in the bull's-eye is 1/21.
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Solve each equation. 25x²+10 x+1=9 .
Bought 4 stamps each week for 3 weeks. he wants to put amps on each page of his album. how many pages will niko use? tell how you can use tools to help solve the problem. choose a tool to represent the problem. explain why you chose that tool. solve the problem. explain how you used the tool you chose
The total number of pages that would be used is, 12
We have to give that,
Bought 4 stamps each week for 3 weeks.
And, he wants to put amps on each page of his album.
Now, the number of stamps in 3 weeks,
1 week = 4 stamps
3 weeks = 4 x 3 stamps
3 weeks = 12 stamps
Since he wants to put stamps on each page of his album.
Hence, the Total number of pages = 12
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